Research ArticleCardinal Basis Piecewise Hermite Interpolation on Fuzzy Data

H Vosoughi and S Abbasbandy

Department of Mathematics Islamic Azad University Science and Research Branch Tehran Iran

Correspondence should be addressed to S Abbasbandy abbasbandyyahoocom

Received 6 June 2016 Accepted 7 December 2016

Academic Editor Katsuhiro Honda

Copyright copy 2016 H Vosoughi and S Abbasbandy This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A numericalmethod alongwith explicit construction to interpolation of fuzzy data through the extension principle results bywidelyused fuzzy-valued piecewise Hermite polynomial in general case based on the cardinal basis functions which satisfy a vanishingproperty on the successive intervals has been introduced here We have provided a numerical method in full detail using the linearspace notions for calculating the presented method In order to illustrate the method in computational examples we take recourseto three prime cases linear cubic and quintic

1 Introduction

Fuzzy interpolation problem was posed by Zadeh [1] Lowenpresented a solution to this problem based on the fundamen-tal polynomial interpolation theorem of Lagrange (see eg[2]) Computational and numerical methods for calculatingthe fuzzy Lagrange interpolate were proposed by Kaleva[3] He introduced an interpolating fuzzy spline of order 119897Important special cases were 119897 = 2 the piecewise linearinterpolant and 119897 = 4 a fuzzy cubic spline Moreover Kalevaobtained an interpolating fuzzy cubic spline with the not-a-knot condition Interpolating of fuzzy data was developedto simple Hermite or osculatory interpolation 119864(3) cubicsplines fuzzy splines complete splines and natural splinesrespectively in [4ndash8] byAbbasbandy et al Later Lodwick andSantos presented the Lagrange fuzzy interpolating functionthat loses smoothness at the knots at every 120572-cut also every120572-cut (120572 = 1) of fuzzy spline with the not-120572-knot boundaryconditions of order 119896 has discontinuous first derivativeson the knots and based on these interpolants some fuzzysurfaces were constructed [9] Zeinali et al [10] presenteda method of interpolation of fuzzy data by Hermite andpiecewise cubic Hermite that was simpler and consistentand also inherited smoothness properties of the generatorinterpolation However probably due to the switching pointsdifficulties the method was expressed in a very specialcase and none of three remaining important cases was not

investigated and this is a fundamental reason for the methodweakness

In total low order versions of piecewise Hermite inter-polation are widely used and when we take more knotsthe error breaks down uniformly to zero Using piecewise-polynomial interpolants instead of high order polynomialinterpolants on the same material and spaced knots is auseful way to diminish the wiggling and to improve theinterpolation These facts as well as cardinal basis functionsperspective motivated us in [11] to patch cubic Hermitepolynomials together to construct piecewise cubic fuzzyHermite polynomial and provide an explicit formula in asuccinct algorithm to calculate the fuzzy interpolant in cubiccase as a new replacement method for [4 10]

Now in this paper in light of our previous work wewant to introduce a wide general class of fuzzy-valuedinterpolation polynomials by extending the same approachin [11] applying a very special case of which general class offuzzy polynomials could be an alternative to fuzzy osculatoryinterpolation in [4] and so its lowest order case (119898 =1) namely the piecewise linear polynomial is an analogyof fuzzy linear spline in [3] Meanwhile when 119898 = 2with exactly the same data we will simply produce thesecond lower order form of mentioned general class that wasintroduced in [11] and the interpolation of fuzzy data in [10]

The paper is organized in five sections In Section 2 wehave reviewed definitions and preliminary results of several

Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2016 Article ID 8127215 8 pageshttpdxdoiorg10115520168127215

2 Advances in Fuzzy Systems

basic concepts and findings next we construct piecewisefuzzy Hermite polynomial in detail based on cardinal basisfunctions and prove some new properties of the introducedgeneral interpolant (Section 3) In Section 4 we have pro-duced three initial linear cubic [11] and quintic cases andshown the relationship between some of the mentioned casesand the newly presented interpolants in [3 4 10] Further-more to illustrate themethod some computational examplesare provided Finally the conclusions of this interpolation arein Section 5

2 Preliminaries

To begin let us introduce some brief account of notionsused throughout the paper We shall denote the set offuzzy numbers by RF the family of all nonempty convexnormal upper semicontinuous and compactly supportedfuzzy subsets defined on the real axis R ObviouslyR sub RFIf 119906 isin RF is a fuzzy number then 119906120572 = 119909 isin R | 119906(119909) ge 1205720 lt 120572 le 1 shows the 120572-cut of 119906 For 120572 = 0 by the closureof the support 1199060 = cl119909 | 119909 isin R 119906(119909) gt 0 It is wellknown the 120572-cuts of 119906 isin RF are closed bounded intervalsin R and we will denote them by 119906120572 = [119906120572 119906120572] functions119906(sdot) 119906(sdot) are the lower and upper branches of 119906 The core of 119906is 1199061 = 119909 | 119909 isin R 119906(119909) = 1 In terms of 120572-cuts we have theaddition and the scalar multiplication

(119906 + V)120572 = 119906120572 + V120572 = 119909 + 119910 | 119909 isin 119906120572 119910 isin V120572(120582119906)120572 = 120582119906120572 = 120582119909 | 119909 isin 119906120572(0)120572 = 0

(1)

for all 0 le 120572 le 1 119906 V isin RF and 120582 isin R119906 = ⟨119886 119887 119888 119889⟩ specifies a trapezoidal fuzzy numberwhere 119886 le 119887 le 119888 le 119889 and if 119887 = 119888 we obtain a triangular fuzzynumber For120572 isin [0 1] we have 119906120572 = [119886+120572(119887minus119886) 119889minus120572(119889minus119888)]In the rest of this paper we will assume that 119906 is a triangularfuzzy number

Definition 1 (see eg [5]) An L-R fuzzy number 119906 = (119898119897 119903)119871119877 is a function from the real numbers into the interval[0 1] satisfying

119906 (119909) =

119877(119909 minus 119898119903 ) for 119898 le 119909 le 119898 + 119903119871 (119898 minus 119909119897 ) for 119898 minus 119897 le 119909 le 1198980 otherwise

(2)

where 119877 and 119871 are continuous and decreasing functions from[0 1] to [0 1] fulfilling the conditions 119877(0) = 119871(0) = 1 and119877(1) = 119871(1) = 0When119877(119909) = 119871(119909) = 1minus119909 wewill have 119871minus119871fuzzy numbers that involve the triangular fuzzy numbers Foran 119871 minus 119871 fuzzy number 119906 = (119898 119897 119903) the support is the closedinterval [119898 minus 119897119898 + 119903] (see eg [6])

The linear space of all polynomials of degree at most 119873will be designated by 119875119873 Full Hermite interpolation problemdefines a unique polynomial called 119901119873(119909) which solves thefollowing problem

Theorem 2 (see [12] (existence and uniqueness)) Let1199090 1199091 119909119899 be 119899 + 1 distinct points 1205720 1205721 120572119899 be positiveintegers 119896 = 0 1 120572119894 and 119873 = 1205720 + 1205721 + sdot sdot sdot + 120572119899 + 119899 Set119908(119909) = prod119899119894=0(119909 minus 119909119894)120572119894+1 and119897119894119896 (119909)= 119908 (119909) (119909 minus 119909119894)119896minus120572119894119896 (119909 minus 119909119894)120572119894+1minus119896

119889(120572119894minus119896)119889119909(120572119894minus119896) [(119909 minus 119909119894)120572119894+1

119908 (119909) ]119909=119909119894

119901119873 (119909) = 119899sum119894=0

1199031198941198971198940 (119909) + 119899sum119894=0

1199031015840119894 1198971198941 (119909) + sdot sdot sdot + 119899sum119894=0

119903(120572119894)119894 119897119894120572119894 (119909)(3)

is a unique member of 119875119873 for which119901119873 (1199090) = 1199030 1199011015840119873 (119909) = 11990310158400 119901(1205720minus1)119873 (1199090) = 119903(1205720)0119901119873 (119909119899) = 119903119899 1199011015840119873 (119909119899) = 1199031015840119899 119901(120572119899minus1)119873 (119909119899) = 119903(120572119899)119899

(4)

When 1205720 = 1205721 = sdot sdot sdot = 120572119899 = 1 the full Hermite interpolationsimplifies into simple Hermite or osculatory interpolation

Definition 3 Given distinct knots 1199090 1199091 119909119899 associatedfunction values 1198910 1198911 119891119899 and a linear space Φ of spe-cific real functions generated by continuous cardinal basisfunctions 120601119895 R rarr R (119895 = 0 1 119899) 120601119895(119909119894) = 120575119894119895(119894 = 0 1 119899) we say that the function 119865 organized in theshape119865(119909) = sum119899119895=0 119891119895120601119895(119909) is an interpolant based on cardinalbasis and such a procedure is the cardinal basis functionsmethod

3 Piecewise Fuzzy HermiteInterpolation Polynomial

A special case of full Hermite interpolation is piecewiseHermite interpolation (see eg [13 14]) Let us assumethroughout the paper that Δ 119886 = 1199090 lt 1199091 lt sdot sdot sdot lt119909119899 = 119887 is a grid of 119868 = [119886 119887] with knots 119909119894 and 119898 isa positive integer All piecewise Hermite polynomials forma certain finite dimensional smooth linear space which wename1198672119898minus1(Δ 119868)Definition 4 1198672119898minus1(Δ 119868) is a collection of all real-valuedpiecewise-polynomial functions 119904(119909) of degree at most 2119898 minus1 defined on 119868 such that 119904(119909) isin 119862119898minus1(119868) The associatedfunction to 119904(119909) on successive intervals [119909119894minus1 119909119894] 1 le 119894 le 119899with knots from Δ is defined by 119904119894(119909) that is a (119898minus 1)-timescontinuously differentiable piecewise Hermite polynomial ofdegree 2119898 minus 1 on 119868Definition 5 (see [15]) Given any real-valued function119891(119909) isin 119862119898minus1(119868) Let its unique 1198672119898minus1(Δ 119868)-interpolate for

Advances in Fuzzy Systems 3

119898 and grid Δ of 119868 be the element 119904(119909) of degree 2119898 minus 1 oneach interval [119909119894 minus 1 119909119894] 1 le 119894 le 119899 such that

119863119896119904 (119909119894) = 119863119896119891 (119909119894)forall0 le 119896 le 119898 minus 1 0 le 119894 le 119899 119863119896 = 119889119896119889119909119896

(5)

Existence and uniqueness of full Hermite interpolation isprovided in [12] Because of this presentation (5) is actually aspecial case of such interpolation on a gridded interval and itfollows that each function belonging to 119862119898minus1(119868) has a uniqueinterpolate in1198672119898minus1(Δ 119868)

A particular cardinal basis for linear space1198672119898minus1(Δ 119868) ofdimension 119898(119899 + 1) is B = 120601119894119896(119909)119899119898minus1119894=0119896=0 (see eg [16])where the basis function 120601119894119896(119909) is defined by

119863119897120601119894119896 (119909119895) = 1205751198961198971205751198941198950 le 119896 119897 le 119898 minus 1 0 le 119894 119895 le 119899 119863119897 = 119889119897119889119909119897

(6)

Some important results based on (6) are simple to see in thesequel as 1206011198940(119909119894) = 1 and 1206011198940(119909119895) = 0 at all knots 119909119895 andsince 119904(119909) outside [119909119894minus1 119909119894] 1 le 119894 le 119899 satisfies zero datathen 1206011198940 equiv 0 for all 1199090 le 119909 le 119909119894minus1 and 119909119894+1 le 119909 le 1199091198991206011198941(119909) is of degree 2119898 minus 1 and 12060110158401198941(119909119894) = 1 but it is zeroat all other knots Moreover because outside the interval[119909119894minus1 119909119894+1]1206011198941(119909) interpolates zero data then 1206011198941(119909) must bevanished identically for all 119909 ge 119909119894+1 and 119909 le 119909119894minus1 (see eg[13 14 17]) Analogous reasoning applies to

1206011198941 (119909)

=

(119909 minus 119909119894minus1)119898 (119909 minus 119909119894)119898minus2sum119895=0

119886119895119909119895 119909119894minus1 le 119909 le 119909119894(119909 minus 119909119894+1)119898 (119909 minus 119909119894)119898minus2sum

119895=0

119887119895119909119895 119909119894 le 119909 le 119909119894+10 otherwise

(7)

In the following theorem we will use the recent features

Theorem 6 Assume that 120601119894119896 isin B and satisfies the piecewiseHermite polynomial cardinal basis function constraints (6)Then

(i) 120601119894 0(119909) + 120601119894+1 0(119909) ge 1 for all 119909 isin (119909119894 119909119894+1) 119894 =0 1 119899 minus 1(ii) For all 119894 = 1 2 119899 minus 1 1206011198941 changes the sign at 119909119894 The

sign of 1206011198941 is not positive on any subinterval [119909119894minus1 119909119894]and that is not negative on [119909119894 119909119894+1]

(iii) The sign of all other elements ofB is not negative on 119868Proof With the assumption of (6) let 120601119894 0(119909) + 120601119894+1 0(119909) bepolynomial of degree 2119898 minus 1 on the interval [119909119894minus1 119909119894+2] andinterpolate the data (119909119895 119891119895) where 119891119895 = 1 for 119895 = 119894 119894 + 1 andzero on the other knots of partition Δ Suppose that 0 lt 119894 lt119899 minus 1 and 120601119894 0(119909) + 120601119894+1 0(119909) lt 1 for some 119909 isin (119909119894 119909119894+1) By

themean value theorem its derivative has a zero on (119909119894 119909119894+1)The derivative has two (119898minus2)th order zeros at 119909119894 and 119909119894+1 andits two other zeros are 119909119894minus1 119909119894+2 Then it has at least 2119898 minus 1zeros on the interval [119909119894minus1 119909119894+2] which is a contradictionThecases 119894 = 0 and 119894 = 119899 minus 1 are treated similarly

In light of representation (7) and condition (6) thepolynomial 1206011198941(119909) is of degree 2119898 minus 1 It has only oneminimum point on [119909119894minus1 119909119894] and a single maximum onthe subinterval [119909119894 119909119894+1] Suppose that each of the abovepoints are one more Then by the mean value theorem firstderivative of 1206011198941(119909) has at least three zeros on (119909119894minus1 119909119894) andthree zeros on (119909119894 119909119894+1) Also the derivative has two (119898minus2)thorder zeros at 119909119894minus1 119909119894+1 Then it has at least 2119898 + 2 zeroswhich is a contradiction Hence 12060110158401198941(119909) has only one zeroon each of the intervals (119909119894minus1 119909119894) and (119909119894 119909119894+1) Now recall12060110158401198941(119909119894) = 1 it follows that 1206011198941(119909) le 0 on [119909119894minus1 119909119894] and1206011198941(119909) ge 0 on [119909119894 119909119894+1] This gives (ii)

A similar proof via definition of basis functions and (6)follows the claim (iii)

For a given 119891(119909) isin 119862119898minus1(119868) and its piecewise Hermiteinterpolate 119904(119909) isin 1198672119898minus1(Δ 119868) an equivalent explicit repre-sentation of 119904(119909) in (5) can be uniquely expressed (see eg[13 17]) namely

119904 (119909) = 119898minus1sum119896=0

119899sum119894=0

119891(119896) (119909119894) 120601119894119896 (119909) (8)

Now we want to construct a fuzzy-valued function as 119904 119868 rarr RF such that 119904(119896)(119909119894) = 119891(119896)(119909119894) = 119906119896119894 isin RF 0 le 119896 le119898minus 1 0 le 119894 le 119899 Also if for all 0 le 119896 le 119898minus 1 0 le 119894 le 119899 119906119896119894 =119910(119896)119894 are crisp numbers in R and 119891(119896)(119909119894) = 120594119910(119896)119894 (see eg[2]) then there is a polynomial of degree 2119898minus1 on successiveintervals [119909119894minus1 119909119894] 0 le 119894 le 119899 with 119904(119896)(119909119894) = 119910(119896)119894 0 le 119896 le119898 minus 1 0 le 119894 le 119899 such that 119904(119909) = 120594119891(119909) for all 119909 isin R where(119909119894 119891119894 1198911015840119894 119891(119898minus1)119894 ) | 119891(119896)119894 isin RF 0 le 119896 le 119898 minus 1 0 le 119894 le119899 is given

We suppose that such a fuzzy function exists and weattempt to find and compute it with respect to interpolationpolynomial presented by Lowen [2] Let for each119909 isin [1199090 119909119899]119904(119909) be a fuzzy piecewise Hermite polynomial and Λ =119910(119896)119894 119899119898minus1119894=0119896=0 then fromKaleva [3] andNguyen [18] we obtainthe 120572-cuts of 119904(119909) in a succinctly algorithm as follows

119904120572 (119909) = 119905 isin R | 120583(119905)119904(119909) ge 120572 = 119905 isin R | exist119910(119896)119894 120583(119910(119896)119894 )119906119896119894ge 120572 0 le 119896 le 119898 minus 1 0 le 119894 le 119899 119904Λ (119909) = 119905 = 119905isin R | 119905 = 119904Λ (119909) 119910(119896)119894 isin 119906120572119896119894 0 le 119896 le 119898 minus 1 0 le 119894le 119899 = 119898minus1sum

119896=0

119899sum119894=0

119906120572119896119894120601119894119896 (119909) (9)

where

119904Λ (119909) = 119898minus1sum119896=0

119899sum119894=0

119910(119896)119894 120601119894119896 (119909) (10)

4 Advances in Fuzzy Systems

is a piecewise Hermite polynomial in crisp case and bydefinition

119904120572 (119909) = 119898minus1sum119896=0

119899sum119894=0

119906120572119896119894120601119894119896 (119909) (11)

we obtain a formula that comprises a simple practical way forcalculating 119904(119909)

119904 (119909) = 119898minus1sum119896=0

119899sum119894=0

119906119896119894120601119894119896 (119909) (12)

