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International Journal of ComputerMathematicsPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/gcom20

A new method based on legendrepolynomials for solution of system offractional order partial differentialequationsRahmat Ali Khan}a & Hammad Khalilaa Department of Mathematics, University of Malakand, ChakadaraDir(L), Khyber Pakhtunkhwa, PakistanAccepted author version posted online: 17 Jan 2014.Publishedonline: 26 Mar 2014.

To cite this article: Rahmat Ali Khan} & Hammad Khalil (2014): A new method based on legendrepolynomials for solution of system of fractional order partial differential equations, InternationalJournal of Computer Mathematics, DOI: 10.1080/00207160.2014.880781

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International Journal of Computer Mathematics, 2014http://dx.doi.org/10.1080/00207160.2014.880781

A new method based on legendre polynomials for solution ofsystem of fractional order partial differential equations

Rahmat Ali Khan and Hammad Khalil∗

Department of Mathematics, University of Malakand, Chakadara Dir(L), Khyber Pakhtunkhwa, Pakistan

(Received 20 August 2013; revised version received 18 November 2013; accepted 19 December 2013 )

We study legendre polynomials in case of more than one variable and develop new operational matricesof fractional order integrations as well as fractional order differentiations. Based on these operationalmatrices, we develop a new sophisticated technique to solve a coupled system of fractional order partialdifferential equations. Our technique reduces the coupled system under consideration to a system of easilysolvable algebraic equations without discretizing the system. As an application, we provide examples andnumerical simulations demonstrating that the results obtained using the new technique matches well withthe exact solutions of the problems. We also study error analysis graphically.

Keywords: legendre polynomials; operational matrices; fractional order partial differential equations;coupled system

2010 AMS Subject Classifications: 35C11; 35E15; 35F10; 35F40; 65M70

1. Introduction

In recent years fractional calculus and fractional differential equations have found interestingapplications in several different disciplines [4,14,23]. It has recently been investigated that manyengineering and physical processes can be modelled and well explained by system of fractionalorder differential equations compared with system of conventional differential equations andthat fractional order differentials and integrals provide more accurate models of systems underconsideration, see, for example [3,9,11,12,24] and the references therein. The analytical resultsbased on the existence and uniqueness of solutions to the fractional differential equations havebeen investigated by many authors, see for example [5,6,19,20,22] and the references therein.

Keeping in view the increasing applications of fractional order differential equations and inmost cases, the non-availability of their explicit analytic solutions, the need to exploit variousefficient and reliable numerical schemes is a problem of fundamental interest. In the literature, anumber of numerical methods have been proposed for obtaining approximate solutions to frac-tional order differential equations such as eigenvector expansion, adomian decomposition method(ADM), fractional differential transform method (FDTM) [1,2] and generalized block pulse oper-ational matrix method [10] to name a few. Recently some interesting numerical schemes based onoperational matrices of fractional order integration with Haar wavelets, legendre wavelets, sinecosine, and Chebyshev wavelets for solutions to fractional order differential equations have been

∗Corresponding author. Email: hammadk310@gmail.com

© 2014 Taylor & Francis

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2 R.A. Khan and H. Khalil

developed in [8,15,17,18,21,26]. In continuation, a new numerical scheme, based on the Haarwavelet operational matrix of integration has been developed in [6] for solutions of fractionalorder multi-point boundary value problems.

In this paper, we use legendre polynomials in two variables and develop new operational matri-ces of fractional order differentiations and integrations to solve coupled system of fractional orderpartial differential equations. The method reduces the system of fractional order differential equa-tions to a coupled system of algebraic equations. Generally, large systems of algebraic equationsmay lead to greater computational complexity and large storage requirements. However, our tech-nique is simple and reduces the computational complexity of the resulting algebraic system. Itis worthwhile to mention that the method based on using the operational matrices of orthogonalfunctions for solving differential equations is computer oriented.

The article is organized as follows: we begin by introducing some necessary definitions andmathematical preliminaries from fractional calculus and legendre polynomials which are requiredfor establishing our results. In Section 3, operational matrices of fractional order derivatives andfractional order integrals are developed. Section 4 is devoted to the application of operational matri-ces of fractional derivatives and fractional integrals to solve a coupled system of fractional orderpartial differential equations. In Section 5, the proposed method is applied to several examples.

2. Preliminaries

For convenience, this section summarizes some concepts, definitions and basic results fromfractional calculus, which are useful for further development in this paper.

Definition 2.1 [7,16] Given an interval [a, b] ⊂ R, the Riemann–Liouville fractional orderintegral of order α ∈ R+ of a function φ ∈ (L1[a, b], R) is defined by

Iαa+φ(t) = 1

�(α)

∫ t

a(t − s)α−1φ(s) ds,

provided that the integral on right-hand side exists.

Definition 2.2 For a given function φ(x) ∈ Cn[a, b], the Caputo fractional order derivative oforder α is defined as

Dαφ(x) = 1

�(n − α)

∫ x

a

φ(n)(t)

(x − t)α+1−ndt, n − 1 ≤ α < n, n ∈ N ,

provided that the right side is pointwise defined on (a, ∞), where n = [α] + 1.

Hence, it follows that

Dαxk = �(1 + k)

�(1 + k − α)xk−α , Iαxk = �(1 + k)

�(1 + k + α)xk+α and DαC = 0, for a constant C.

(1)

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International Journal of Computer Mathematics 3

2.1 The shifted legendre polynomials

The legendre polynomials defined on [−1, 1] are given by the following recurrence relation:

Łi+1(z) = 2i + 1

i + 1zŁi(z) − i

i + 1Łi−1(z), i = 1, 2 . . . where Ł0(z) = 0, Ł1(z) = z.

The transformation x = (z + 1)/2 transforms the interval [−1, 1] to [0, 1], and the shifted legendrepolynomials are given by

Pi(x) =i∑

k=0

(−1)i+k (i + k)!(i − k)!

xk

(k!)2, i = 0, 1, 2, 3 . . . , (2)

where Pi(0) = (−1)i, Pi(1) = 1. The orthogonality condition is

∫ 1

0Pi(x)Pj(x) dx =

⎧⎨⎩

1

2i + 1if i = j,

0 if i �= j,

which implies that any f (x) ∈ C[0, 1] can be approximated by legendre polynomials as follows:

f (x) ≈m∑

a=0

CaPa(x) where Ca = 〈f (x), Pa(x)〉 = (2a + 1)

∫ 1

0f (x)Pa(x) dx. (3)

In vector notation, we write

f (x) = KTMP̂M , (4)

where M = m + 1, K is the coefficient vector and P̂ is M terms vector function. We extend thenotion to two dimension space and define two dimension legendre polynomials of order M as aproduct function of two legendre polynomials

Pn(x, y) = Pa(x)Pb(y), n = Ma + b + 1, a = 0, 1, 2, . . . , m, b = 0, 1, 2, . . . , m. (5)

The orthogonality condition of Pn(x, y) is found to be

∫ 1

0

∫ 1

0Pa(x)Pb(y)Pc(x)Pd(y) dx dy =

⎧⎨⎩

1

(2a + 1)(2b + 1)if a = c, b = d,

0 otherwise.

Any f (x, y) ∈ C([0, 1] × [0, 1]) can be approximated by the polynomials Pn(x, y) as follows:

f (x, y) ≈m∑

a=0

m∑b=0

CabPa(x)Pb(y), (6)

where Cab can be obtained by the relation

Cab = (2a + 1)(2b + 1)

∫ 1

0

∫ 1

0f (x, y)Pa(x)Pb(y) dx dy. (7)

For simplicity, use the notation Cn = Cab where n = Ma + b + 1, and rewrite Equation (6) asfollows:

f (x, y) ≈M2∑n=1

CnPn(x, y) = KTM2�(x, y) (8)

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4 R.A. Khan and H. Khalil

in vector notation, where KM2 is the 1 × M2 coefficient row vector and �(x, y) is the M2 × 1column vector of functions defined by

�(x, y) = (ψ11(x, y) · · · ψ1M(x, y)ψ21(x, y) · · · ψ2M(x, y) · · · ψMM(x, y))T, (9)

where ψi+1,j+1(x, y) = Pi(x)Pj(y), i, j = 0, 1, 2, . . . , m.