Since for each 0 le 119896 le 119898 minus 1 0 le 119894 le 119899 119906120572119896119894 = [119906120572119896119894 119906120572119896119894]then we will have 119904120572(119909) by solving the following optimizationproblems

max amp min119898minus1sum119896=0

119899sum119894=0

119910(119896)119894 120601119894119896 (119909)subject to 119906120572119896119894 le 119910(119896)119894 le 119906120572119896119894

0 le 119896 le 119898 minus 1 0 le 119894 le 119899(13)

From the 120601119894119895rsquos sign that we represented in Theorem 6 theseproblems have the following optimal solutions

Maximization is as follows

119910(119896)119894 = 119906120572119896119894 if 120601119894119896 (119909) ge 0119906120572119896119894 if 120601119894119896 (119909) lt 0

0 le 119896 le 119898 minus 1 0 le 119894 le 119899(14)

Minimization is as follows

119910(119896)119894 = 119906120572119896119894 if 120601119894119896 (119909) ge 0119906120572119896119894 if 120601119894119896 (119909) lt 0

0 le 119896 le 119898 minus 1 0 le 119894 le 119899(15)

Theorem 7 If 119904(119909) = sum119898minus1119896=0 sum119899119894=0 119906119896119894120601119894119896(119909) is an interpolatingpiecewise fuzzy Hermite polynomial then for all 120572 isin [0 1]119894 isin 0 1 119899 minus 1 119909 isin [119909119894 119909119894+1]

119897119890119899 119904120572 (119909) ge min 119897119890119899 119904120572 (119909119894) 119897119890119899 119904120572 (119909119894+1) (16)

where

119904120572 (119909) = [119904120572 (119909) 119904120572 (119909)] 119897119890119899 119904120572 (119909) = 119904120572 (119909) minus 119904120572 (119909) (17)

Proof By using Theorem 6 and (11) we have 119904120572(119909119894) = 1199061205720 119894119904120572(119909119894+1) = 1199061205720 119894+1 and len 119904120572(119909119894) = len 1199061205720119894 len 119904120572(119909119894+1) =

len 1199061205720 119894+1 Since the addition does not decrease thelength of an interval from (11) we can write 119904120572(119909) =sum119898minus1119896=0 sum119899119895=0 119906120572119896119895120601119895119896(119909) then

len 119904120572 (119909) ge 119898minus1sum119896=0

119899sum119895=0

10038161003816100381610038161003816120601119895119896 (119909)10038161003816100381610038161003816 len 119906120572119896119895ge 119899sum119895=0

100381610038161003816100381610038161206011198950 (119909)10038161003816100381610038161003816 len 1199060119895ge 10038161003816100381610038161206011198940 (119909)1003816100381610038161003816 len 1199060119894 + 1003816100381610038161003816120601119894+1 0 (119909)1003816100381610038161003816 len 1199060 119894+1ge min len 1199060 119894 len 1199060 119894+1 (10038161003816100381610038161206011198940 (119909)1003816100381610038161003816 + 1003816100381610038161003816120601119894+1 0 (119909)1003816100381610038161003816)ge min len 1199060 119894 len 1199060 119894+1= min len 119904120572 (119909119894) len 119904120572 (119909119894+1)

(18)

Theorem 8 Let 119906119896119894 = (119898119896119894 119897119896119894 119903119896119894) 0 le 119896 le 119898 minus 1 0 le119894 le 119899 be a triangular 119871 minus 119871 fuzzy number then also 119904(119909)the piecewise fuzzy Hermite polynomial interpolation is sucha fuzzy number for each 119909Proof The closed interval [119898 minus 119897119898 + 119903] is the support of 119906 =(119898 119897 119903) a triangular 119871minus119871 fuzzy number then for each 119909 and119906119896119894 we have119904 (119909) = (119898minus1sum

119896=0

119899sum119894=0

119906119896119894120601119894119896 (119909))

= [[sum sum120601119894119896ge0

(119898119896119894 minus 119897119896119894) 120601119894119896 (119909)+sum sum120601119894119896lt0

(119898119896119894 + 119903119896119894) 120601119894119896 (119909) sum sum120601119894119896ge0

(119898119896119894 + 119903119896119894) 120601119894119896 (119909)

+sum sum120601119894119896lt0

(119898119896119894 minus 119897119896119894) 120601119894119896 (119909)]] = [[119898minus1sum119896=0

119899sum119894=0

119898119896119894120601119894119896 (119909)

minus (sum sum120601119894119896ge0

119897119896119894120601119894119896 (119909) minussum sum120601119894119896lt0

119903119896119894120601119894119896 (119909))

+ 119898minus1sum119896=0

119899sum119894=0

119898119896119894120601119894119896 (119909)

+ (sum sum120601119894119896ge0

119903119896119894 (119909) 120601119894119896 (119909) minussum sum120601119894119896lt0

119897119894119896120601119894119896 (119909))]]= (119898 (119909) minus 119897 (119909) 119898 (119909) + 119903 (119909))

(19)

Advances in Fuzzy Systems 5

It follows that if 119904(119909) = (119898(119909) 119897(119909) 119903(119909)) is a triangular119871 minus 119871 fuzzy number for each 119909 then

119898(119909) = 119898minus1sum119896=0

119899sum119894=0

119898119896119894120601119896119894119897 (119909) = sum sum

120601119894119896ge0

119897119896119894120601119894119896 (119909) minussum sum120601119894119896lt0

119903119896119894120601119894119896 (119909) 119903 (119909) = sum sum

120601119894119896ge0

119903119896119894120601119894119896 minussum sum120601119894119896lt0

119897119894119896120601119894119896 (119909) (20)

4 Piecewise-Polynomial Linear Cubic andQuintic Fuzzy Hermite Interpolation

We consider 119898 = 1 and compute the piecewise fuzzy linearinterpolant as the initial case of the presented method basedon (12) and for a given set of fuzzy data (119909119894 119891119894) | 119891119894 isinRF 0 le 119894 le 119899 as follows

119904 (119909) = 119899sum119894=0

11990601198941206011198940 (119909) (21)

where 1199060119894 = 119891119894 0 le 119894 le 119899 and subject to conditions (6)

12060100 (119909) = 0 119909 ge 1199091( 1199091 minus 1199091199091 minus 1199090) 1199090 le 119909 le 1199091

1206011198940 (119909) =

0 119909 le 119909119894minus1 119909 ge 119909119894+1( 119909 minus 119909119894minus1119909119894 minus 119909119894minus1) 119909119894minus1 le 119909 le 119909119894( 119909119894+1 minus 119909119909119894+1 minus 119909119894) 119909119894 le 119909 le 119909119894+1

1 le 119894 le 119899 minus 11206011198990 (119909) =

0 119909 le 119909119899minus1( 119909 minus 119909119899minus1119909119899 minus 119909119899minus1) 119909119899minus1 le 119909 le 119909119899

(22)

The obtained 119904(119909) is the same as fuzzy spline of order 119897 = 2that had been introduced in [3] because the basic splines andthe cardinal basis functions in two interpolants are equal

Example 9 (see [4]) Suppose the data (1 (0 2 1) (1 0 3))(13 (5 1 2) (0 2 1)) (22 (1 0 3) (4 4 3)) (3 (4 4 3)(5 1 2)) (35 (0 3 2) (1 1 1)) (4 (1 1 1) and (0 3 2))In Figure 1 the dashed line is the 05-cut set of piecewisecubic fuzzy interpolation 119904(119909) 119909 isin [1 4] and the solid linesrepresent the support and the core of 119904(119909)

10 15 20 25 30 35 40

minus2

0

2

4

6

8

Figure 1 Graph of Example 9

10 15 20 25 30 35 40

minus2

0

2

4

6

Figure 2 Graph of Example 10

When 119898 = 2 we get the piecewise cubic fuzzy Her-mite polynomial interpolant in [11] for a given set of data(119909119894 119891119894 1198911015840119894 ) | 119891119894 1198911015840119894 isin RF 0 le 119894 le 119899

119904 (119909) = 119899sum119894=0

11990601198941206011198940 (119909) + 119899sum119894=0

11990611198941206011198941 (119909) (23)

where 119906119896119894 = 119891(119896)119894 119896 = 0 1 0 le 119894 le 119899An outstanding feature of this study is that by simply

applying the second case of the presented general methodand exactly the same data we have produced an alternativeto simple fuzzy Hermite polynomial interpolation in [4]Heretofore thementioned cubic case (23) was independentlyintroduced in [10] but only in very weak conditions andwithout using the extension principle

The cardinal basis functions 120601119894119896(119909) 119896 = 0 1 0 le 119894 le 119899were computed in [17]

Example 10 Suppose the data (1 (0 2 1) (1 0 3)) (15 (51 2) (0 2 1)) (27 (1 0 3) (4 4 3)) (3 (4 4 3) (5 1 2))(37 (0 3 2) (1 1 1)) and (4 (1 1 1) (0 3 2)) In Figure 2the dashed line is the 05-cut set of piecewise cubic fuzzyinterpolations 119904(119909) 119909 isin [1 4] and the solid lines representthe support and the core of 119904(119909)

Let 119898 = 3 from (6) we shall construct the cardi-nal basis for 1198675(Δ 119868) The quintic Hermite polynomials

6 Advances in Fuzzy Systems

1206011198940(119909) 1206011198941(119909) and 1206011198942(119909) are solving the interpolationproblem

119863119897120601119894119896 (119909119895) = 120575119896119897120575119894119895 0 le 119896 119897 le 2 0 le 119894 119895 le 119899 (24)

To this end we determine uniquely all the pervious 120601119894119895rsquos bythe (24)

For 1 le 119894 le 119899 minus 1 let

1206011198940 (119909) =

(119909119894minus1 minus 119909)3(119909119894minus1 minus 119909119894)5 [(119909119894minus1 + 3119909) (119909119894minus1 minus 5119909119894) + 61199092 + 101199092119894 ] 119909119894minus1 le 119909 le 119909119894

(119909119894+1 minus 119909)3(119909119894+1 minus 119909119894)5 [(119909119894+1 + 3119909) (119909119894+1 minus 5119909119894) + 61199092 + 101199092119894 ] 119909119894 le 119909 le 119909119894+1

0 otherwise

1206011198941 (119909) =

( 119909119894minus1 minus 119909119909119894minus1 minus 119909119894)3 (119909 minus 119909119894) [1 + 3 119909 minus 119909119894119909119894minus1 minus 119909119894 ] 119909119894minus1 le 119909 le 119909119894

( 119909119894+1 minus 119909119909119894 minus 119909119894+1)3 (119909119894 minus 119909) [1 + 3 119909119894 minus 119909119909119894 minus 119909119894+1 ] 119909119894 le 119909 le 119909119894+1

0 otherwise

1206011198942 (119909) =

( 119909119894minus1 minus 119909119909119894minus1 minus 119909119894)3 (119909119894 minus 119909)22 119909119894minus1 le 119909 le 119909119894

( 119909 minus 119909119894+1119909119894 minus 119909119894+1)3 (119909119894 minus 119909)22 119909119894 le 119909 le 119909119894+1

0 otherwise

(25)

The six next functions are similarly defined In particular

12060100 (119909) =

(1199091 minus 119909)3(1199091 minus 1199090)5 [(1199091 + 3119909) (1199091 minus 51199090) + 61199092 + 1011990930] 1199090 le 119909 le 1199091

0 1199091 le 119909 le 119909119899

1206011198990 (119909) =

(119909119899minus1 minus 119909)3(119909119899minus1 minus 119909119899)5 [(119909119899minus1 + 3119909) (119909119899minus1 minus 5119909119899) + 61199092 + 101199092119899] 119909119899minus1 le 119909 le 119909119899

0 1199090 le 119909 le 119909119899minus112060101 (119909) =

( 1199091 minus 1199091199090 minus 1199091)3 (1199090 minus 119909) [1 + 3 1199090 minus 1199091199090 minus 1199091 ] 1199090 le 119909 le 1199091

0 1199091 le 119909 le 1199091198991206011198991 (119909) =

( 119909119899minus1 minus 119909119909119899minus1 minus 119909119899)3 (119909 minus 119909119899) [1 + 3 119909 minus 119909119899119909119899minus1 minus 119909119899 ] 119909119899minus1 le 119909 le 119909119899

0 1199090 le 119909 le 119909119899minus112060102 (119909) =

( 119909 minus 11990911199090 minus 1199091)

3 (1199090 minus 119909)22 1199090 le 119909 le 11990910 1199091 le 119909 le 119909119899

Advances in Fuzzy Systems 7

0 1 2 3 4 5 6minus10

0

10

20

30

Figure 3 Graph of Example 11

1206011198992 (119909) =( 119909119899minus1 minus 119909119909119899minus1 minus 119909119899)

3 (119909119899 minus 119909)22 119909119899minus1 le 119909 le 1199091198990 1199090 le 119909 le 119909119899minus1

(26)

Thus we can immediately write down piecewise quintic fuzzyHermite interpolation polynomial 119904(119909) using

119904 (119909) = 119899sum119894=0

11990601198941206011198940 (119909) + 119899sum119894=0

11990611198941206011198941 (119909) + 119899sum119894=0

11990621198941206011198942 (119909) (27)

where (119909119894 119891119894 1198911015840119894 11989110158401015840119894 ) | 119891(119896)119894 isin BF 0 le 119896 le 2 0 le 119894 le 119899 isgiven and 119906119896119894 = 119891(119896)119894

Example 11 Suppose that (0 (0 1 3) (0 2 2) (1 4 4)) (13(005 19 35) (03 32 08) (1 31 3)) (2 (2 67 53) (2 0535) (1 26 24)) (4 (8 101 99) (4 4 0) (1 06 05)) (53(14 13 12) (53 02 38) (1 15 17)) (6 (18 132 148) (6 093) and (1 34 32)) are the interpolation data In Figure 3the solid lines denote the support and the core of piecewisequintic fuzzy Hermite interpolation 119904(119909) 119909 isin [0 6] and thedashed line is the 05-cut set of 119904(119909)5 Conclusions and Further Work

Based on the cardinal basis functions for119898(119899+ 1) dimension1198672119898minus1(Δ 119868) linear space interpolation of fuzzy data by thefuzzy-valued piecewise Hermite polynomial as the extensionof same approach in [11] has been successfully introduced ingeneral case and provided a succinct formula for calculatingthe new fuzzy interpolant Moreover two first cases ofthe presented method have been applied as an analogy tofuzzy spline of order two in [3] and an alternative to fuzzyosculatory interpolation in [4] respectively In the guise of aremarkable achievement the piecewise fuzzy cubic Hermitepolynomial interpolation that was constructed with poorconditions and without using the extension principle in [10]has been produced in the role of a very special subdivision forthe presented general method in this study Finally the thirdinitial case piecewise fuzzy quintic Hermite polynomial

has been described in detail The next step to improve thismethod is interpolation of fuzzy data including switchingpoints by a fuzzy differentiable piecewise interpolant

Competing Interests

The authors declare that they have no competing interests

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] R Lowen ldquoA fuzzy Lagrange interpolation theoremrdquo Fuzzy Setsand Systems vol 34 no 1 pp 33ndash38 1990

[3] O Kaleva ldquoInterpolation of fuzzy datardquo Fuzzy Sets and SystemsAn International Journal in Information Science andEngineeringvol 61 no 1 pp 63ndash70 1994

[4] H S Goghary and S Abbasbandy ldquoInterpolation of fuzzy databy Hermite polynomialrdquo International Journal of ComputerMathematics vol 82 no 5 pp 595ndash600 2005

[5] H Behforooz R Ezzati and S Abbasbandy ldquoInterpolation offuzzy data by using E(3) cubic splinesrdquo International Journal ofPure and Applied Mathematics vol 60 no 4 pp 383ndash392 2010

[6] S Abbasbandy R Ezzati and H Behforooz ldquoInterpolation offuzzy data by using fuzzy splinesrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 16 no1 pp 107ndash115 2008

[7] S Abbasbandy ldquoInterpolation of fuzzy data by completesplinesrdquoKorean Journal of Computational ampAppliedMathemat-ics vol 8 no 3 pp 587ndash594 2001

[8] S Abbasbandy and E Babolian ldquoInterpolation of fuzzy data bynatural splinesrdquoTheKorean Journal of Computational ampAppliedMathematics vol 5 no 2 pp 457ndash463 1998

8 Advances in Fuzzy Systems

[9] W A Lodwick and J Santos ldquoConstructing consistent fuzzysurfaces from fuzzy datardquo Fuzzy Sets and Systems An Interna-tional Journal in Information Science and Engineering vol 135no 2 pp 259ndash277 2003

[10] M Zeinali S Shahmorad and K Mirnia ldquoHermite and piece-wise cubic Hermite interpolation of fuzzy datardquo Journal ofIntelligent amp Fuzzy Systems vol 26 no 6 pp 2889ndash2898 2014

[11] H Vosoughi and S Abbasbandy ldquoInterpolation of fuzzy data bycubic and piecewise-polynomial cubic hermitesrdquo Indian Journalof Science and Technology vol 9 no 8 2016

[12] P J Davis Interpolation and Approximation Dover New YorkNY USA 1975

[13] B Wendroff Theoretical Numerical Analysis Academic PressNew York NY USA 1966

[14] P G Ciarlet M H Schultz and R S Varga ldquoNumerical meth-ods of high-order accuracy for nonlinear boundary valueproblemsrdquo Numerische Mathematik vol 9 pp 397ndash430 1967

[15] G Birkhoff M H Schultz and R S Varga ldquoPiecewise Hermiteinterpolation in one and two variables with applications topartial differential equationsrdquo Numerische Mathematik vol 11pp 232ndash256 1968

[16] R S Varga ldquoHermite interpolation-type Ritz methods for two-point boundary value problemsrdquo in Numerical Solution ofPartial Differential Equations J H Bramble Ed pp 365ndash373Academic Press New York NY USA 1966