2.2 Error analysis

For sufficiently smooth function f (x, y) on [0, 1] × [0, 1], the error of the approximation is givenby

‖f (x, y) − Pn(x, y)‖2 ≤(

C1 + C2 + C31

MM+1

)1

MM+1,

where

C1 = 1

4max

(x,y)∈[0,1]×[0,1]

∣∣∣∣ ∂M+1

∂xM+1f (x, y)

∣∣∣∣ , C2 = 1

4max

(x,y)∈[0,1]×[0,1]

∣∣∣∣ ∂M+1

∂yM+1f (x, y)

∣∣∣∣ ,

C3 = 1

16max

(x,y)∈[0,1]×[0,1]

∣∣∣∣ ∂2M+2

∂xM+1∂yM+1f (x, y)

∣∣∣∣ .

We refer the reader to [13] for the proof of the above result. As an example, consider a sufficientlysmooth function f (x, y) = sin(x) + cos(y) and its approximation by the two-dimension legendrepolynomials with M = 5. The comparison between the exact function and its approximation isshown in Figure 1 which demonstrate that the two-dimension legendre polynomials approximationis the best approximation of the function.

00.2

0.40.6

0.81

00.2

0.40.6

0.81

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

xy

Exact function

Approximated with two dimension legender polynomial

Figure 1. Comparison of the function with its approximation by two dimensional legendre polynomials.

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International Journal of Computer Mathematics 5

3. Operational matrices of integrations and differentiations

In case of a single variable, the operational matrices of fractional order integration and differentia-tion via legendre polynomials are available in [17,25]. We generalize the notion to the case of twovariables and develop operational matrices of fractional order integrations and differentiations ofarbitrary order.

Lemma 3.1 Let �(x, y) be as defined in Equation (9), then the integration of order α of �(x, y)w.r.t. x is given by

Iαx (�(x, y)) � Pα

M2×M2�(x, y), (10)

where PαM2×M2 is the operational matrix of integration of order α and is defined as

Pα,xM2×M2 =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

�1,1,k �1,2,k · · · �1,r,k · · · �1,M2,k

�2,1,k �2,2,k · · · �2,r,k · · · �2,M2,k...

......

......

...�q,1,k �q,2,k · · · �q,r,k · · · �q,M2,k

......

......

......

�M2,1,k �M2,2,k · · · �M2,r,k · · · �M2,M2,k

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (11)

and r = Mi + j + 1, q = Ma + b + 1, �q,r,k = Si,j,b,a,k for i, j, a, b = 0, 1, 2, . . . , m and

Si,j,b,a,k =a∑

k=0

δi,b(2i + 1)

i∑l=0

(−1)i+l+a+k(i + l)!(a + k)!(i − l)!(l!)2(k + l + α + 1)(a − k)!k!�(k + α + 1)

. (12)

Proof In order to prove the result, take Pn(x, y) as defined by Equation (5), then the fractionalintegral of order α of Pn(x, y) with respect to x is given by the relation

Iαx Pn(x, y) = Iα

x Pa(x)Pb(y) = Iαx Pa(x)Pb(y) =

a∑k=0

(−1)a+k (a + k)!(a − k)!(k!)2

Iαx xkPb(y)

which in view of the definition of fractional derivative takes the form

Iαx Pa(x)Pb(y) =

a∑k=0

(−1)a+k (a + k)!(a − k)!(k!)�(k + α + 1)

Pb(y)xk+α , b = 1, . . . , M.

Approximating Pb(y)xk+α by legendre polynomials in two variables, we obtain

Pb(x)xk+α ≈

m∑i=0

m∑j=0

SijPi(x)Pj(y), (13)

where Sij = (2i + 1)(2j + 1)∫ 1

0

∫ 10 Pb(y)xk+αPi(x)Pj(y) dx dy. From the orthogonality relation,

we obtain

Sij =

⎧⎪⎨⎪⎩

0 if j �= b;

(2i + 1)

j∑l=0

(−1)i+l(i + l)!(i − l)!(l!)2(k + l + α + 1)

if j = b,

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6 R.A. Khan and H. Khalil

which can be rewritten as

Sij,b = δj,b(2i + 1)

i∑l=0

(−1)i+l(i + l)!(i − l)!(l!)2(k + l + α + 1)

, (14)

where

δj,b ={

1 if j = b;

0 if j �= b.