[17] P M Prenter Splines and Variational Methods Wiley-Intersci-ence 1975

[18] H T Nguyen ldquoA note on the extension principle for fuzzy setsrdquoJournal of Mathematical Analysis and Applications vol 64 no2 pp 369ndash380 1978

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2 Advances in Fuzzy Systems

basic concepts and findings next we construct piecewisefuzzy Hermite polynomial in detail based on cardinal basisfunctions and prove some new properties of the introducedgeneral interpolant (Section 3) In Section 4 we have pro-duced three initial linear cubic [11] and quintic cases andshown the relationship between some of the mentioned casesand the newly presented interpolants in [3 4 10] Further-more to illustrate themethod some computational examplesare provided Finally the conclusions of this interpolation arein Section 5

2 Preliminaries

To begin let us introduce some brief account of notionsused throughout the paper We shall denote the set offuzzy numbers by RF the family of all nonempty convexnormal upper semicontinuous and compactly supportedfuzzy subsets defined on the real axis R ObviouslyR sub RFIf 119906 isin RF is a fuzzy number then 119906120572 = 119909 isin R | 119906(119909) ge 1205720 lt 120572 le 1 shows the 120572-cut of 119906 For 120572 = 0 by the closureof the support 1199060 = cl119909 | 119909 isin R 119906(119909) gt 0 It is wellknown the 120572-cuts of 119906 isin RF are closed bounded intervalsin R and we will denote them by 119906120572 = [119906120572 119906120572] functions119906(sdot) 119906(sdot) are the lower and upper branches of 119906 The core of 119906is 1199061 = 119909 | 119909 isin R 119906(119909) = 1 In terms of 120572-cuts we have theaddition and the scalar multiplication

(119906 + V)120572 = 119906120572 + V120572 = 119909 + 119910 | 119909 isin 119906120572 119910 isin V120572(120582119906)120572 = 120582119906120572 = 120582119909 | 119909 isin 119906120572(0)120572 = 0

(1)

for all 0 le 120572 le 1 119906 V isin RF and 120582 isin R119906 = ⟨119886 119887 119888 119889⟩ specifies a trapezoidal fuzzy numberwhere 119886 le 119887 le 119888 le 119889 and if 119887 = 119888 we obtain a triangular fuzzynumber For120572 isin [0 1] we have 119906120572 = [119886+120572(119887minus119886) 119889minus120572(119889minus119888)]In the rest of this paper we will assume that 119906 is a triangularfuzzy number

Definition 1 (see eg [5]) An L-R fuzzy number 119906 = (119898119897 119903)119871119877 is a function from the real numbers into the interval[0 1] satisfying

119906 (119909) =

119877(119909 minus 119898119903 ) for 119898 le 119909 le 119898 + 119903119871 (119898 minus 119909119897 ) for 119898 minus 119897 le 119909 le 1198980 otherwise

(2)

where 119877 and 119871 are continuous and decreasing functions from[0 1] to [0 1] fulfilling the conditions 119877(0) = 119871(0) = 1 and119877(1) = 119871(1) = 0When119877(119909) = 119871(119909) = 1minus119909 wewill have 119871minus119871fuzzy numbers that involve the triangular fuzzy numbers Foran 119871 minus 119871 fuzzy number 119906 = (119898 119897 119903) the support is the closedinterval [119898 minus 119897119898 + 119903] (see eg [6])

The linear space of all polynomials of degree at most 119873will be designated by 119875119873 Full Hermite interpolation problemdefines a unique polynomial called 119901119873(119909) which solves thefollowing problem

Theorem 2 (see [12] (existence and uniqueness)) Let1199090 1199091 119909119899 be 119899 + 1 distinct points 1205720 1205721 120572119899 be positiveintegers 119896 = 0 1 120572119894 and 119873 = 1205720 + 1205721 + sdot sdot sdot + 120572119899 + 119899 Set119908(119909) = prod119899119894=0(119909 minus 119909119894)120572119894+1 and119897119894119896 (119909)= 119908 (119909) (119909 minus 119909119894)119896minus120572119894119896 (119909 minus 119909119894)120572119894+1minus119896

119889(120572119894minus119896)119889119909(120572119894minus119896) [(119909 minus 119909119894)120572119894+1

119908 (119909) ]119909=119909119894

119901119873 (119909) = 119899sum119894=0

1199031198941198971198940 (119909) + 119899sum119894=0

1199031015840119894 1198971198941 (119909) + sdot sdot sdot + 119899sum119894=0

119903(120572119894)119894 119897119894120572119894 (119909)(3)

is a unique member of 119875119873 for which119901119873 (1199090) = 1199030 1199011015840119873 (119909) = 11990310158400 119901(1205720minus1)119873 (1199090) = 119903(1205720)0119901119873 (119909119899) = 119903119899 1199011015840119873 (119909119899) = 1199031015840119899 119901(120572119899minus1)119873 (119909119899) = 119903(120572119899)119899

(4)

When 1205720 = 1205721 = sdot sdot sdot = 120572119899 = 1 the full Hermite interpolationsimplifies into simple Hermite or osculatory interpolation

Definition 3 Given distinct knots 1199090 1199091 119909119899 associatedfunction values 1198910 1198911 119891119899 and a linear space Φ of spe-cific real functions generated by continuous cardinal basisfunctions 120601119895 R rarr R (119895 = 0 1 119899) 120601119895(119909119894) = 120575119894119895(119894 = 0 1 119899) we say that the function 119865 organized in theshape119865(119909) = sum119899119895=0 119891119895120601119895(119909) is an interpolant based on cardinalbasis and such a procedure is the cardinal basis functionsmethod

3 Piecewise Fuzzy HermiteInterpolation Polynomial

A special case of full Hermite interpolation is piecewiseHermite interpolation (see eg [13 14]) Let us assumethroughout the paper that Δ 119886 = 1199090 lt 1199091 lt sdot sdot sdot lt119909119899 = 119887 is a grid of 119868 = [119886 119887] with knots 119909119894 and 119898 isa positive integer All piecewise Hermite polynomials forma certain finite dimensional smooth linear space which wename1198672119898minus1(Δ 119868)Definition 4 1198672119898minus1(Δ 119868) is a collection of all real-valuedpiecewise-polynomial functions 119904(119909) of degree at most 2119898 minus1 defined on 119868 such that 119904(119909) isin 119862119898minus1(119868) The associatedfunction to 119904(119909) on successive intervals [119909119894minus1 119909119894] 1 le 119894 le 119899with knots from Δ is defined by 119904119894(119909) that is a (119898minus 1)-timescontinuously differentiable piecewise Hermite polynomial ofdegree 2119898 minus 1 on 119868Definition 5 (see [15]) Given any real-valued function119891(119909) isin 119862119898minus1(119868) Let its unique 1198672119898minus1(Δ 119868)-interpolate for

Advances in Fuzzy Systems 3

119898 and grid Δ of 119868 be the element 119904(119909) of degree 2119898 minus 1 oneach interval [119909119894 minus 1 119909119894] 1 le 119894 le 119899 such that

119863119896119904 (119909119894) = 119863119896119891 (119909119894)forall0 le 119896 le 119898 minus 1 0 le 119894 le 119899 119863119896 = 119889119896119889119909119896

(5)

Existence and uniqueness of full Hermite interpolation isprovided in [12] Because of this presentation (5) is actually aspecial case of such interpolation on a gridded interval and itfollows that each function belonging to 119862119898minus1(119868) has a uniqueinterpolate in1198672119898minus1(Δ 119868)

A particular cardinal basis for linear space1198672119898minus1(Δ 119868) ofdimension 119898(119899 + 1) is B = 120601119894119896(119909)119899119898minus1119894=0119896=0 (see eg [16])where the basis function 120601119894119896(119909) is defined by

119863119897120601119894119896 (119909119895) = 1205751198961198971205751198941198950 le 119896 119897 le 119898 minus 1 0 le 119894 119895 le 119899 119863119897 = 119889119897119889119909119897

(6)

Some important results based on (6) are simple to see in thesequel as 1206011198940(119909119894) = 1 and 1206011198940(119909119895) = 0 at all knots 119909119895 andsince 119904(119909) outside [119909119894minus1 119909119894] 1 le 119894 le 119899 satisfies zero datathen 1206011198940 equiv 0 for all 1199090 le 119909 le 119909119894minus1 and 119909119894+1 le 119909 le 1199091198991206011198941(119909) is of degree 2119898 minus 1 and 12060110158401198941(119909119894) = 1 but it is zeroat all other knots Moreover because outside the interval[119909119894minus1 119909119894+1]1206011198941(119909) interpolates zero data then 1206011198941(119909) must bevanished identically for all 119909 ge 119909119894+1 and 119909 le 119909119894minus1 (see eg[13 14 17]) Analogous reasoning applies to

1206011198941 (119909)

=

(119909 minus 119909119894minus1)119898 (119909 minus 119909119894)119898minus2sum119895=0

119886119895119909119895 119909119894minus1 le 119909 le 119909119894(119909 minus 119909119894+1)119898 (119909 minus 119909119894)119898minus2sum

119895=0

119887119895119909119895 119909119894 le 119909 le 119909119894+10 otherwise

(7)

In the following theorem we will use the recent features

Theorem 6 Assume that 120601119894119896 isin B and satisfies the piecewiseHermite polynomial cardinal basis function constraints (6)Then

(i) 120601119894 0(119909) + 120601119894+1 0(119909) ge 1 for all 119909 isin (119909119894 119909119894+1) 119894 =0 1 119899 minus 1(ii) For all 119894 = 1 2 119899 minus 1 1206011198941 changes the sign at 119909119894 The

sign of 1206011198941 is not positive on any subinterval [119909119894minus1 119909119894]and that is not negative on [119909119894 119909119894+1]

(iii) The sign of all other elements ofB is not negative on 119868Proof With the assumption of (6) let 120601119894 0(119909) + 120601119894+1 0(119909) bepolynomial of degree 2119898 minus 1 on the interval [119909119894minus1 119909119894+2] andinterpolate the data (119909119895 119891119895) where 119891119895 = 1 for 119895 = 119894 119894 + 1 andzero on the other knots of partition Δ Suppose that 0 lt 119894 lt119899 minus 1 and 120601119894 0(119909) + 120601119894+1 0(119909) lt 1 for some 119909 isin (119909119894 119909119894+1) By

themean value theorem its derivative has a zero on (119909119894 119909119894+1)The derivative has two (119898minus2)th order zeros at 119909119894 and 119909119894+1 andits two other zeros are 119909119894minus1 119909119894+2 Then it has at least 2119898 minus 1zeros on the interval [119909119894minus1 119909119894+2] which is a contradictionThecases 119894 = 0 and 119894 = 119899 minus 1 are treated similarly

In light of representation (7) and condition (6) thepolynomial 1206011198941(119909) is of degree 2119898 minus 1 It has only oneminimum point on [119909119894minus1 119909119894] and a single maximum onthe subinterval [119909119894 119909119894+1] Suppose that each of the abovepoints are one more Then by the mean value theorem firstderivative of 1206011198941(119909) has at least three zeros on (119909119894minus1 119909119894) andthree zeros on (119909119894 119909119894+1) Also the derivative has two (119898minus2)thorder zeros at 119909119894minus1 119909119894+1 Then it has at least 2119898 + 2 zeroswhich is a contradiction Hence 12060110158401198941(119909) has only one zeroon each of the intervals (119909119894minus1 119909119894) and (119909119894 119909119894+1) Now recall12060110158401198941(119909119894) = 1 it follows that 1206011198941(119909) le 0 on [119909119894minus1 119909119894] and1206011198941(119909) ge 0 on [119909119894 119909119894+1] This gives (ii)

A similar proof via definition of basis functions and (6)follows the claim (iii)

For a given 119891(119909) isin 119862119898minus1(119868) and its piecewise Hermiteinterpolate 119904(119909) isin 1198672119898minus1(Δ 119868) an equivalent explicit repre-sentation of 119904(119909) in (5) can be uniquely expressed (see eg[13 17]) namely

119904 (119909) = 119898minus1sum119896=0

119899sum119894=0

119891(119896) (119909119894) 120601119894119896 (119909) (8)

Now we want to construct a fuzzy-valued function as 119904 119868 rarr RF such that 119904(119896)(119909119894) = 119891(119896)(119909119894) = 119906119896119894 isin RF 0 le 119896 le119898minus 1 0 le 119894 le 119899 Also if for all 0 le 119896 le 119898minus 1 0 le 119894 le 119899 119906119896119894 =119910(119896)119894 are crisp numbers in R and 119891(119896)(119909119894) = 120594119910(119896)119894 (see eg[2]) then there is a polynomial of degree 2119898minus1 on successiveintervals [119909119894minus1 119909119894] 0 le 119894 le 119899 with 119904(119896)(119909119894) = 119910(119896)119894 0 le 119896 le119898 minus 1 0 le 119894 le 119899 such that 119904(119909) = 120594119891(119909) for all 119909 isin R where(119909119894 119891119894 1198911015840119894 119891(119898minus1)119894 ) | 119891(119896)119894 isin RF 0 le 119896 le 119898 minus 1 0 le 119894 le119899 is given

We suppose that such a fuzzy function exists and weattempt to find and compute it with respect to interpolationpolynomial presented by Lowen [2] Let for each119909 isin [1199090 119909119899]119904(119909) be a fuzzy piecewise Hermite polynomial and Λ =119910(119896)119894 119899119898minus1119894=0119896=0 then fromKaleva [3] andNguyen [18] we obtainthe 120572-cuts of 119904(119909) in a succinctly algorithm as follows

119904120572 (119909) = 119905 isin R | 120583(119905)119904(119909) ge 120572 = 119905 isin R | exist119910(119896)119894 120583(119910(119896)119894 )119906119896119894ge 120572 0 le 119896 le 119898 minus 1 0 le 119894 le 119899 119904Λ (119909) = 119905 = 119905isin R | 119905 = 119904Λ (119909) 119910(119896)119894 isin 119906120572119896119894 0 le 119896 le 119898 minus 1 0 le 119894le 119899 = 119898minus1sum

119896=0

119899sum119894=0

119906120572119896119894120601119894119896 (119909) (9)

where

119904Λ (119909) = 119898minus1sum119896=0

119899sum119894=0

119910(119896)119894 120601119894119896 (119909) (10)

4 Advances in Fuzzy Systems

is a piecewise Hermite polynomial in crisp case and bydefinition

119904120572 (119909) = 119898minus1sum119896=0

119899sum119894=0

119906120572119896119894120601119894119896 (119909) (11)

we obtain a formula that comprises a simple practical way forcalculating 119904(119909)

119904 (119909) = 119898minus1sum119896=0

119899sum119894=0

119906119896119894120601119894119896 (119909) (12)

Since for each 0 le 119896 le 119898 minus 1 0 le 119894 le 119899 119906120572119896119894 = [119906120572119896119894 119906120572119896119894]then we will have 119904120572(119909) by solving the following optimizationproblems

max amp min119898minus1sum119896=0

119899sum119894=0

119910(119896)119894 120601119894119896 (119909)subject to 119906120572119896119894 le 119910(119896)119894 le 119906120572119896119894

0 le 119896 le 119898 minus 1 0 le 119894 le 119899(13)

From the 120601119894119895rsquos sign that we represented in Theorem 6 theseproblems have the following optimal solutions

Maximization is as follows

119910(119896)119894 = 119906120572119896119894 if 120601119894119896 (119909) ge 0119906120572119896119894 if 120601119894119896 (119909) lt 0

0 le 119896 le 119898 minus 1 0 le 119894 le 119899(14)

Minimization is as follows

119910(119896)119894 = 119906120572119896119894 if 120601119894119896 (119909) ge 0119906120572119896119894 if 120601119894119896 (119909) lt 0

0 le 119896 le 119898 minus 1 0 le 119894 le 119899(15)

Theorem 7 If 119904(119909) = sum119898minus1119896=0 sum119899119894=0 119906119896119894120601119894119896(119909) is an interpolatingpiecewise fuzzy Hermite polynomial then for all 120572 isin [0 1]119894 isin 0 1 119899 minus 1 119909 isin [119909119894 119909119894+1]

119897119890119899 119904120572 (119909) ge min 119897119890119899 119904120572 (119909119894) 119897119890119899 119904120572 (119909119894+1) (16)

where

119904120572 (119909) = [119904120572 (119909) 119904120572 (119909)] 119897119890119899 119904120572 (119909) = 119904120572 (119909) minus 119904120572 (119909) (17)

Proof By using Theorem 6 and (11) we have 119904120572(119909119894) = 1199061205720 119894119904120572(119909119894+1) = 1199061205720 119894+1 and len 119904120572(119909119894) = len 1199061205720119894 len 119904120572(119909119894+1) =

len 1199061205720 119894+1 Since the addition does not decrease thelength of an interval from (11) we can write 119904120572(119909) =sum119898minus1119896=0 sum119899119895=0 119906120572119896119895120601119895119896(119909) then

len 119904120572 (119909) ge 119898minus1sum119896=0

119899sum119895=0

10038161003816100381610038161003816120601119895119896 (119909)10038161003816100381610038161003816 len 119906120572119896119895ge 119899sum119895=0

100381610038161003816100381610038161206011198950 (119909)10038161003816100381610038161003816 len 1199060119895ge 10038161003816100381610038161206011198940 (119909)1003816100381610038161003816 len 1199060119894 + 1003816100381610038161003816120601119894+1 0 (119909)1003816100381610038161003816 len 1199060 119894+1ge min len 1199060 119894 len 1199060 119894+1 (10038161003816100381610038161206011198940 (119909)1003816100381610038161003816 + 1003816100381610038161003816120601119894+1 0 (119909)1003816100381610038161003816)ge min len 1199060 119894 len 1199060 119894+1= min len 119904120572 (119909119894) len 119904120572 (119909119894+1)

(18)

Theorem 8 Let 119906119896119894 = (119898119896119894 119897119896119894 119903119896119894) 0 le 119896 le 119898 minus 1 0 le119894 le 119899 be a triangular 119871 minus 119871 fuzzy number then also 119904(119909)the piecewise fuzzy Hermite polynomial interpolation is sucha fuzzy number for each 119909Proof The closed interval [119898 minus 119897119898 + 119903] is the support of 119906 =(119898 119897 119903) a triangular 119871minus119871 fuzzy number then for each 119909 and119906119896119894 we have119904 (119909) = (119898minus1sum

119896=0

119899sum119894=0

119906119896119894120601119894119896 (119909))

= [[sum sum120601119894119896ge0

(119898119896119894 minus 119897119896119894) 120601119894119896 (119909)+sum sum120601119894119896lt0

(119898119896119894 + 119903119896119894) 120601119894119896 (119909) sum sum120601119894119896ge0

(119898119896119894 + 119903119896119894) 120601119894119896 (119909)

+sum sum120601119894119896lt0

(119898119896119894 minus 119897119896119894) 120601119894119896 (119909)]] = [[119898minus1sum119896=0

119899sum119894=0

119898119896119894120601119894119896 (119909)

minus (sum sum120601119894119896ge0

119897119896119894120601119894119896 (119909) minussum sum120601119894119896lt0

119903119896119894120601119894119896 (119909))

+ 119898minus1sum119896=0

119899sum119894=0

119898119896119894120601119894119896 (119909)

+ (sum sum120601119894119896ge0

119903119896119894 (119909) 120601119894119896 (119909) minussum sum120601119894119896lt0

119897119894119896120601119894119896 (119909))]]= (119898 (119909) minus 119897 (119909) 119898 (119909) + 119903 (119909))

(19)

Advances in Fuzzy Systems 5

It follows that if 119904(119909) = (119898(119909) 119897(119909) 119903(119909)) is a triangular119871 minus 119871 fuzzy number for each 119909 then

119898(119909) = 119898minus1sum119896=0

119899sum119894=0

119898119896119894120601119896119894119897 (119909) = sum sum

120601119894119896ge0

119897119896119894120601119894119896 (119909) minussum sum120601119894119896lt0

119903119896119894120601119894119896 (119909) 119903 (119909) = sum sum

120601119894119896ge0

119903119896119894120601119894119896 minussum sum120601119894119896lt0

119897119894119896120601119894119896 (119909) (20)

4 Piecewise-Polynomial Linear Cubic andQuintic Fuzzy Hermite Interpolation

We consider 119898 = 1 and compute the piecewise fuzzy linearinterpolant as the initial case of the presented method basedon (12) and for a given set of fuzzy data (119909119894 119891119894) | 119891119894 isinRF 0 le 119894 le 119899 as follows

119904 (119909) = 119899sum119894=0

11990601198941206011198940 (119909) (21)

where 1199060119894 = 119891119894 0 le 119894 le 119899 and subject to conditions (6)

12060100 (119909) = 0 119909 ge 1199091( 1199091 minus 1199091199091 minus 1199090) 1199090 le 119909 le 1199091

1206011198940 (119909) =

0 119909 le 119909119894minus1 119909 ge 119909119894+1( 119909 minus 119909119894minus1119909119894 minus 119909119894minus1) 119909119894minus1 le 119909 le 119909119894( 119909119894+1 minus 119909119909119894+1 minus 119909119894) 119909119894 le 119909 le 119909119894+1

1 le 119894 le 119899 minus 11206011198990 (119909) =

0 119909 le 119909119899minus1( 119909 minus 119909119899minus1119909119899 minus 119909119899minus1) 119909119899minus1 le 119909 le 119909119899

(22)

The obtained 119904(119909) is the same as fuzzy spline of order 119897 = 2that had been introduced in [3] because the basic splines andthe cardinal basis functions in two interpolants are equal

Example 9 (see [4]) Suppose the data (1 (0 2 1) (1 0 3))(13 (5 1 2) (0 2 1)) (22 (1 0 3) (4 4 3)) (3 (4 4 3)(5 1 2)) (35 (0 3 2) (1 1 1)) (4 (1 1 1) and (0 3 2))In Figure 1 the dashed line is the 05-cut set of piecewisecubic fuzzy interpolation 119904(119909) 119909 isin [1 4] and the solid linesrepresent the support and the core of 119904(119909)

10 15 20 25 30 35 40

minus2

0

2

4

6

8

Figure 1 Graph of Example 9

10 15 20 25 30 35 40

minus2

0

2

4

6

Figure 2 Graph of Example 10

When 119898 = 2 we get the piecewise cubic fuzzy Her-mite polynomial interpolant in [11] for a given set of data(119909119894 119891119894 1198911015840119894 ) | 119891119894 1198911015840119894 isin RF 0 le 119894 le 119899

119904 (119909) = 119899sum119894=0

11990601198941206011198940 (119909) + 119899sum119894=0

11990611198941206011198941 (119909) (23)

where 119906119896119894 = 119891(119896)119894 119896 = 0 1 0 le 119894 le 119899An outstanding feature of this study is that by simply

applying the second case of the presented general methodand exactly the same data we have produced an alternativeto simple fuzzy Hermite polynomial interpolation in [4]Heretofore thementioned cubic case (23) was independentlyintroduced in [10] but only in very weak conditions andwithout using the extension principle

The cardinal basis functions 120601119894119896(119909) 119896 = 0 1 0 le 119894 le 119899were computed in [17]

Example 10 Suppose the data (1 (0 2 1) (1 0 3)) (15 (51 2) (0 2 1)) (27 (1 0 3) (4 4 3)) (3 (4 4 3) (5 1 2))(37 (0 3 2) (1 1 1)) and (4 (1 1 1) (0 3 2)) In Figure 2the dashed line is the 05-cut set of piecewise cubic fuzzyinterpolations 119904(119909) 119909 isin [1 4] and the solid lines representthe support and the core of 119904(119909)

Let 119898 = 3 from (6) we shall construct the cardi-nal basis for 1198675(Δ 119868) The quintic Hermite polynomials

6 Advances in Fuzzy Systems

1206011198940(119909) 1206011198941(119909) and 1206011198942(119909) are solving the interpolationproblem

119863119897120601119894119896 (119909119895) = 120575119896119897120575119894119895 0 le 119896 119897 le 2 0 le 119894 119895 le 119899 (24)

To this end we determine uniquely all the pervious 120601119894119895rsquos bythe (24)

For 1 le 119894 le 119899 minus 1 let

1206011198940 (119909) =

(119909119894minus1 minus 119909)3(119909119894minus1 minus 119909119894)5 [(119909119894minus1 + 3119909) (119909119894minus1 minus 5119909119894) + 61199092 + 101199092119894 ] 119909119894minus1 le 119909 le 119909119894

(119909119894+1 minus 119909)3(119909119894+1 minus 119909119894)5 [(119909119894+1 + 3119909) (119909119894+1 minus 5119909119894) + 61199092 + 101199092119894 ] 119909119894 le 119909 le 119909119894+1

0 otherwise

1206011198941 (119909) =

( 119909119894minus1 minus 119909119909119894minus1 minus 119909119894)3 (119909 minus 119909119894) [1 + 3 119909 minus 119909119894119909119894minus1 minus 119909119894 ] 119909119894minus1 le 119909 le 119909119894

( 119909119894+1 minus 119909119909119894 minus 119909119894+1)3 (119909119894 minus 119909) [1 + 3 119909119894 minus 119909119909119894 minus 119909119894+1 ] 119909119894 le 119909 le 119909119894+1

0 otherwise

1206011198942 (119909) =

( 119909119894minus1 minus 119909119909119894minus1 minus 119909119894)3 (119909119894 minus 119909)22 119909119894minus1 le 119909 le 119909119894

( 119909 minus 119909119894+1119909119894 minus 119909119894+1)3 (119909119894 minus 119909)22 119909119894 le 119909 le 119909119894+1

0 otherwise

(25)

The six next functions are similarly defined In particular

12060100 (119909) =

(1199091 minus 119909)3(1199091 minus 1199090)5 [(1199091 + 3119909) (1199091 minus 51199090) + 61199092 + 1011990930] 1199090 le 119909 le 1199091

0 1199091 le 119909 le 119909119899

1206011198990 (119909) =

(119909119899minus1 minus 119909)3(119909119899minus1 minus 119909119899)5 [(119909119899minus1 + 3119909) (119909119899minus1 minus 5119909119899) + 61199092 + 101199092119899] 119909119899minus1 le 119909 le 119909119899

0 1199090 le 119909 le 119909119899minus112060101 (119909) =

( 1199091 minus 1199091199090 minus 1199091)3 (1199090 minus 119909) [1 + 3 1199090 minus 1199091199090 minus 1199091 ] 1199090 le 119909 le 1199091

0 1199091 le 119909 le 1199091198991206011198991 (119909) =

( 119909119899minus1 minus 119909119909119899minus1 minus 119909119899)3 (119909 minus 119909119899) [1 + 3 119909 minus 119909119899119909119899minus1 minus 119909119899 ] 119909119899minus1 le 119909 le 119909119899

0 1199090 le 119909 le 119909119899minus112060102 (119909) =

( 119909 minus 11990911199090 minus 1199091)

3 (1199090 minus 119909)22 1199090 le 119909 le 11990910 1199091 le 119909 le 119909119899

Advances in Fuzzy Systems 7

0 1 2 3 4 5 6minus10

0

10

20

30

Figure 3 Graph of Example 11

1206011198992 (119909) =( 119909119899minus1 minus 119909119909119899minus1 minus 119909119899)

3 (119909119899 minus 119909)22 119909119899minus1 le 119909 le 1199091198990 1199090 le 119909 le 119909119899minus1

(26)

Thus we can immediately write down piecewise quintic fuzzyHermite interpolation polynomial 119904(119909) using

119904 (119909) = 119899sum119894=0

11990601198941206011198940 (119909) + 119899sum119894=0

11990611198941206011198941 (119909) + 119899sum119894=0

11990621198941206011198942 (119909) (27)

where (119909119894 119891119894 1198911015840119894 11989110158401015840119894 ) | 119891(119896)119894 isin BF 0 le 119896 le 2 0 le 119894 le 119899 isgiven and 119906119896119894 = 119891(119896)119894

Example 11 Suppose that (0 (0 1 3) (0 2 2) (1 4 4)) (13(005 19 35) (03 32 08) (1 31 3)) (2 (2 67 53) (2 0535) (1 26 24)) (4 (8 101 99) (4 4 0) (1 06 05)) (53(14 13 12) (53 02 38) (1 15 17)) (6 (18 132 148) (6 093) and (1 34 32)) are the interpolation data In Figure 3the solid lines denote the support and the core of piecewisequintic fuzzy Hermite interpolation 119904(119909) 119909 isin [0 6] and thedashed line is the 05-cut set of 119904(119909)5 Conclusions and Further Work

Based on the cardinal basis functions for119898(119899+ 1) dimension1198672119898minus1(Δ 119868) linear space interpolation of fuzzy data by thefuzzy-valued piecewise Hermite polynomial as the extensionof same approach in [11] has been successfully introduced ingeneral case and provided a succinct formula for calculatingthe new fuzzy interpolant Moreover two first cases ofthe presented method have been applied as an analogy tofuzzy spline of order two in [3] and an alternative to fuzzyosculatory interpolation in [4] respectively In the guise of aremarkable achievement the piecewise fuzzy cubic Hermitepolynomial interpolation that was constructed with poorconditions and without using the extension principle in [10]has been produced in the role of a very special subdivision forthe presented general method in this study Finally the thirdinitial case piecewise fuzzy quintic Hermite polynomial

has been described in detail The next step to improve thismethod is interpolation of fuzzy data including switchingpoints by a fuzzy differentiable piecewise interpolant

Competing Interests

The authors declare that they have no competing interests

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] R Lowen ldquoA fuzzy Lagrange interpolation theoremrdquo Fuzzy Setsand Systems vol 34 no 1 pp 33ndash38 1990

[3] O Kaleva ldquoInterpolation of fuzzy datardquo Fuzzy Sets and SystemsAn International Journal in Information Science andEngineeringvol 61 no 1 pp 63ndash70 1994

[4] H S Goghary and S Abbasbandy ldquoInterpolation of fuzzy databy Hermite polynomialrdquo International Journal of ComputerMathematics vol 82 no 5 pp 595ndash600 2005

[5] H Behforooz R Ezzati and S Abbasbandy ldquoInterpolation offuzzy data by using E(3) cubic splinesrdquo International Journal ofPure and Applied Mathematics vol 60 no 4 pp 383ndash392 2010

[6] S Abbasbandy R Ezzati and H Behforooz ldquoInterpolation offuzzy data by using fuzzy splinesrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 16 no1 pp 107ndash115 2008

[7] S Abbasbandy ldquoInterpolation of fuzzy data by completesplinesrdquoKorean Journal of Computational ampAppliedMathemat-ics vol 8 no 3 pp 587ndash594 2001

[8] S Abbasbandy and E Babolian ldquoInterpolation of fuzzy data bynatural splinesrdquoTheKorean Journal of Computational ampAppliedMathematics vol 5 no 2 pp 457ndash463 1998

8 Advances in Fuzzy Systems

[9] W A Lodwick and J Santos ldquoConstructing consistent fuzzysurfaces from fuzzy datardquo Fuzzy Sets and Systems An Interna-tional Journal in Information Science and Engineering vol 135no 2 pp 259ndash277 2003

[10] M Zeinali S Shahmorad and K Mirnia ldquoHermite and piece-wise cubic Hermite interpolation of fuzzy datardquo Journal ofIntelligent amp Fuzzy Systems vol 26 no 6 pp 2889ndash2898 2014

[11] H Vosoughi and S Abbasbandy ldquoInterpolation of fuzzy data bycubic and piecewise-polynomial cubic hermitesrdquo Indian Journalof Science and Technology vol 9 no 8 2016

[12] P J Davis Interpolation and Approximation Dover New YorkNY USA 1975

[13] B Wendroff Theoretical Numerical Analysis Academic PressNew York NY USA 1966

[14] P G Ciarlet M H Schultz and R S Varga ldquoNumerical meth-ods of high-order accuracy for nonlinear boundary valueproblemsrdquo Numerische Mathematik vol 9 pp 397ndash430 1967

[15] G Birkhoff M H Schultz and R S Varga ldquoPiecewise Hermiteinterpolation in one and two variables with applications topartial differential equationsrdquo Numerische Mathematik vol 11pp 232ndash256 1968

[16] R S Varga ldquoHermite interpolation-type Ritz methods for two-point boundary value problemsrdquo in Numerical Solution ofPartial Differential Equations J H Bramble Ed pp 365ndash373Academic Press New York NY USA 1966

[17] P M Prenter Splines and Variational Methods Wiley-Intersci-ence 1975

[18] H T Nguyen ldquoA note on the extension principle for fuzzy setsrdquoJournal of Mathematical Analysis and Applications vol 64 no2 pp 369ndash380 1978

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Advances in Fuzzy Systems 3

119898 and grid Δ of 119868 be the element 119904(119909) of degree 2119898 minus 1 oneach interval [119909119894 minus 1 119909119894] 1 le 119894 le 119899 such that

119863119896119904 (119909119894) = 119863119896119891 (119909119894)forall0 le 119896 le 119898 minus 1 0 le 119894 le 119899 119863119896 = 119889119896119889119909119896

(5)

Existence and uniqueness of full Hermite interpolation isprovided in [12] Because of this presentation (5) is actually aspecial case of such interpolation on a gridded interval and itfollows that each function belonging to 119862119898minus1(119868) has a uniqueinterpolate in1198672119898minus1(Δ 119868)

A particular cardinal basis for linear space1198672119898minus1(Δ 119868) ofdimension 119898(119899 + 1) is B = 120601119894119896(119909)119899119898minus1119894=0119896=0 (see eg [16])where the basis function 120601119894119896(119909) is defined by

119863119897120601119894119896 (119909119895) = 1205751198961198971205751198941198950 le 119896 119897 le 119898 minus 1 0 le 119894 119895 le 119899 119863119897 = 119889119897119889119909119897

(6)

Some important results based on (6) are simple to see in thesequel as 1206011198940(119909119894) = 1 and 1206011198940(119909119895) = 0 at all knots 119909119895 andsince 119904(119909) outside [119909119894minus1 119909119894] 1 le 119894 le 119899 satisfies zero datathen 1206011198940 equiv 0 for all 1199090 le 119909 le 119909119894minus1 and 119909119894+1 le 119909 le 1199091198991206011198941(119909) is of degree 2119898 minus 1 and 12060110158401198941(119909119894) = 1 but it is zeroat all other knots Moreover because outside the interval[119909119894minus1 119909119894+1]1206011198941(119909) interpolates zero data then 1206011198941(119909) must bevanished identically for all 119909 ge 119909119894+1 and 119909 le 119909119894minus1 (see eg[13 14 17]) Analogous reasoning applies to

1206011198941 (119909)

=

(119909 minus 119909119894minus1)119898 (119909 minus 119909119894)119898minus2sum119895=0

119886119895119909119895 119909119894minus1 le 119909 le 119909119894(119909 minus 119909119894+1)119898 (119909 minus 119909119894)119898minus2sum

119895=0

119887119895119909119895 119909119894 le 119909 le 119909119894+10 otherwise

(7)

In the following theorem we will use the recent features

Theorem 6 Assume that 120601119894119896 isin B and satisfies the piecewiseHermite polynomial cardinal basis function constraints (6)Then

(i) 120601119894 0(119909) + 120601119894+1 0(119909) ge 1 for all 119909 isin (119909119894 119909119894+1) 119894 =0 1 119899 minus 1(ii) For all 119894 = 1 2 119899 minus 1 1206011198941 changes the sign at 119909119894 The

sign of 1206011198941 is not positive on any subinterval [119909119894minus1 119909119894]and that is not negative on [119909119894 119909119894+1]