It follows that

Iαx Pa(x)Pb(y) =

a∑k=0

(−1)a+k (a + k)!(a − k)!(k!)�(k + α + 1)

m∑i=0

m∑j=0

Sij,bPi(x)Pj(y)

=m∑

i=0

m∑j=0

a∑k=0

(−1)a+k (a + k)!(a − k)!(k!)�(k + α + 1)

Sij,bPi(x)Pj(y)

=m∑

i=0

m∑j=0

Sij,b,a,kPi(x)Pj(y), (15)

where

Sij,b,a,k =a∑

k=0

δi,b(2i + 1)

i∑l=0

(−1)i+l+a+k(i + l)!(a + k)!(i − l)!(l!)2(k + l + α + 1)(a − k)!k!�(k + α + 1)

. (16)

Using the notations r = Mi + j + 1, q = Ma + b + 1, �q,r,k = Si,j,b,a,k for i, j, a, b =0, 1, 2, 3, . . . , m, we obtain

Iαx Pa(x)Pb(y) =

M2∑q=1

M2∑r=1

�q,r,kPi(x)Pj(y) for a, b, i, j = 0, 1, 2, . . . , m. (17)

Consequently, the desired result follows. �

Lemma 3.2 Let �(x, y) be the function vector as defined in Equation (11) then the fractionalderivative of order α of �(x, y) w.r.t. y is given by

Dαy (�(x, y)) � Hα

M2×M2�(x, y), (18)

where HαM2×M2 is the operational matrix of differentiation of order α and is defined as

Hα,xM2×M2 =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

1,1,k 1,2,k · · · 1,r,k · · · 1,M2,k

2,1,k 2,2,k · · · 2,r,k · · · 2,M2,k...

......

......

...q,1,k q,2,k · · · q,r,k · · · q,M2,k

......

......

......

M2,1,k M2,2,k · · · M2,r,k · · · M2,M2,k

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (19)

where q = Mi + j + 1, r = Ma + b + 1, i, j, a, b = 0, 1, 2, . . . , m and

q,r,k = Ci,j,b,a,k =a∑

k= α�δi,a(2j + 1)

j∑l=0

(−1)j+l+b+k(j + l)!(b + k)!(j − l)!(l!)2(k + l − α + 1)(b − k)!k!�(k − α + 1)

(20)with ci,j,b,a,k = 0 if b < α.

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International Journal of Computer Mathematics 7

Proof Taking the element Pn(x, y) defined by Equation (5), then the fractional order partialderivative of Pn(x, y) w.r.t. y is given by the relation

Dαy (Pn(x, y)) = Pa(x)

b∑k=0

(−1)b+k (b + k)!(b − k)!(k!)2

Dαy yk =

b∑k=0

(−1)b+k (b + k)!(b − k)!(k!)2

Pa(x)Dαy yk .

Using the definition of fractional derivative, we obtain

Dαy Pa(x)Pb(y) =

b∑k= α�

(−1)b+k (b + k)!(b − k)!(k!)�(k − α + 1)

Pa(x)yk−α , b = α�, . . . , M. (21)

Approximating Pa(x)yk−α by legendre polynomials in two variables, we obtain

Pa(x)yk−α ≈

m∑i=0

m∑j=0

CijPi(x)Pj(y), (22)

where Cij = (2i + 1)(2j + 1)∫ 1

0

∫ 10 Pa(x)yk−αPi(x)Pj(y) dx dy, which in view of the orthogonal-

ity conditions implies that

Cij,a = δi,a(2j + 1)

j∑l=0

(−1)j+l(j + l)!(j − l)!(l!)2(k + l − α + 1)

, (23)

where

δia ={

1 if i = a;

0 if i �= a.

Hence it follows that

Dαy Pa(x)Pb(y) =

b∑k= α�

(−1)b+k (b + k)!(b − k)!(k!)�(k − α + 1)

m∑i=0

m∑j=0

Cij,aPi(x)Pj(y)

=m∑

i=0

m∑j=0

b∑k= α�

(−1)b+k (b + k)!(b − k)!(k!)�(k − α + 1)

Cij,aPi(x)Pj(y)

=m∑

i=0

m∑j=0

Cij,a,b,kPi(x)Pj(y), b = α�, . . . , M, (24)

where

Cij,a,b,k =b∑

k= α�δi,a(2j + 1)

j∑l=0

(−1)j+l+b+k(j + l)!(b + k)!(j − l)!(l!)2(k + l − α + 1)(b − k)!k!�(k − α + 1)

. (25)

Using the notations, q = Mi + j + 1, r = Ma + b + 1 and q,r,k = Ci,j,b,a,k for i, j, a, b =0, 1, 2, 3, . . . m, we obtain the desired result. �

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8 R.A. Khan and H. Khalil

4. Application of the operational matrices of integration and derivative to system ofequations

In order to show the applicability of our technique, we consider a coupled system of fractionalorder partial differential equations of the type

C1∂αU(x, y)

∂xα+ C2

∂αV(x, y)

∂yα= 0, 0 < α ≤ 1,

C3∂βV(x, y)

∂xβ+ C4

∂βU(x, y)

∂yβ= 0, 0 < β ≤ 1

(26)

subject to the initial conditions

U(0, y) = f (y), V(0, y) = g(y). (27)

We approximate the fractional derivatives by legendre polynomials of order M in two variables

∂αU(x, y)

∂xα= KM2�(x, y),

∂βV(x, y)

∂xβ= LM2�(x, y), (28)

applying integration of order α and β, respectively, with respect to x and the initial conditions(27), we obtain

U(x, y) = KM2 PαM2×M2�(x, y) + f (y), V(x, y) = LM2 Pβ

M2×M2�(x, y) + g(y). (29)

Approximating f (y) and g(y) as follows:

f (y) = F1�(x, y), g(y) = F2�(x, y).

Substituting in Equation (29) and applying fractional derivatives of order α, β , we obtain

∂αU(x, y)

∂yα= KM2 Pα

M2×M2 HαM2×M2�(x, y) + F1Hα

M2×M2�(x, y),

∂βV(x, y)

∂yβ= LM2 Pβ

M2×M2 Hβ

M2×M2�(x, y) + F2Hβ

M2×M2�(x, y).

(30)

Using Equations (28)and (30) in Equation (26), we obtain

KM2�(x, y) = − C2

C1LM2 Pβ

M2×M2 Hβ

M2×M2�(x, y) − C2

C1F2Hβ

M2×M2�(x, y),

LM2�(x, y) = −C4

C3KM2 Pα

M2×M2 HαM2×M2�(x, y) − C4

C3F1Hα

M2×M2�(x, y),

(31)

which can be rewritten in the matrix form as

(KT

M2�(x, y)

LTM2�(x, y)

)=

⎛⎜⎜⎝

−C2

C1(LT

M2 Pβ

M2×M2 Hβ

M2×M2�(x, y))

−C3

C4(KT

M2 PαM2×M2 Hα

M2×M2�(x, y))

⎞⎟⎟⎠ +

⎛⎜⎜⎝

−C2

C1F2Hβ

M2×M2�(x, y)

−C4

C3F1Hα

M2×M2�(x, y)

⎞⎟⎟⎠ . (32)

Taking transpose of the system, we obtain

(KTM2 LT

M2)

(�(x, y) 0

0 �(x, y)

)= −

(C2

C1LT

M2 PβHβC4

C3KT

M2 PαHα

) (�(x, y) 0

0 �(x, y)

)

−(

C2

C1F2Hβ

C4

C3F1Hα

) (�(x, y) 0

0 ψ(x, y)

), (33)

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International Journal of Computer Mathematics 9

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

10

2

4

6

xy

U(x

,y)

alpha=0.6alpha=0.7alpha=0.8alpha=0.9exact solution alpha=1approximate alpha=1

Figure 2. Comparison U(x, y) with the component U(x, y) of the exact solution of the system for different values of α

keeping M = 5 and β = 1.