(iii) The sign of all other elements ofB is not negative on 119868Proof With the assumption of (6) let 120601119894 0(119909) + 120601119894+1 0(119909) bepolynomial of degree 2119898 minus 1 on the interval [119909119894minus1 119909119894+2] andinterpolate the data (119909119895 119891119895) where 119891119895 = 1 for 119895 = 119894 119894 + 1 andzero on the other knots of partition Δ Suppose that 0 lt 119894 lt119899 minus 1 and 120601119894 0(119909) + 120601119894+1 0(119909) lt 1 for some 119909 isin (119909119894 119909119894+1) By

themean value theorem its derivative has a zero on (119909119894 119909119894+1)The derivative has two (119898minus2)th order zeros at 119909119894 and 119909119894+1 andits two other zeros are 119909119894minus1 119909119894+2 Then it has at least 2119898 minus 1zeros on the interval [119909119894minus1 119909119894+2] which is a contradictionThecases 119894 = 0 and 119894 = 119899 minus 1 are treated similarly

In light of representation (7) and condition (6) thepolynomial 1206011198941(119909) is of degree 2119898 minus 1 It has only oneminimum point on [119909119894minus1 119909119894] and a single maximum onthe subinterval [119909119894 119909119894+1] Suppose that each of the abovepoints are one more Then by the mean value theorem firstderivative of 1206011198941(119909) has at least three zeros on (119909119894minus1 119909119894) andthree zeros on (119909119894 119909119894+1) Also the derivative has two (119898minus2)thorder zeros at 119909119894minus1 119909119894+1 Then it has at least 2119898 + 2 zeroswhich is a contradiction Hence 12060110158401198941(119909) has only one zeroon each of the intervals (119909119894minus1 119909119894) and (119909119894 119909119894+1) Now recall12060110158401198941(119909119894) = 1 it follows that 1206011198941(119909) le 0 on [119909119894minus1 119909119894] and1206011198941(119909) ge 0 on [119909119894 119909119894+1] This gives (ii)

A similar proof via definition of basis functions and (6)follows the claim (iii)

For a given 119891(119909) isin 119862119898minus1(119868) and its piecewise Hermiteinterpolate 119904(119909) isin 1198672119898minus1(Δ 119868) an equivalent explicit repre-sentation of 119904(119909) in (5) can be uniquely expressed (see eg[13 17]) namely

119904 (119909) = 119898minus1sum119896=0

119899sum119894=0

119891(119896) (119909119894) 120601119894119896 (119909) (8)

Now we want to construct a fuzzy-valued function as 119904 119868 rarr RF such that 119904(119896)(119909119894) = 119891(119896)(119909119894) = 119906119896119894 isin RF 0 le 119896 le119898minus 1 0 le 119894 le 119899 Also if for all 0 le 119896 le 119898minus 1 0 le 119894 le 119899 119906119896119894 =119910(119896)119894 are crisp numbers in R and 119891(119896)(119909119894) = 120594119910(119896)119894 (see eg[2]) then there is a polynomial of degree 2119898minus1 on successiveintervals [119909119894minus1 119909119894] 0 le 119894 le 119899 with 119904(119896)(119909119894) = 119910(119896)119894 0 le 119896 le119898 minus 1 0 le 119894 le 119899 such that 119904(119909) = 120594119891(119909) for all 119909 isin R where(119909119894 119891119894 1198911015840119894 119891(119898minus1)119894 ) | 119891(119896)119894 isin RF 0 le 119896 le 119898 minus 1 0 le 119894 le119899 is given

We suppose that such a fuzzy function exists and weattempt to find and compute it with respect to interpolationpolynomial presented by Lowen [2] Let for each119909 isin [1199090 119909119899]119904(119909) be a fuzzy piecewise Hermite polynomial and Λ =119910(119896)119894 119899119898minus1119894=0119896=0 then fromKaleva [3] andNguyen [18] we obtainthe 120572-cuts of 119904(119909) in a succinctly algorithm as follows

119904120572 (119909) = 119905 isin R | 120583(119905)119904(119909) ge 120572 = 119905 isin R | exist119910(119896)119894 120583(119910(119896)119894 )119906119896119894ge 120572 0 le 119896 le 119898 minus 1 0 le 119894 le 119899 119904Λ (119909) = 119905 = 119905isin R | 119905 = 119904Λ (119909) 119910(119896)119894 isin 119906120572119896119894 0 le 119896 le 119898 minus 1 0 le 119894le 119899 = 119898minus1sum

119896=0

119899sum119894=0

119906120572119896119894120601119894119896 (119909) (9)

where

119904Λ (119909) = 119898minus1sum119896=0

119899sum119894=0

119910(119896)119894 120601119894119896 (119909) (10)

4 Advances in Fuzzy Systems

is a piecewise Hermite polynomial in crisp case and bydefinition

119904120572 (119909) = 119898minus1sum119896=0

119899sum119894=0

119906120572119896119894120601119894119896 (119909) (11)

we obtain a formula that comprises a simple practical way forcalculating 119904(119909)

119904 (119909) = 119898minus1sum119896=0

119899sum119894=0

119906119896119894120601119894119896 (119909) (12)

Since for each 0 le 119896 le 119898 minus 1 0 le 119894 le 119899 119906120572119896119894 = [119906120572119896119894 119906120572119896119894]then we will have 119904120572(119909) by solving the following optimizationproblems

max amp min119898minus1sum119896=0

119899sum119894=0

119910(119896)119894 120601119894119896 (119909)subject to 119906120572119896119894 le 119910(119896)119894 le 119906120572119896119894

0 le 119896 le 119898 minus 1 0 le 119894 le 119899(13)

From the 120601119894119895rsquos sign that we represented in Theorem 6 theseproblems have the following optimal solutions

Maximization is as follows

119910(119896)119894 = 119906120572119896119894 if 120601119894119896 (119909) ge 0119906120572119896119894 if 120601119894119896 (119909) lt 0

0 le 119896 le 119898 minus 1 0 le 119894 le 119899(14)

Minimization is as follows

119910(119896)119894 = 119906120572119896119894 if 120601119894119896 (119909) ge 0119906120572119896119894 if 120601119894119896 (119909) lt 0

0 le 119896 le 119898 minus 1 0 le 119894 le 119899(15)

Theorem 7 If 119904(119909) = sum119898minus1119896=0 sum119899119894=0 119906119896119894120601119894119896(119909) is an interpolatingpiecewise fuzzy Hermite polynomial then for all 120572 isin [0 1]119894 isin 0 1 119899 minus 1 119909 isin [119909119894 119909119894+1]

119897119890119899 119904120572 (119909) ge min 119897119890119899 119904120572 (119909119894) 119897119890119899 119904120572 (119909119894+1) (16)

where

119904120572 (119909) = [119904120572 (119909) 119904120572 (119909)] 119897119890119899 119904120572 (119909) = 119904120572 (119909) minus 119904120572 (119909) (17)

Proof By using Theorem 6 and (11) we have 119904120572(119909119894) = 1199061205720 119894119904120572(119909119894+1) = 1199061205720 119894+1 and len 119904120572(119909119894) = len 1199061205720119894 len 119904120572(119909119894+1) =

len 1199061205720 119894+1 Since the addition does not decrease thelength of an interval from (11) we can write 119904120572(119909) =sum119898minus1119896=0 sum119899119895=0 119906120572119896119895120601119895119896(119909) then

len 119904120572 (119909) ge 119898minus1sum119896=0

119899sum119895=0

10038161003816100381610038161003816120601119895119896 (119909)10038161003816100381610038161003816 len 119906120572119896119895ge 119899sum119895=0

100381610038161003816100381610038161206011198950 (119909)10038161003816100381610038161003816 len 1199060119895ge 10038161003816100381610038161206011198940 (119909)1003816100381610038161003816 len 1199060119894 + 1003816100381610038161003816120601119894+1 0 (119909)1003816100381610038161003816 len 1199060 119894+1ge min len 1199060 119894 len 1199060 119894+1 (10038161003816100381610038161206011198940 (119909)1003816100381610038161003816 + 1003816100381610038161003816120601119894+1 0 (119909)1003816100381610038161003816)ge min len 1199060 119894 len 1199060 119894+1= min len 119904120572 (119909119894) len 119904120572 (119909119894+1)

(18)

Theorem 8 Let 119906119896119894 = (119898119896119894 119897119896119894 119903119896119894) 0 le 119896 le 119898 minus 1 0 le119894 le 119899 be a triangular 119871 minus 119871 fuzzy number then also 119904(119909)the piecewise fuzzy Hermite polynomial interpolation is sucha fuzzy number for each 119909Proof The closed interval [119898 minus 119897119898 + 119903] is the support of 119906 =(119898 119897 119903) a triangular 119871minus119871 fuzzy number then for each 119909 and119906119896119894 we have119904 (119909) = (119898minus1sum

119896=0

119899sum119894=0

119906119896119894120601119894119896 (119909))

= [[sum sum120601119894119896ge0

(119898119896119894 minus 119897119896119894) 120601119894119896 (119909)+sum sum120601119894119896lt0

(119898119896119894 + 119903119896119894) 120601119894119896 (119909) sum sum120601119894119896ge0

(119898119896119894 + 119903119896119894) 120601119894119896 (119909)

+sum sum120601119894119896lt0

(119898119896119894 minus 119897119896119894) 120601119894119896 (119909)]] = [[119898minus1sum119896=0

119899sum119894=0

119898119896119894120601119894119896 (119909)

minus (sum sum120601119894119896ge0

119897119896119894120601119894119896 (119909) minussum sum120601119894119896lt0

119903119896119894120601119894119896 (119909))

+ 119898minus1sum119896=0

119899sum119894=0

119898119896119894120601119894119896 (119909)

+ (sum sum120601119894119896ge0

119903119896119894 (119909) 120601119894119896 (119909) minussum sum120601119894119896lt0

119897119894119896120601119894119896 (119909))]]= (119898 (119909) minus 119897 (119909) 119898 (119909) + 119903 (119909))

(19)

Advances in Fuzzy Systems 5

It follows that if 119904(119909) = (119898(119909) 119897(119909) 119903(119909)) is a triangular119871 minus 119871 fuzzy number for each 119909 then

119898(119909) = 119898minus1sum119896=0

119899sum119894=0

119898119896119894120601119896119894119897 (119909) = sum sum

120601119894119896ge0

119897119896119894120601119894119896 (119909) minussum sum120601119894119896lt0

119903119896119894120601119894119896 (119909) 119903 (119909) = sum sum

120601119894119896ge0

119903119896119894120601119894119896 minussum sum120601119894119896lt0

119897119894119896120601119894119896 (119909) (20)

4 Piecewise-Polynomial Linear Cubic andQuintic Fuzzy Hermite Interpolation

We consider 119898 = 1 and compute the piecewise fuzzy linearinterpolant as the initial case of the presented method basedon (12) and for a given set of fuzzy data (119909119894 119891119894) | 119891119894 isinRF 0 le 119894 le 119899 as follows

119904 (119909) = 119899sum119894=0

11990601198941206011198940 (119909) (21)

where 1199060119894 = 119891119894 0 le 119894 le 119899 and subject to conditions (6)

12060100 (119909) = 0 119909 ge 1199091( 1199091 minus 1199091199091 minus 1199090) 1199090 le 119909 le 1199091

1206011198940 (119909) =

0 119909 le 119909119894minus1 119909 ge 119909119894+1( 119909 minus 119909119894minus1119909119894 minus 119909119894minus1) 119909119894minus1 le 119909 le 119909119894( 119909119894+1 minus 119909119909119894+1 minus 119909119894) 119909119894 le 119909 le 119909119894+1

1 le 119894 le 119899 minus 11206011198990 (119909) =

0 119909 le 119909119899minus1( 119909 minus 119909119899minus1119909119899 minus 119909119899minus1) 119909119899minus1 le 119909 le 119909119899

(22)

The obtained 119904(119909) is the same as fuzzy spline of order 119897 = 2that had been introduced in [3] because the basic splines andthe cardinal basis functions in two interpolants are equal

Example 9 (see [4]) Suppose the data (1 (0 2 1) (1 0 3))(13 (5 1 2) (0 2 1)) (22 (1 0 3) (4 4 3)) (3 (4 4 3)(5 1 2)) (35 (0 3 2) (1 1 1)) (4 (1 1 1) and (0 3 2))In Figure 1 the dashed line is the 05-cut set of piecewisecubic fuzzy interpolation 119904(119909) 119909 isin [1 4] and the solid linesrepresent the support and the core of 119904(119909)

10 15 20 25 30 35 40

minus2

0

2

4

6

8

Figure 1 Graph of Example 9

10 15 20 25 30 35 40

minus2

0

2

4

6

Figure 2 Graph of Example 10

When 119898 = 2 we get the piecewise cubic fuzzy Her-mite polynomial interpolant in [11] for a given set of data(119909119894 119891119894 1198911015840119894 ) | 119891119894 1198911015840119894 isin RF 0 le 119894 le 119899

119904 (119909) = 119899sum119894=0

11990601198941206011198940 (119909) + 119899sum119894=0

11990611198941206011198941 (119909) (23)

where 119906119896119894 = 119891(119896)119894 119896 = 0 1 0 le 119894 le 119899An outstanding feature of this study is that by simply

applying the second case of the presented general methodand exactly the same data we have produced an alternativeto simple fuzzy Hermite polynomial interpolation in [4]Heretofore thementioned cubic case (23) was independentlyintroduced in [10] but only in very weak conditions andwithout using the extension principle

The cardinal basis functions 120601119894119896(119909) 119896 = 0 1 0 le 119894 le 119899were computed in [17]

Example 10 Suppose the data (1 (0 2 1) (1 0 3)) (15 (51 2) (0 2 1)) (27 (1 0 3) (4 4 3)) (3 (4 4 3) (5 1 2))(37 (0 3 2) (1 1 1)) and (4 (1 1 1) (0 3 2)) In Figure 2the dashed line is the 05-cut set of piecewise cubic fuzzyinterpolations 119904(119909) 119909 isin [1 4] and the solid lines representthe support and the core of 119904(119909)

Let 119898 = 3 from (6) we shall construct the cardi-nal basis for 1198675(Δ 119868) The quintic Hermite polynomials

6 Advances in Fuzzy Systems

1206011198940(119909) 1206011198941(119909) and 1206011198942(119909) are solving the interpolationproblem

119863119897120601119894119896 (119909119895) = 120575119896119897120575119894119895 0 le 119896 119897 le 2 0 le 119894 119895 le 119899 (24)

To this end we determine uniquely all the pervious 120601119894119895rsquos bythe (24)

For 1 le 119894 le 119899 minus 1 let

1206011198940 (119909) =

(119909119894minus1 minus 119909)3(119909119894minus1 minus 119909119894)5 [(119909119894minus1 + 3119909) (119909119894minus1 minus 5119909119894) + 61199092 + 101199092119894 ] 119909119894minus1 le 119909 le 119909119894

(119909119894+1 minus 119909)3(119909119894+1 minus 119909119894)5 [(119909119894+1 + 3119909) (119909119894+1 minus 5119909119894) + 61199092 + 101199092119894 ] 119909119894 le 119909 le 119909119894+1

0 otherwise

1206011198941 (119909) =

( 119909119894minus1 minus 119909119909119894minus1 minus 119909119894)3 (119909 minus 119909119894) [1 + 3 119909 minus 119909119894119909119894minus1 minus 119909119894 ] 119909119894minus1 le 119909 le 119909119894

( 119909119894+1 minus 119909119909119894 minus 119909119894+1)3 (119909119894 minus 119909) [1 + 3 119909119894 minus 119909119909119894 minus 119909119894+1 ] 119909119894 le 119909 le 119909119894+1

0 otherwise

1206011198942 (119909) =

( 119909119894minus1 minus 119909119909119894minus1 minus 119909119894)3 (119909119894 minus 119909)22 119909119894minus1 le 119909 le 119909119894

( 119909 minus 119909119894+1119909119894 minus 119909119894+1)3 (119909119894 minus 119909)22 119909119894 le 119909 le 119909119894+1

0 otherwise

(25)

The six next functions are similarly defined In particular

12060100 (119909) =

(1199091 minus 119909)3(1199091 minus 1199090)5 [(1199091 + 3119909) (1199091 minus 51199090) + 61199092 + 1011990930] 1199090 le 119909 le 1199091

0 1199091 le 119909 le 119909119899

1206011198990 (119909) =

(119909119899minus1 minus 119909)3(119909119899minus1 minus 119909119899)5 [(119909119899minus1 + 3119909) (119909119899minus1 minus 5119909119899) + 61199092 + 101199092119899] 119909119899minus1 le 119909 le 119909119899

0 1199090 le 119909 le 119909119899minus112060101 (119909) =

( 1199091 minus 1199091199090 minus 1199091)3 (1199090 minus 119909) [1 + 3 1199090 minus 1199091199090 minus 1199091 ] 1199090 le 119909 le 1199091

0 1199091 le 119909 le 1199091198991206011198991 (119909) =

( 119909119899minus1 minus 119909119909119899minus1 minus 119909119899)3 (119909 minus 119909119899) [1 + 3 119909 minus 119909119899119909119899minus1 minus 119909119899 ] 119909119899minus1 le 119909 le 119909119899

0 1199090 le 119909 le 119909119899minus112060102 (119909) =

( 119909 minus 11990911199090 minus 1199091)

3 (1199090 minus 119909)22 1199090 le 119909 le 11990910 1199091 le 119909 le 119909119899

Advances in Fuzzy Systems 7

0 1 2 3 4 5 6minus10

0

10

20

30

Figure 3 Graph of Example 11

1206011198992 (119909) =( 119909119899minus1 minus 119909119909119899minus1 minus 119909119899)

3 (119909119899 minus 119909)22 119909119899minus1 le 119909 le 1199091198990 1199090 le 119909 le 119909119899minus1

(26)

Thus we can immediately write down piecewise quintic fuzzyHermite interpolation polynomial 119904(119909) using

119904 (119909) = 119899sum119894=0

11990601198941206011198940 (119909) + 119899sum119894=0

11990611198941206011198941 (119909) + 119899sum119894=0

11990621198941206011198942 (119909) (27)

where (119909119894 119891119894 1198911015840119894 11989110158401015840119894 ) | 119891(119896)119894 isin BF 0 le 119896 le 2 0 le 119894 le 119899 isgiven and 119906119896119894 = 119891(119896)119894

Example 11 Suppose that (0 (0 1 3) (0 2 2) (1 4 4)) (13(005 19 35) (03 32 08) (1 31 3)) (2 (2 67 53) (2 0535) (1 26 24)) (4 (8 101 99) (4 4 0) (1 06 05)) (53(14 13 12) (53 02 38) (1 15 17)) (6 (18 132 148) (6 093) and (1 34 32)) are the interpolation data In Figure 3the solid lines denote the support and the core of piecewisequintic fuzzy Hermite interpolation 119904(119909) 119909 isin [0 6] and thedashed line is the 05-cut set of 119904(119909)5 Conclusions and Further Work

Based on the cardinal basis functions for119898(119899+ 1) dimension1198672119898minus1(Δ 119868) linear space interpolation of fuzzy data by thefuzzy-valued piecewise Hermite polynomial as the extensionof same approach in [11] has been successfully introduced ingeneral case and provided a succinct formula for calculatingthe new fuzzy interpolant Moreover two first cases ofthe presented method have been applied as an analogy tofuzzy spline of order two in [3] and an alternative to fuzzyosculatory interpolation in [4] respectively In the guise of aremarkable achievement the piecewise fuzzy cubic Hermitepolynomial interpolation that was constructed with poorconditions and without using the extension principle in [10]has been produced in the role of a very special subdivision forthe presented general method in this study Finally the thirdinitial case piecewise fuzzy quintic Hermite polynomial

has been described in detail The next step to improve thismethod is interpolation of fuzzy data including switchingpoints by a fuzzy differentiable piecewise interpolant

Competing Interests

The authors declare that they have no competing interests

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] R Lowen ldquoA fuzzy Lagrange interpolation theoremrdquo Fuzzy Setsand Systems vol 34 no 1 pp 33ndash38 1990

[3] O Kaleva ldquoInterpolation of fuzzy datardquo Fuzzy Sets and SystemsAn International Journal in Information Science andEngineeringvol 61 no 1 pp 63ndash70 1994

[4] H S Goghary and S Abbasbandy ldquoInterpolation of fuzzy databy Hermite polynomialrdquo International Journal of ComputerMathematics vol 82 no 5 pp 595ndash600 2005

[5] H Behforooz R Ezzati and S Abbasbandy ldquoInterpolation offuzzy data by using E(3) cubic splinesrdquo International Journal ofPure and Applied Mathematics vol 60 no 4 pp 383ndash392 2010

[6] S Abbasbandy R Ezzati and H Behforooz ldquoInterpolation offuzzy data by using fuzzy splinesrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 16 no1 pp 107ndash115 2008

[7] S Abbasbandy ldquoInterpolation of fuzzy data by completesplinesrdquoKorean Journal of Computational ampAppliedMathemat-ics vol 8 no 3 pp 587ndash594 2001

[8] S Abbasbandy and E Babolian ldquoInterpolation of fuzzy data bynatural splinesrdquoTheKorean Journal of Computational ampAppliedMathematics vol 5 no 2 pp 457ndash463 1998

8 Advances in Fuzzy Systems

[9] W A Lodwick and J Santos ldquoConstructing consistent fuzzysurfaces from fuzzy datardquo Fuzzy Sets and Systems An Interna-tional Journal in Information Science and Engineering vol 135no 2 pp 259ndash277 2003

[10] M Zeinali S Shahmorad and K Mirnia ldquoHermite and piece-wise cubic Hermite interpolation of fuzzy datardquo Journal ofIntelligent amp Fuzzy Systems vol 26 no 6 pp 2889ndash2898 2014

[11] H Vosoughi and S Abbasbandy ldquoInterpolation of fuzzy data bycubic and piecewise-polynomial cubic hermitesrdquo Indian Journalof Science and Technology vol 9 no 8 2016

[12] P J Davis Interpolation and Approximation Dover New YorkNY USA 1975

[13] B Wendroff Theoretical Numerical Analysis Academic PressNew York NY USA 1966

[14] P G Ciarlet M H Schultz and R S Varga ldquoNumerical meth-ods of high-order accuracy for nonlinear boundary valueproblemsrdquo Numerische Mathematik vol 9 pp 397ndash430 1967

[15] G Birkhoff M H Schultz and R S Varga ldquoPiecewise Hermiteinterpolation in one and two variables with applications topartial differential equationsrdquo Numerische Mathematik vol 11pp 232ndash256 1968

[16] R S Varga ldquoHermite interpolation-type Ritz methods for two-point boundary value problemsrdquo in Numerical Solution ofPartial Differential Equations J H Bramble Ed pp 365ndash373Academic Press New York NY USA 1966

[17] P M Prenter Splines and Variational Methods Wiley-Intersci-ence 1975

[18] H T Nguyen ldquoA note on the extension principle for fuzzy setsrdquoJournal of Mathematical Analysis and Applications vol 64 no2 pp 369ndash380 1978

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4 Advances in Fuzzy Systems

is a piecewise Hermite polynomial in crisp case and bydefinition

119904120572 (119909) = 119898minus1sum119896=0

119899sum119894=0

119906120572119896119894120601119894119896 (119909) (11)

we obtain a formula that comprises a simple practical way forcalculating 119904(119909)

119904 (119909) = 119898minus1sum119896=0

119899sum119894=0

119906119896119894120601119894119896 (119909) (12)

Since for each 0 le 119896 le 119898 minus 1 0 le 119894 le 119899 119906120572119896119894 = [119906120572119896119894 119906120572119896119894]then we will have 119904120572(119909) by solving the following optimizationproblems

max amp min119898minus1sum119896=0

119899sum119894=0

119910(119896)119894 120601119894119896 (119909)subject to 119906120572119896119894 le 119910(119896)119894 le 119906120572119896119894

0 le 119896 le 119898 minus 1 0 le 119894 le 119899(13)

From the 120601119894119895rsquos sign that we represented in Theorem 6 theseproblems have the following optimal solutions

Maximization is as follows

119910(119896)119894 = 119906120572119896119894 if 120601119894119896 (119909) ge 0119906120572119896119894 if 120601119894119896 (119909) lt 0

0 le 119896 le 119898 minus 1 0 le 119894 le 119899(14)

Minimization is as follows

119910(119896)119894 = 119906120572119896119894 if 120601119894119896 (119909) ge 0119906120572119896119894 if 120601119894119896 (119909) lt 0

0 le 119896 le 119898 minus 1 0 le 119894 le 119899(15)

Theorem 7 If 119904(119909) = sum119898minus1119896=0 sum119899119894=0 119906119896119894120601119894119896(119909) is an interpolatingpiecewise fuzzy Hermite polynomial then for all 120572 isin [0 1]119894 isin 0 1 119899 minus 1 119909 isin [119909119894 119909119894+1]

119897119890119899 119904120572 (119909) ge min 119897119890119899 119904120572 (119909119894) 119897119890119899 119904120572 (119909119894+1) (16)

where

119904120572 (119909) = [119904120572 (119909) 119904120572 (119909)] 119897119890119899 119904120572 (119909) = 119904120572 (119909) minus 119904120572 (119909) (17)

Proof By using Theorem 6 and (11) we have 119904120572(119909119894) = 1199061205720 119894119904120572(119909119894+1) = 1199061205720 119894+1 and len 119904120572(119909119894) = len 1199061205720119894 len 119904120572(119909119894+1) =

len 1199061205720 119894+1 Since the addition does not decrease thelength of an interval from (11) we can write 119904120572(119909) =sum119898minus1119896=0 sum119899119895=0 119906120572119896119895120601119895119896(119909) then

len 119904120572 (119909) ge 119898minus1sum119896=0

119899sum119895=0

10038161003816100381610038161003816120601119895119896 (119909)10038161003816100381610038161003816 len 119906120572119896119895ge 119899sum119895=0

100381610038161003816100381610038161206011198950 (119909)10038161003816100381610038161003816 len 1199060119895ge 10038161003816100381610038161206011198940 (119909)1003816100381610038161003816 len 1199060119894 + 1003816100381610038161003816120601119894+1 0 (119909)1003816100381610038161003816 len 1199060 119894+1ge min len 1199060 119894 len 1199060 119894+1 (10038161003816100381610038161206011198940 (119909)1003816100381610038161003816 + 1003816100381610038161003816120601119894+1 0 (119909)1003816100381610038161003816)ge min len 1199060 119894 len 1199060 119894+1= min len 119904120572 (119909119894) len 119904120572 (119909119894+1)

(18)

Theorem 8 Let 119906119896119894 = (119898119896119894 119897119896119894 119903119896119894) 0 le 119896 le 119898 minus 1 0 le119894 le 119899 be a triangular 119871 minus 119871 fuzzy number then also 119904(119909)the piecewise fuzzy Hermite polynomial interpolation is sucha fuzzy number for each 119909Proof The closed interval [119898 minus 119897119898 + 119903] is the support of 119906 =(119898 119897 119903) a triangular 119871minus119871 fuzzy number then for each 119909 and119906119896119894 we have119904 (119909) = (119898minus1sum

119896=0

119899sum119894=0

119906119896119894120601119894119896 (119909))

= [[sum sum120601119894119896ge0

(119898119896119894 minus 119897119896119894) 120601119894119896 (119909)+sum sum120601119894119896lt0

(119898119896119894 + 119903119896119894) 120601119894119896 (119909) sum sum120601119894119896ge0

(119898119896119894 + 119903119896119894) 120601119894119896 (119909)

+sum sum120601119894119896lt0

(119898119896119894 minus 119897119896119894) 120601119894119896 (119909)]] = [[119898minus1sum119896=0

119899sum119894=0

119898119896119894120601119894119896 (119909)

minus (sum sum120601119894119896ge0

119897119896119894120601119894119896 (119909) minussum sum120601119894119896lt0

119903119896119894120601119894119896 (119909))

+ 119898minus1sum119896=0

119899sum119894=0

119898119896119894120601119894119896 (119909)

+ (sum sum120601119894119896ge0

119903119896119894 (119909) 120601119894119896 (119909) minussum sum120601119894119896lt0

119897119894119896120601119894119896 (119909))]]= (119898 (119909) minus 119897 (119909) 119898 (119909) + 119903 (119909))

(19)

Advances in Fuzzy Systems 5

It follows that if 119904(119909) = (119898(119909) 119897(119909) 119903(119909)) is a triangular119871 minus 119871 fuzzy number for each 119909 then

119898(119909) = 119898minus1sum119896=0

119899sum119894=0

119898119896119894120601119896119894119897 (119909) = sum sum

120601119894119896ge0

119897119896119894120601119894119896 (119909) minussum sum120601119894119896lt0

119903119896119894120601119894119896 (119909) 119903 (119909) = sum sum

120601119894119896ge0

119903119896119894120601119894119896 minussum sum120601119894119896lt0

119897119894119896120601119894119896 (119909) (20)

4 Piecewise-Polynomial Linear Cubic andQuintic Fuzzy Hermite Interpolation

We consider 119898 = 1 and compute the piecewise fuzzy linearinterpolant as the initial case of the presented method basedon (12) and for a given set of fuzzy data (119909119894 119891119894) | 119891119894 isinRF 0 le 119894 le 119899 as follows

119904 (119909) = 119899sum119894=0

11990601198941206011198940 (119909) (21)

where 1199060119894 = 119891119894 0 le 119894 le 119899 and subject to conditions (6)

12060100 (119909) = 0 119909 ge 1199091( 1199091 minus 1199091199091 minus 1199090) 1199090 le 119909 le 1199091

1206011198940 (119909) =

0 119909 le 119909119894minus1 119909 ge 119909119894+1( 119909 minus 119909119894minus1119909119894 minus 119909119894minus1) 119909119894minus1 le 119909 le 119909119894( 119909119894+1 minus 119909119909119894+1 minus 119909119894) 119909119894 le 119909 le 119909119894+1

1 le 119894 le 119899 minus 11206011198990 (119909) =

0 119909 le 119909119899minus1( 119909 minus 119909119899minus1119909119899 minus 119909119899minus1) 119909119899minus1 le 119909 le 119909119899

(22)

The obtained 119904(119909) is the same as fuzzy spline of order 119897 = 2that had been introduced in [3] because the basic splines andthe cardinal basis functions in two interpolants are equal

Example 9 (see [4]) Suppose the data (1 (0 2 1) (1 0 3))(13 (5 1 2) (0 2 1)) (22 (1 0 3) (4 4 3)) (3 (4 4 3)(5 1 2)) (35 (0 3 2) (1 1 1)) (4 (1 1 1) and (0 3 2))In Figure 1 the dashed line is the 05-cut set of piecewisecubic fuzzy interpolation 119904(119909) 119909 isin [1 4] and the solid linesrepresent the support and the core of 119904(119909)

10 15 20 25 30 35 40

minus2

0

2

4

6

8

Figure 1 Graph of Example 9

10 15 20 25 30 35 40

minus2

0

2

4

6

Figure 2 Graph of Example 10

When 119898 = 2 we get the piecewise cubic fuzzy Her-mite polynomial interpolant in [11] for a given set of data(119909119894 119891119894 1198911015840119894 ) | 119891119894 1198911015840119894 isin RF 0 le 119894 le 119899

119904 (119909) = 119899sum119894=0

11990601198941206011198940 (119909) + 119899sum119894=0

11990611198941206011198941 (119909) (23)

where 119906119896119894 = 119891(119896)119894 119896 = 0 1 0 le 119894 le 119899An outstanding feature of this study is that by simply

applying the second case of the presented general methodand exactly the same data we have produced an alternativeto simple fuzzy Hermite polynomial interpolation in [4]Heretofore thementioned cubic case (23) was independentlyintroduced in [10] but only in very weak conditions andwithout using the extension principle

The cardinal basis functions 120601119894119896(119909) 119896 = 0 1 0 le 119894 le 119899were computed in [17]

Example 10 Suppose the data (1 (0 2 1) (1 0 3)) (15 (51 2) (0 2 1)) (27 (1 0 3) (4 4 3)) (3 (4 4 3) (5 1 2))(37 (0 3 2) (1 1 1)) and (4 (1 1 1) (0 3 2)) In Figure 2the dashed line is the 05-cut set of piecewise cubic fuzzyinterpolations 119904(119909) 119909 isin [1 4] and the solid lines representthe support and the core of 119904(119909)

Let 119898 = 3 from (6) we shall construct the cardi-nal basis for 1198675(Δ 119868) The quintic Hermite polynomials

6 Advances in Fuzzy Systems

1206011198940(119909) 1206011198941(119909) and 1206011198942(119909) are solving the interpolationproblem

119863119897120601119894119896 (119909119895) = 120575119896119897120575119894119895 0 le 119896 119897 le 2 0 le 119894 119895 le 119899 (24)

To this end we determine uniquely all the pervious 120601119894119895rsquos bythe (24)

For 1 le 119894 le 119899 minus 1 let

1206011198940 (119909) =

(119909119894minus1 minus 119909)3(119909119894minus1 minus 119909119894)5 [(119909119894minus1 + 3119909) (119909119894minus1 minus 5119909119894) + 61199092 + 101199092119894 ] 119909119894minus1 le 119909 le 119909119894

(119909119894+1 minus 119909)3(119909119894+1 minus 119909119894)5 [(119909119894+1 + 3119909) (119909119894+1 minus 5119909119894) + 61199092 + 101199092119894 ] 119909119894 le 119909 le 119909119894+1

0 otherwise

1206011198941 (119909) =

( 119909119894minus1 minus 119909119909119894minus1 minus 119909119894)3 (119909 minus 119909119894) [1 + 3 119909 minus 119909119894119909119894minus1 minus 119909119894 ] 119909119894minus1 le 119909 le 119909119894

( 119909119894+1 minus 119909119909119894 minus 119909119894+1)3 (119909119894 minus 119909) [1 + 3 119909119894 minus 119909119909119894 minus 119909119894+1 ] 119909119894 le 119909 le 119909119894+1

0 otherwise

1206011198942 (119909) =

( 119909119894minus1 minus 119909119909119894minus1 minus 119909119894)3 (119909119894 minus 119909)22 119909119894minus1 le 119909 le 119909119894

( 119909 minus 119909119894+1119909119894 minus 119909119894+1)3 (119909119894 minus 119909)22 119909119894 le 119909 le 119909119894+1

0 otherwise

(25)

The six next functions are similarly defined In particular

12060100 (119909) =

(1199091 minus 119909)3(1199091 minus 1199090)5 [(1199091 + 3119909) (1199091 minus 51199090) + 61199092 + 1011990930] 1199090 le 119909 le 1199091

0 1199091 le 119909 le 119909119899

1206011198990 (119909) =

(119909119899minus1 minus 119909)3(119909119899minus1 minus 119909119899)5 [(119909119899minus1 + 3119909) (119909119899minus1 minus 5119909119899) + 61199092 + 101199092119899] 119909119899minus1 le 119909 le 119909119899

0 1199090 le 119909 le 119909119899minus112060101 (119909) =

( 1199091 minus 1199091199090 minus 1199091)3 (1199090 minus 119909) [1 + 3 1199090 minus 1199091199090 minus 1199091 ] 1199090 le 119909 le 1199091

0 1199091 le 119909 le 1199091198991206011198991 (119909) =

( 119909119899minus1 minus 119909119909119899minus1 minus 119909119899)3 (119909 minus 119909119899) [1 + 3 119909 minus 119909119899119909119899minus1 minus 119909119899 ] 119909119899minus1 le 119909 le 119909119899

0 1199090 le 119909 le 119909119899minus112060102 (119909) =

( 119909 minus 11990911199090 minus 1199091)

3 (1199090 minus 119909)22 1199090 le 119909 le 11990910 1199091 le 119909 le 119909119899