0

0.2

0.4

0.6

0.8

1

00.10.20.30.40.50.60.70.80.91

−0.5

0

0.5

1

1.5

2

y

x

V(x

,y)

beta=0.6beta=0.7beta=0.8beta=0.9exact solution beta=1approximate beta=1

Figure 3. Comparison V(x, y) with the component V(x, y) of the exact solution of the system for different values of β

keeping M = 5 and α = 1.

where 0 is the M2 × 1 zero vector. For simplicity, taking A =(

�(x,y) 00 ψ(x,y)

), then, we have

(KTM2 LT

M2)A = −(

C2

C1LT

M2 PβHβC4

C3KT

M2 PαHα

)A −

(C2

C1F2Hβ

C4

C3F1Hα

)A, (34)

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10 R.A. Khan and H. Khalil

00.2

0.40.6

0.81

00.2

0.40.6

0.81

0.5

1

1.5

2

2.5

3

xy

U(x

,y)

Exact U(x,y)Approximate U(x,y)

Figure 4. Comparison U(x, y) with the component U(x, y) of the exact solution of the system for keeping α = 1, β = 1and M = 4.

0

0.5

1

00.20.40.60.81

−0.5

0

0.5

1

1.5

2

2.5

y

x

V(x

,y)

Exact V(x,y)Approximate V(x,y)

Figure 5. Comparison V(x, y) with the component V(x, y) of the exact solution of the system for keeping α = 1, β = 1and M = 4.

which implies that

(KTM2 LT

M2) +(

C2

C1LM2 TPβHβ

C4

C3KT

M2 PαHα

)+

(C2

C1F2Hβ

C4

C3F1Hα

)= 0, (35)

which is a system of algebraic equations. Note that the system is of the form of the Lyapanovmatrix equation and can be easily solved using computer program. We use Matlab for numericalsimulations.

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International Journal of Computer Mathematics 11

00.10.20.30.40.50.60.70.80.91

00.2

0.40.6

0.81

−4

−3

−2

−1

0

1

2

x 10−3

xy

error in U(x,y)

Figure 6. Error of approximation in U(x, y) at α = 1, M = 5, and β = 1.

0

0.5

1

00.2

0.40.6

0.81

−2

−1

0

1

2

3

4

x 10−3

xy

error in V(x,y)

Figure 7. Error of approximation in V(x, y) at α = 1, M = 5, β = 1.

Example 1 Consider the system (26) with C1 = 1, C2 = −1, C3 = 1, C4 = 1 and the initialcondition f (y) = sin(y), g(y) = cos(y). Note that for α = 1 and β = 1, the system (26) is asystem of laplace equations whose exact solution is U(x, y) = ex sin(y) and V(x, y) = ex cos(y).Comparisons of our results obtained via the new method with the exact solution of the system

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12 R.A. Khan and H. Khalil

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

x

Y

U(x

,y)

exact solution

approximate solution

Figure 8. Comparison of U(x, y) with exact solution for keeping α = 1, β = 1 and M = 4.

0

0.5

1

00.10.20.30.40.50.60.70.80.91

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

y

x

V(x

,y)

exact resultapproximate

Figure 9. Comparison of V(x, y) with exact solution for keeping α = 1, β = 1 and M = 4.

for different values of α and β are shown. Comparison of U(x, y) with the exact solution of thesystem for different values of α and β = 1 is shown in Figure 2, while comparison of V(x, y)with the exact solution of the system for different values of β and α = 1 is shown in Figure 3. We

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International Journal of Computer Mathematics 13

0

0.5

1

00.2

0.40.6

0.81

−2

−1

0

1

2

3

4

x 10−3

xy

error in V(x,y)

Figure 10. Error of approximation in V(x, y) for α = 1, M = 5 and β = 1.

see that as α → 1, β → 1, the numerical result approaches the exact solution of the system asit evident from Figures 4 and 5. We also study error analysis numerically and see that error fallsbelow 10−3 which is very much acceptable number for a choice of M ≤ 5. The error can furtherbe reduced as we increase the value M. The results are shown in Figures 6 and 7.

Example 2 Consider the system with the initial condition f (y) = −y2 and g(y) = 0 and C1 = 1,C2 = −1, C3 = 1, and C4 = 1. Note that for α = 1 and β = 1 the system is just system oflaplace equations whose exact solution is U(x, y) = x2 − y2 and V(x, y) = 2xy. Repeating thesame process as was done in the previous example, we see that our results match with the exactsolutions, Figures 8 and 9 and the same conclusion also holds for error of approximation, Figure 10.

Acknowledgements

We are grateful to the reviewers for their valuable comments and suggestions which led to improvement of the manuscript.

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