Advances in Fuzzy Systems 7

0 1 2 3 4 5 6minus10

0

10

20

30

Figure 3 Graph of Example 11

1206011198992 (119909) =( 119909119899minus1 minus 119909119909119899minus1 minus 119909119899)

3 (119909119899 minus 119909)22 119909119899minus1 le 119909 le 1199091198990 1199090 le 119909 le 119909119899minus1

(26)

Thus we can immediately write down piecewise quintic fuzzyHermite interpolation polynomial 119904(119909) using

119904 (119909) = 119899sum119894=0

11990601198941206011198940 (119909) + 119899sum119894=0

11990611198941206011198941 (119909) + 119899sum119894=0

11990621198941206011198942 (119909) (27)

where (119909119894 119891119894 1198911015840119894 11989110158401015840119894 ) | 119891(119896)119894 isin BF 0 le 119896 le 2 0 le 119894 le 119899 isgiven and 119906119896119894 = 119891(119896)119894

Example 11 Suppose that (0 (0 1 3) (0 2 2) (1 4 4)) (13(005 19 35) (03 32 08) (1 31 3)) (2 (2 67 53) (2 0535) (1 26 24)) (4 (8 101 99) (4 4 0) (1 06 05)) (53(14 13 12) (53 02 38) (1 15 17)) (6 (18 132 148) (6 093) and (1 34 32)) are the interpolation data In Figure 3the solid lines denote the support and the core of piecewisequintic fuzzy Hermite interpolation 119904(119909) 119909 isin [0 6] and thedashed line is the 05-cut set of 119904(119909)5 Conclusions and Further Work

Based on the cardinal basis functions for119898(119899+ 1) dimension1198672119898minus1(Δ 119868) linear space interpolation of fuzzy data by thefuzzy-valued piecewise Hermite polynomial as the extensionof same approach in [11] has been successfully introduced ingeneral case and provided a succinct formula for calculatingthe new fuzzy interpolant Moreover two first cases ofthe presented method have been applied as an analogy tofuzzy spline of order two in [3] and an alternative to fuzzyosculatory interpolation in [4] respectively In the guise of aremarkable achievement the piecewise fuzzy cubic Hermitepolynomial interpolation that was constructed with poorconditions and without using the extension principle in [10]has been produced in the role of a very special subdivision forthe presented general method in this study Finally the thirdinitial case piecewise fuzzy quintic Hermite polynomial

has been described in detail The next step to improve thismethod is interpolation of fuzzy data including switchingpoints by a fuzzy differentiable piecewise interpolant

Competing Interests

The authors declare that they have no competing interests

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] R Lowen ldquoA fuzzy Lagrange interpolation theoremrdquo Fuzzy Setsand Systems vol 34 no 1 pp 33ndash38 1990

[3] O Kaleva ldquoInterpolation of fuzzy datardquo Fuzzy Sets and SystemsAn International Journal in Information Science andEngineeringvol 61 no 1 pp 63ndash70 1994

[4] H S Goghary and S Abbasbandy ldquoInterpolation of fuzzy databy Hermite polynomialrdquo International Journal of ComputerMathematics vol 82 no 5 pp 595ndash600 2005

[5] H Behforooz R Ezzati and S Abbasbandy ldquoInterpolation offuzzy data by using E(3) cubic splinesrdquo International Journal ofPure and Applied Mathematics vol 60 no 4 pp 383ndash392 2010

[6] S Abbasbandy R Ezzati and H Behforooz ldquoInterpolation offuzzy data by using fuzzy splinesrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 16 no1 pp 107ndash115 2008

[7] S Abbasbandy ldquoInterpolation of fuzzy data by completesplinesrdquoKorean Journal of Computational ampAppliedMathemat-ics vol 8 no 3 pp 587ndash594 2001

[8] S Abbasbandy and E Babolian ldquoInterpolation of fuzzy data bynatural splinesrdquoTheKorean Journal of Computational ampAppliedMathematics vol 5 no 2 pp 457ndash463 1998

8 Advances in Fuzzy Systems

[9] W A Lodwick and J Santos ldquoConstructing consistent fuzzysurfaces from fuzzy datardquo Fuzzy Sets and Systems An Interna-tional Journal in Information Science and Engineering vol 135no 2 pp 259ndash277 2003

[10] M Zeinali S Shahmorad and K Mirnia ldquoHermite and piece-wise cubic Hermite interpolation of fuzzy datardquo Journal ofIntelligent amp Fuzzy Systems vol 26 no 6 pp 2889ndash2898 2014

[11] H Vosoughi and S Abbasbandy ldquoInterpolation of fuzzy data bycubic and piecewise-polynomial cubic hermitesrdquo Indian Journalof Science and Technology vol 9 no 8 2016

[12] P J Davis Interpolation and Approximation Dover New YorkNY USA 1975

[13] B Wendroff Theoretical Numerical Analysis Academic PressNew York NY USA 1966

[14] P G Ciarlet M H Schultz and R S Varga ldquoNumerical meth-ods of high-order accuracy for nonlinear boundary valueproblemsrdquo Numerische Mathematik vol 9 pp 397ndash430 1967

[15] G Birkhoff M H Schultz and R S Varga ldquoPiecewise Hermiteinterpolation in one and two variables with applications topartial differential equationsrdquo Numerische Mathematik vol 11pp 232ndash256 1968

[16] R S Varga ldquoHermite interpolation-type Ritz methods for two-point boundary value problemsrdquo in Numerical Solution ofPartial Differential Equations J H Bramble Ed pp 365ndash373Academic Press New York NY USA 1966

[17] P M Prenter Splines and Variational Methods Wiley-Intersci-ence 1975

[18] H T Nguyen ldquoA note on the extension principle for fuzzy setsrdquoJournal of Mathematical Analysis and Applications vol 64 no2 pp 369ndash380 1978

Submit your manuscripts athttpwwwhindawicom

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Human-ComputerInteraction

Advances in

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Advances in Fuzzy Systems 5

It follows that if 119904(119909) = (119898(119909) 119897(119909) 119903(119909)) is a triangular119871 minus 119871 fuzzy number for each 119909 then

119898(119909) = 119898minus1sum119896=0

119899sum119894=0

119898119896119894120601119896119894119897 (119909) = sum sum

120601119894119896ge0

119897119896119894120601119894119896 (119909) minussum sum120601119894119896lt0

119903119896119894120601119894119896 (119909) 119903 (119909) = sum sum

120601119894119896ge0

119903119896119894120601119894119896 minussum sum120601119894119896lt0

119897119894119896120601119894119896 (119909) (20)

4 Piecewise-Polynomial Linear Cubic andQuintic Fuzzy Hermite Interpolation

We consider 119898 = 1 and compute the piecewise fuzzy linearinterpolant as the initial case of the presented method basedon (12) and for a given set of fuzzy data (119909119894 119891119894) | 119891119894 isinRF 0 le 119894 le 119899 as follows

119904 (119909) = 119899sum119894=0

11990601198941206011198940 (119909) (21)

where 1199060119894 = 119891119894 0 le 119894 le 119899 and subject to conditions (6)

12060100 (119909) = 0 119909 ge 1199091( 1199091 minus 1199091199091 minus 1199090) 1199090 le 119909 le 1199091

1206011198940 (119909) =

0 119909 le 119909119894minus1 119909 ge 119909119894+1( 119909 minus 119909119894minus1119909119894 minus 119909119894minus1) 119909119894minus1 le 119909 le 119909119894( 119909119894+1 minus 119909119909119894+1 minus 119909119894) 119909119894 le 119909 le 119909119894+1

1 le 119894 le 119899 minus 11206011198990 (119909) =

0 119909 le 119909119899minus1( 119909 minus 119909119899minus1119909119899 minus 119909119899minus1) 119909119899minus1 le 119909 le 119909119899

(22)

The obtained 119904(119909) is the same as fuzzy spline of order 119897 = 2that had been introduced in [3] because the basic splines andthe cardinal basis functions in two interpolants are equal

Example 9 (see [4]) Suppose the data (1 (0 2 1) (1 0 3))(13 (5 1 2) (0 2 1)) (22 (1 0 3) (4 4 3)) (3 (4 4 3)(5 1 2)) (35 (0 3 2) (1 1 1)) (4 (1 1 1) and (0 3 2))In Figure 1 the dashed line is the 05-cut set of piecewisecubic fuzzy interpolation 119904(119909) 119909 isin [1 4] and the solid linesrepresent the support and the core of 119904(119909)

10 15 20 25 30 35 40

minus2

0

2

4

6

8

Figure 1 Graph of Example 9

10 15 20 25 30 35 40

minus2

0

2

4

6

Figure 2 Graph of Example 10

When 119898 = 2 we get the piecewise cubic fuzzy Her-mite polynomial interpolant in [11] for a given set of data(119909119894 119891119894 1198911015840119894 ) | 119891119894 1198911015840119894 isin RF 0 le 119894 le 119899

119904 (119909) = 119899sum119894=0

11990601198941206011198940 (119909) + 119899sum119894=0

11990611198941206011198941 (119909) (23)

where 119906119896119894 = 119891(119896)119894 119896 = 0 1 0 le 119894 le 119899An outstanding feature of this study is that by simply

applying the second case of the presented general methodand exactly the same data we have produced an alternativeto simple fuzzy Hermite polynomial interpolation in [4]Heretofore thementioned cubic case (23) was independentlyintroduced in [10] but only in very weak conditions andwithout using the extension principle

The cardinal basis functions 120601119894119896(119909) 119896 = 0 1 0 le 119894 le 119899were computed in [17]

Example 10 Suppose the data (1 (0 2 1) (1 0 3)) (15 (51 2) (0 2 1)) (27 (1 0 3) (4 4 3)) (3 (4 4 3) (5 1 2))(37 (0 3 2) (1 1 1)) and (4 (1 1 1) (0 3 2)) In Figure 2the dashed line is the 05-cut set of piecewise cubic fuzzyinterpolations 119904(119909) 119909 isin [1 4] and the solid lines representthe support and the core of 119904(119909)

Let 119898 = 3 from (6) we shall construct the cardi-nal basis for 1198675(Δ 119868) The quintic Hermite polynomials

6 Advances in Fuzzy Systems

1206011198940(119909) 1206011198941(119909) and 1206011198942(119909) are solving the interpolationproblem

119863119897120601119894119896 (119909119895) = 120575119896119897120575119894119895 0 le 119896 119897 le 2 0 le 119894 119895 le 119899 (24)

To this end we determine uniquely all the pervious 120601119894119895rsquos bythe (24)

For 1 le 119894 le 119899 minus 1 let

1206011198940 (119909) =

(119909119894minus1 minus 119909)3(119909119894minus1 minus 119909119894)5 [(119909119894minus1 + 3119909) (119909119894minus1 minus 5119909119894) + 61199092 + 101199092119894 ] 119909119894minus1 le 119909 le 119909119894

(119909119894+1 minus 119909)3(119909119894+1 minus 119909119894)5 [(119909119894+1 + 3119909) (119909119894+1 minus 5119909119894) + 61199092 + 101199092119894 ] 119909119894 le 119909 le 119909119894+1

0 otherwise

1206011198941 (119909) =

( 119909119894minus1 minus 119909119909119894minus1 minus 119909119894)3 (119909 minus 119909119894) [1 + 3 119909 minus 119909119894119909119894minus1 minus 119909119894 ] 119909119894minus1 le 119909 le 119909119894

( 119909119894+1 minus 119909119909119894 minus 119909119894+1)3 (119909119894 minus 119909) [1 + 3 119909119894 minus 119909119909119894 minus 119909119894+1 ] 119909119894 le 119909 le 119909119894+1

0 otherwise

1206011198942 (119909) =

( 119909119894minus1 minus 119909119909119894minus1 minus 119909119894)3 (119909119894 minus 119909)22 119909119894minus1 le 119909 le 119909119894

( 119909 minus 119909119894+1119909119894 minus 119909119894+1)3 (119909119894 minus 119909)22 119909119894 le 119909 le 119909119894+1

0 otherwise

(25)

The six next functions are similarly defined In particular

12060100 (119909) =

(1199091 minus 119909)3(1199091 minus 1199090)5 [(1199091 + 3119909) (1199091 minus 51199090) + 61199092 + 1011990930] 1199090 le 119909 le 1199091

0 1199091 le 119909 le 119909119899

1206011198990 (119909) =

(119909119899minus1 minus 119909)3(119909119899minus1 minus 119909119899)5 [(119909119899minus1 + 3119909) (119909119899minus1 minus 5119909119899) + 61199092 + 101199092119899] 119909119899minus1 le 119909 le 119909119899

0 1199090 le 119909 le 119909119899minus112060101 (119909) =

( 1199091 minus 1199091199090 minus 1199091)3 (1199090 minus 119909) [1 + 3 1199090 minus 1199091199090 minus 1199091 ] 1199090 le 119909 le 1199091

0 1199091 le 119909 le 1199091198991206011198991 (119909) =

( 119909119899minus1 minus 119909119909119899minus1 minus 119909119899)3 (119909 minus 119909119899) [1 + 3 119909 minus 119909119899119909119899minus1 minus 119909119899 ] 119909119899minus1 le 119909 le 119909119899

0 1199090 le 119909 le 119909119899minus112060102 (119909) =

( 119909 minus 11990911199090 minus 1199091)

3 (1199090 minus 119909)22 1199090 le 119909 le 11990910 1199091 le 119909 le 119909119899

Advances in Fuzzy Systems 7

0 1 2 3 4 5 6minus10

0

10

20

30

Figure 3 Graph of Example 11

1206011198992 (119909) =( 119909119899minus1 minus 119909119909119899minus1 minus 119909119899)

3 (119909119899 minus 119909)22 119909119899minus1 le 119909 le 1199091198990 1199090 le 119909 le 119909119899minus1

(26)

Thus we can immediately write down piecewise quintic fuzzyHermite interpolation polynomial 119904(119909) using

119904 (119909) = 119899sum119894=0

11990601198941206011198940 (119909) + 119899sum119894=0

11990611198941206011198941 (119909) + 119899sum119894=0

11990621198941206011198942 (119909) (27)

where (119909119894 119891119894 1198911015840119894 11989110158401015840119894 ) | 119891(119896)119894 isin BF 0 le 119896 le 2 0 le 119894 le 119899 isgiven and 119906119896119894 = 119891(119896)119894

Example 11 Suppose that (0 (0 1 3) (0 2 2) (1 4 4)) (13(005 19 35) (03 32 08) (1 31 3)) (2 (2 67 53) (2 0535) (1 26 24)) (4 (8 101 99) (4 4 0) (1 06 05)) (53(14 13 12) (53 02 38) (1 15 17)) (6 (18 132 148) (6 093) and (1 34 32)) are the interpolation data In Figure 3the solid lines denote the support and the core of piecewisequintic fuzzy Hermite interpolation 119904(119909) 119909 isin [0 6] and thedashed line is the 05-cut set of 119904(119909)5 Conclusions and Further Work

Based on the cardinal basis functions for119898(119899+ 1) dimension1198672119898minus1(Δ 119868) linear space interpolation of fuzzy data by thefuzzy-valued piecewise Hermite polynomial as the extensionof same approach in [11] has been successfully introduced ingeneral case and provided a succinct formula for calculatingthe new fuzzy interpolant Moreover two first cases ofthe presented method have been applied as an analogy tofuzzy spline of order two in [3] and an alternative to fuzzyosculatory interpolation in [4] respectively In the guise of aremarkable achievement the piecewise fuzzy cubic Hermitepolynomial interpolation that was constructed with poorconditions and without using the extension principle in [10]has been produced in the role of a very special subdivision forthe presented general method in this study Finally the thirdinitial case piecewise fuzzy quintic Hermite polynomial

has been described in detail The next step to improve thismethod is interpolation of fuzzy data including switchingpoints by a fuzzy differentiable piecewise interpolant

Competing Interests

The authors declare that they have no competing interests

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] R Lowen ldquoA fuzzy Lagrange interpolation theoremrdquo Fuzzy Setsand Systems vol 34 no 1 pp 33ndash38 1990

[3] O Kaleva ldquoInterpolation of fuzzy datardquo Fuzzy Sets and SystemsAn International Journal in Information Science andEngineeringvol 61 no 1 pp 63ndash70 1994

[4] H S Goghary and S Abbasbandy ldquoInterpolation of fuzzy databy Hermite polynomialrdquo International Journal of ComputerMathematics vol 82 no 5 pp 595ndash600 2005

[5] H Behforooz R Ezzati and S Abbasbandy ldquoInterpolation offuzzy data by using E(3) cubic splinesrdquo International Journal ofPure and Applied Mathematics vol 60 no 4 pp 383ndash392 2010

[6] S Abbasbandy R Ezzati and H Behforooz ldquoInterpolation offuzzy data by using fuzzy splinesrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 16 no1 pp 107ndash115 2008

[7] S Abbasbandy ldquoInterpolation of fuzzy data by completesplinesrdquoKorean Journal of Computational ampAppliedMathemat-ics vol 8 no 3 pp 587ndash594 2001

[8] S Abbasbandy and E Babolian ldquoInterpolation of fuzzy data bynatural splinesrdquoTheKorean Journal of Computational ampAppliedMathematics vol 5 no 2 pp 457ndash463 1998

8 Advances in Fuzzy Systems

[9] W A Lodwick and J Santos ldquoConstructing consistent fuzzysurfaces from fuzzy datardquo Fuzzy Sets and Systems An Interna-tional Journal in Information Science and Engineering vol 135no 2 pp 259ndash277 2003

[10] M Zeinali S Shahmorad and K Mirnia ldquoHermite and piece-wise cubic Hermite interpolation of fuzzy datardquo Journal ofIntelligent amp Fuzzy Systems vol 26 no 6 pp 2889ndash2898 2014

[11] H Vosoughi and S Abbasbandy ldquoInterpolation of fuzzy data bycubic and piecewise-polynomial cubic hermitesrdquo Indian Journalof Science and Technology vol 9 no 8 2016

[12] P J Davis Interpolation and Approximation Dover New YorkNY USA 1975

[13] B Wendroff Theoretical Numerical Analysis Academic PressNew York NY USA 1966

[14] P G Ciarlet M H Schultz and R S Varga ldquoNumerical meth-ods of high-order accuracy for nonlinear boundary valueproblemsrdquo Numerische Mathematik vol 9 pp 397ndash430 1967

[15] G Birkhoff M H Schultz and R S Varga ldquoPiecewise Hermiteinterpolation in one and two variables with applications topartial differential equationsrdquo Numerische Mathematik vol 11pp 232ndash256 1968

[16] R S Varga ldquoHermite interpolation-type Ritz methods for two-point boundary value problemsrdquo in Numerical Solution ofPartial Differential Equations J H Bramble Ed pp 365ndash373Academic Press New York NY USA 1966

[17] P M Prenter Splines and Variational Methods Wiley-Intersci-ence 1975

[18] H T Nguyen ldquoA note on the extension principle for fuzzy setsrdquoJournal of Mathematical Analysis and Applications vol 64 no2 pp 369ndash380 1978

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 Advances in Fuzzy Systems

1206011198940(119909) 1206011198941(119909) and 1206011198942(119909) are solving the interpolationproblem

119863119897120601119894119896 (119909119895) = 120575119896119897120575119894119895 0 le 119896 119897 le 2 0 le 119894 119895 le 119899 (24)

To this end we determine uniquely all the pervious 120601119894119895rsquos bythe (24)

For 1 le 119894 le 119899 minus 1 let

1206011198940 (119909) =

(119909119894minus1 minus 119909)3(119909119894minus1 minus 119909119894)5 [(119909119894minus1 + 3119909) (119909119894minus1 minus 5119909119894) + 61199092 + 101199092119894 ] 119909119894minus1 le 119909 le 119909119894

(119909119894+1 minus 119909)3(119909119894+1 minus 119909119894)5 [(119909119894+1 + 3119909) (119909119894+1 minus 5119909119894) + 61199092 + 101199092119894 ] 119909119894 le 119909 le 119909119894+1

0 otherwise

1206011198941 (119909) =

( 119909119894minus1 minus 119909119909119894minus1 minus 119909119894)3 (119909 minus 119909119894) [1 + 3 119909 minus 119909119894119909119894minus1 minus 119909119894 ] 119909119894minus1 le 119909 le 119909119894

( 119909119894+1 minus 119909119909119894 minus 119909119894+1)3 (119909119894 minus 119909) [1 + 3 119909119894 minus 119909119909119894 minus 119909119894+1 ] 119909119894 le 119909 le 119909119894+1

0 otherwise

1206011198942 (119909) =

( 119909119894minus1 minus 119909119909119894minus1 minus 119909119894)3 (119909119894 minus 119909)22 119909119894minus1 le 119909 le 119909119894

( 119909 minus 119909119894+1119909119894 minus 119909119894+1)3 (119909119894 minus 119909)22 119909119894 le 119909 le 119909119894+1

0 otherwise

(25)

The six next functions are similarly defined In particular

12060100 (119909) =

(1199091 minus 119909)3(1199091 minus 1199090)5 [(1199091 + 3119909) (1199091 minus 51199090) + 61199092 + 1011990930] 1199090 le 119909 le 1199091

0 1199091 le 119909 le 119909119899

1206011198990 (119909) =

(119909119899minus1 minus 119909)3(119909119899minus1 minus 119909119899)5 [(119909119899minus1 + 3119909) (119909119899minus1 minus 5119909119899) + 61199092 + 101199092119899] 119909119899minus1 le 119909 le 119909119899

0 1199090 le 119909 le 119909119899minus112060101 (119909) =

( 1199091 minus 1199091199090 minus 1199091)3 (1199090 minus 119909) [1 + 3 1199090 minus 1199091199090 minus 1199091 ] 1199090 le 119909 le 1199091

0 1199091 le 119909 le 1199091198991206011198991 (119909) =

( 119909119899minus1 minus 119909119909119899minus1 minus 119909119899)3 (119909 minus 119909119899) [1 + 3 119909 minus 119909119899119909119899minus1 minus 119909119899 ] 119909119899minus1 le 119909 le 119909119899

0 1199090 le 119909 le 119909119899minus112060102 (119909) =

( 119909 minus 11990911199090 minus 1199091)

3 (1199090 minus 119909)22 1199090 le 119909 le 11990910 1199091 le 119909 le 119909119899

Advances in Fuzzy Systems 7

0 1 2 3 4 5 6minus10

0

10

20

30

Figure 3 Graph of Example 11

1206011198992 (119909) =( 119909119899minus1 minus 119909119909119899minus1 minus 119909119899)

3 (119909119899 minus 119909)22 119909119899minus1 le 119909 le 1199091198990 1199090 le 119909 le 119909119899minus1

(26)

Thus we can immediately write down piecewise quintic fuzzyHermite interpolation polynomial 119904(119909) using

119904 (119909) = 119899sum119894=0

11990601198941206011198940 (119909) + 119899sum119894=0

11990611198941206011198941 (119909) + 119899sum119894=0

11990621198941206011198942 (119909) (27)

where (119909119894 119891119894 1198911015840119894 11989110158401015840119894 ) | 119891(119896)119894 isin BF 0 le 119896 le 2 0 le 119894 le 119899 isgiven and 119906119896119894 = 119891(119896)119894

Example 11 Suppose that (0 (0 1 3) (0 2 2) (1 4 4)) (13(005 19 35) (03 32 08) (1 31 3)) (2 (2 67 53) (2 0535) (1 26 24)) (4 (8 101 99) (4 4 0) (1 06 05)) (53(14 13 12) (53 02 38) (1 15 17)) (6 (18 132 148) (6 093) and (1 34 32)) are the interpolation data In Figure 3the solid lines denote the support and the core of piecewisequintic fuzzy Hermite interpolation 119904(119909) 119909 isin [0 6] and thedashed line is the 05-cut set of 119904(119909)5 Conclusions and Further Work

Based on the cardinal basis functions for119898(119899+ 1) dimension1198672119898minus1(Δ 119868) linear space interpolation of fuzzy data by thefuzzy-valued piecewise Hermite polynomial as the extensionof same approach in [11] has been successfully introduced ingeneral case and provided a succinct formula for calculatingthe new fuzzy interpolant Moreover two first cases ofthe presented method have been applied as an analogy tofuzzy spline of order two in [3] and an alternative to fuzzyosculatory interpolation in [4] respectively In the guise of aremarkable achievement the piecewise fuzzy cubic Hermitepolynomial interpolation that was constructed with poorconditions and without using the extension principle in [10]has been produced in the role of a very special subdivision forthe presented general method in this study Finally the thirdinitial case piecewise fuzzy quintic Hermite polynomial

has been described in detail The next step to improve thismethod is interpolation of fuzzy data including switchingpoints by a fuzzy differentiable piecewise interpolant

Competing Interests

The authors declare that they have no competing interests

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] R Lowen ldquoA fuzzy Lagrange interpolation theoremrdquo Fuzzy Setsand Systems vol 34 no 1 pp 33ndash38 1990

[3] O Kaleva ldquoInterpolation of fuzzy datardquo Fuzzy Sets and SystemsAn International Journal in Information Science andEngineeringvol 61 no 1 pp 63ndash70 1994

[4] H S Goghary and S Abbasbandy ldquoInterpolation of fuzzy databy Hermite polynomialrdquo International Journal of ComputerMathematics vol 82 no 5 pp 595ndash600 2005

[5] H Behforooz R Ezzati and S Abbasbandy ldquoInterpolation offuzzy data by using E(3) cubic splinesrdquo International Journal ofPure and Applied Mathematics vol 60 no 4 pp 383ndash392 2010

[6] S Abbasbandy R Ezzati and H Behforooz ldquoInterpolation offuzzy data by using fuzzy splinesrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 16 no1 pp 107ndash115 2008

[7] S Abbasbandy ldquoInterpolation of fuzzy data by completesplinesrdquoKorean Journal of Computational ampAppliedMathemat-ics vol 8 no 3 pp 587ndash594 2001

[8] S Abbasbandy and E Babolian ldquoInterpolation of fuzzy data bynatural splinesrdquoTheKorean Journal of Computational ampAppliedMathematics vol 5 no 2 pp 457ndash463 1998

8 Advances in Fuzzy Systems

[9] W A Lodwick and J Santos ldquoConstructing consistent fuzzysurfaces from fuzzy datardquo Fuzzy Sets and Systems An Interna-tional Journal in Information Science and Engineering vol 135no 2 pp 259ndash277 2003

[10] M Zeinali S Shahmorad and K Mirnia ldquoHermite and piece-wise cubic Hermite interpolation of fuzzy datardquo Journal ofIntelligent amp Fuzzy Systems vol 26 no 6 pp 2889ndash2898 2014

[11] H Vosoughi and S Abbasbandy ldquoInterpolation of fuzzy data bycubic and piecewise-polynomial cubic hermitesrdquo Indian Journalof Science and Technology vol 9 no 8 2016

[12] P J Davis Interpolation and Approximation Dover New YorkNY USA 1975

[13] B Wendroff Theoretical Numerical Analysis Academic PressNew York NY USA 1966

[14] P G Ciarlet M H Schultz and R S Varga ldquoNumerical meth-ods of high-order accuracy for nonlinear boundary valueproblemsrdquo Numerische Mathematik vol 9 pp 397ndash430 1967

[15] G Birkhoff M H Schultz and R S Varga ldquoPiecewise Hermiteinterpolation in one and two variables with applications topartial differential equationsrdquo Numerische Mathematik vol 11pp 232ndash256 1968

[16] R S Varga ldquoHermite interpolation-type Ritz methods for two-point boundary value problemsrdquo in Numerical Solution ofPartial Differential Equations J H Bramble Ed pp 365ndash373Academic Press New York NY USA 1966

[17] P M Prenter Splines and Variational Methods Wiley-Intersci-ence 1975

[18] H T Nguyen ldquoA note on the extension principle for fuzzy setsrdquoJournal of Mathematical Analysis and Applications vol 64 no2 pp 369ndash380 1978

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Fuzzy Systems 7

0 1 2 3 4 5 6minus10

0

10

20

30

Figure 3 Graph of Example 11

1206011198992 (119909) =( 119909119899minus1 minus 119909119909119899minus1 minus 119909119899)

3 (119909119899 minus 119909)22 119909119899minus1 le 119909 le 1199091198990 1199090 le 119909 le 119909119899minus1

(26)

Thus we can immediately write down piecewise quintic fuzzyHermite interpolation polynomial 119904(119909) using

119904 (119909) = 119899sum119894=0

11990601198941206011198940 (119909) + 119899sum119894=0

11990611198941206011198941 (119909) + 119899sum119894=0

11990621198941206011198942 (119909) (27)

where (119909119894 119891119894 1198911015840119894 11989110158401015840119894 ) | 119891(119896)119894 isin BF 0 le 119896 le 2 0 le 119894 le 119899 isgiven and 119906119896119894 = 119891(119896)119894

Example 11 Suppose that (0 (0 1 3) (0 2 2) (1 4 4)) (13(005 19 35) (03 32 08) (1 31 3)) (2 (2 67 53) (2 0535) (1 26 24)) (4 (8 101 99) (4 4 0) (1 06 05)) (53(14 13 12) (53 02 38) (1 15 17)) (6 (18 132 148) (6 093) and (1 34 32)) are the interpolation data In Figure 3the solid lines denote the support and the core of piecewisequintic fuzzy Hermite interpolation 119904(119909) 119909 isin [0 6] and thedashed line is the 05-cut set of 119904(119909)5 Conclusions and Further Work

Based on the cardinal basis functions for119898(119899+ 1) dimension1198672119898minus1(Δ 119868) linear space interpolation of fuzzy data by thefuzzy-valued piecewise Hermite polynomial as the extensionof same approach in [11] has been successfully introduced ingeneral case and provided a succinct formula for calculatingthe new fuzzy interpolant Moreover two first cases ofthe presented method have been applied as an analogy tofuzzy spline of order two in [3] and an alternative to fuzzyosculatory interpolation in [4] respectively In the guise of aremarkable achievement the piecewise fuzzy cubic Hermitepolynomial interpolation that was constructed with poorconditions and without using the extension principle in [10]has been produced in the role of a very special subdivision forthe presented general method in this study Finally the thirdinitial case piecewise fuzzy quintic Hermite polynomial

has been described in detail The next step to improve thismethod is interpolation of fuzzy data including switchingpoints by a fuzzy differentiable piecewise interpolant

Competing Interests

The authors declare that they have no competing interests

References

[1] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[2] R Lowen ldquoA fuzzy Lagrange interpolation theoremrdquo Fuzzy Setsand Systems vol 34 no 1 pp 33ndash38 1990

[3] O Kaleva ldquoInterpolation of fuzzy datardquo Fuzzy Sets and SystemsAn International Journal in Information Science andEngineeringvol 61 no 1 pp 63ndash70 1994

[4] H S Goghary and S Abbasbandy ldquoInterpolation of fuzzy databy Hermite polynomialrdquo International Journal of ComputerMathematics vol 82 no 5 pp 595ndash600 2005

[5] H Behforooz R Ezzati and S Abbasbandy ldquoInterpolation offuzzy data by using E(3) cubic splinesrdquo International Journal ofPure and Applied Mathematics vol 60 no 4 pp 383ndash392 2010

[6] S Abbasbandy R Ezzati and H Behforooz ldquoInterpolation offuzzy data by using fuzzy splinesrdquo International Journal ofUncertainty Fuzziness andKnowledge-Based Systems vol 16 no1 pp 107ndash115 2008

[7] S Abbasbandy ldquoInterpolation of fuzzy data by completesplinesrdquoKorean Journal of Computational ampAppliedMathemat-ics vol 8 no 3 pp 587ndash594 2001

[8] S Abbasbandy and E Babolian ldquoInterpolation of fuzzy data bynatural splinesrdquoTheKorean Journal of Computational ampAppliedMathematics vol 5 no 2 pp 457ndash463 1998

8 Advances in Fuzzy Systems

[9] W A Lodwick and J Santos ldquoConstructing consistent fuzzysurfaces from fuzzy datardquo Fuzzy Sets and Systems An Interna-tional Journal in Information Science and Engineering vol 135no 2 pp 259ndash277 2003

[10] M Zeinali S Shahmorad and K Mirnia ldquoHermite and piece-wise cubic Hermite interpolation of fuzzy datardquo Journal ofIntelligent amp Fuzzy Systems vol 26 no 6 pp 2889ndash2898 2014

[11] H Vosoughi and S Abbasbandy ldquoInterpolation of fuzzy data bycubic and piecewise-polynomial cubic hermitesrdquo Indian Journalof Science and Technology vol 9 no 8 2016

[12] P J Davis Interpolation and Approximation Dover New YorkNY USA 1975

[13] B Wendroff Theoretical Numerical Analysis Academic PressNew York NY USA 1966

[14] P G Ciarlet M H Schultz and R S Varga ldquoNumerical meth-ods of high-order accuracy for nonlinear boundary valueproblemsrdquo Numerische Mathematik vol 9 pp 397ndash430 1967

[15] G Birkhoff M H Schultz and R S Varga ldquoPiecewise Hermiteinterpolation in one and two variables with applications topartial differential equationsrdquo Numerische Mathematik vol 11pp 232ndash256 1968

[16] R S Varga ldquoHermite interpolation-type Ritz methods for two-point boundary value problemsrdquo in Numerical Solution ofPartial Differential Equations J H Bramble Ed pp 365ndash373Academic Press New York NY USA 1966

[17] P M Prenter Splines and Variational Methods Wiley-Intersci-ence 1975

[18] H T Nguyen ldquoA note on the extension principle for fuzzy setsrdquoJournal of Mathematical Analysis and Applications vol 64 no2 pp 369ndash380 1978

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

8 Advances in Fuzzy Systems

[9] W A Lodwick and J Santos ldquoConstructing consistent fuzzysurfaces from fuzzy datardquo Fuzzy Sets and Systems An Interna-tional Journal in Information Science and Engineering vol 135no 2 pp 259ndash277 2003

[10] M Zeinali S Shahmorad and K Mirnia ldquoHermite and piece-wise cubic Hermite interpolation of fuzzy datardquo Journal ofIntelligent amp Fuzzy Systems vol 26 no 6 pp 2889ndash2898 2014

[11] H Vosoughi and S Abbasbandy ldquoInterpolation of fuzzy data bycubic and piecewise-polynomial cubic hermitesrdquo Indian Journalof Science and Technology vol 9 no 8 2016

[12] P J Davis Interpolation and Approximation Dover New YorkNY USA 1975

[13] B Wendroff Theoretical Numerical Analysis Academic PressNew York NY USA 1966

[14] P G Ciarlet M H Schultz and R S Varga ldquoNumerical meth-ods of high-order accuracy for nonlinear boundary valueproblemsrdquo Numerische Mathematik vol 9 pp 397ndash430 1967

[15] G Birkhoff M H Schultz and R S Varga ldquoPiecewise Hermiteinterpolation in one and two variables with applications topartial differential equationsrdquo Numerische Mathematik vol 11pp 232ndash256 1968

[16] R S Varga ldquoHermite interpolation-type Ritz methods for two-point boundary value problemsrdquo in Numerical Solution ofPartial Differential Equations J H Bramble Ed pp 365ndash373Academic Press New York NY USA 1966

[17] P M Prenter Splines and Variational Methods Wiley-Intersci-ence 1975

[18] H T Nguyen ldquoA note on the extension principle for fuzzy setsrdquoJournal of Mathematical Analysis and Applications vol 64 no2 pp 369ndash380 1978

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

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Artificial Intelligence

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Electrical and Computer Engineering

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Computer Networks and Communications

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Biomedical Imaging

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ArtificialNeural Systems

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RoboticsJournal of

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Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014