and norazak senu - universiti putra malaysiaeinspem.upm.edu.my/sawona2018/presentation for...
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Fudziah Ismail, Sufia Zulfa and Norazak Senu
Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor,
Malaysia
Extra Derivative Multistep Collocation Methods with Trigonometrically-fitting for Oscillatory Problems
Contents
โข Introduction
โข Derivation of the method
โข Trigonometrically fitting the method
โข Numerical Results
โข Discussion and Conclusion
Objectives 1. To derive extra derivative multistep methods of order 3 and 4,
using collocation technique. 2. Trigonometrically-fitting the methods. 3. Validating the methods using a set of special second order differential equations which have oscillatory solutions. 4. Comparing the numerical results with other well known existing methods in the scientific literature.
1. Introduction
The special second order ordinary differential equations (ODEs) can be
represented by
๐ฆ" = ๐(๐ฅ, ๐ฆ), ๐ฆ ๐ฅ0 = ๐ฅ0, ๐ฆโฒ ๐ฅ0 = ๐ฆ0โฒ (1)
in which the first derivative does not appear explicitly.
This type of problems often appears in the field of science, mathematics and
engineering such as quantum mechanics, spatial semi-discretizations of wave
equations, populations modeling and celestial mechanics.
Currently, equation (1) can be solved directly without converting it first to a
system of first order ODEs. Such methods are direct multistep method, Runge-
Kutta Nystrำงm (RKN) method, and hybrid method.
Work RKN can be seen can be seen in Dormand et al. [1] and Sommeijer [2].
Work on hybrid method can be seen in Franco [3] and Coleman [4],where they
have developed hybrid algorithm and constructed the order condition of the
hybrid methods for directly solving equation (1).
Research on directly solving equation (1) using methods which involved, the
interpolation and collocation techniques has been done by various researchers
such as the work of authors in [5]-[6]. As well as the work of Awoyemi [7] and
Jator [8].
The special second order ODEs also often exhibit oscillatory solutions which cannot
be solved efficiently using conventional methods.
To obtain a more efficient process for solving oscillatory problems, numerical
methods are constructed by taking into account the nature of the problem.
This results in methods in which the coefficients depend on the problem.
Numerical methods can adapted to the special structure of the problem, such
techniques for this purpose are trigonometrically-fitted and phase-fitted
techniques.
A good theoretical foundation of exponentially fitted technique was given by
Lyche [9]. Since then a lot of exponentially fitted linear multistep have been
constructed mostly for special second order ODEs such as (1).
Vanden Berghe et. al [10]. Introduced an explicit exponentially fitted explicit
Runge-Kutta which integrate exactly first order systems whose solution can be
expressed as linear combination of functions of the form ๐๐๐ก, ๐โ๐๐ก or
cos ๐๐ก , sin(๐๐ก) .
This idea is extended to Runge-Kutta method by Simos [11] and Franco [12].
Fang and Wu [13] , developed a Trigonometrically fitted explicit Numerov-type
method for second-order initial value problems with oscillating solutions.
In this research, we are going to construct extra derivative multistep methods of
order 3 and 4 using collocation technique and Chebyshev polynomial will be used
as the basis function
In order to improve the efficiency of the methods, we trigonometrically fitted the
methods so that the coefficients will depend on the fitted frequency and step size
of the problems. And used the methods for solving oscillatory Differential
Equations
2. Derivation of Linear Multistep Methods (LMM) Using Collocation Technique
The general ๐-step LMM for solving special second order ODEs is given as
๐ผ๐๐ฆ๐+๐๐๐=0 = โ2 ๐ฝ๐๐๐+๐
๐๐=0 ,
Here, we will use Chebyshev Polynomials as basis function. The following are the
first five terms of the sequence from Chebyshev Polynomials :
ฮค0 ๐ฅ = 1, ฮค1 ๐ฅ = ๐ฅ, ฮค2 ๐ฅ = 2๐ฅ2 โ 1, ฮค3 ๐ฅ = 4๐ฅ3 โ 3๐ฅ, ฮค4 ๐ฅ = 8๐ฅ4 โ
8๐ฅ2 + 1 (2)
Here, we are going to develop linear multistep method with extra derivatives of the
form of
๐ผ๐๐ฆ๐+๐๐๐=0 = โ2 ๐๐๐๐+๐
๐๐=0 + โ3 ๐๐๐๐+๐
๐๐=0 , (3)
where ๐ผ๐, ๐๐ and ๐๐ are constant values, ๐๐+๐ = ๐ฆโฒโฒ๐+๐ and ๐๐+๐ = ๐ฆโฒโฒโฒ๐+๐.
We proceed to approximate the exact solution ๐ฆ(๐ฅ) by the interpolating
function of the form
๐ฆ ๐ฅ = ๐๐๐ ๐ (๐ฅ โ ๐ฅ๐)
๐
๐=0
, (4)
which is the polynomial of degree n and satisfied the equations
๐ฆโฒโฒ ๐ฅ = ๐ ๐ฅ, ๐ฆ ๐ฅ , ๐ฅ๐ โค ๐ฅ โค ๐ฅ๐+๐ , ๐ฆ ๐ฅ๐ = ๐ฆ๐ . (5)
2.1. Derivation of LMMC(3)
For ๐ = 4, equation (4) can be written as
๐ฆ ๐ฅ = ๐0 + ๐1 ๐ฅ โ ๐ฅ๐ + ๐2 ๐ฅ โ ๐ฅ๐2 โ 1 + ๐3 ๐ฅ โ ๐ฅ๐
3 โ 3 ๐ฅ โ ๐ฅ๐
+๐4 ๐ฅ โ ๐ฅ๐4 โ 8 ๐ฅ โ ๐ฅ๐
2 + 1 . (6)
Differentiating equation (6) three times and we get the first, second, and third
derivatives of the equations as follows:
๐ฆโฒ ๐ฅ = ๐1 + ๐22 ๐ฅ โ ๐ฅ๐ + ๐33 ๐ฅ โ ๐ฅ๐2 โ 1
+๐44 ๐ฅ โ ๐ฅ๐3 โ 4 ๐ฅ โ ๐ฅ๐ , (7)
๐ฆโฒโฒ ๐ฅ = 2๐2 + ๐36 ๐ฅ โ ๐ฅ๐ + ๐44 3 ๐ฅ โ ๐ฅ๐2 โ 4 , (8)
๐ฆโฒโฒโฒ ๐ฅ = 6๐3 + ๐424 ๐ฅ โ ๐ฅ๐ , (9)
Next, equations (6) and (8) are collocated at ๐ฅ = ๐ฅ๐+1, ๐ฅ๐+2 and interpolated equation
(9) at ๐ฅ = ๐ฅ๐+1 which yields
๐ฆ ๐ฅ๐+1 = ๐0 + ๐1 ๐ฅ๐+1 โ ๐ฅ๐ + ๐2 ๐ฅ๐+1 โ ๐ฅ๐2 โ 1
+ ๐3 ๐ฅ๐+1 โ ๐ฅ๐3 โ 3 ๐ฅ๐+1 โ ๐ฅ๐
+๐4 ๐ฅ๐+1 โ ๐ฅ๐4 โ 8 ๐ฅ๐+1 โ ๐ฅ๐
2 + 1 = ๐ฆ๐+1, (10)
๐ฆ ๐ฅ๐+2 = ๐0 + ๐1 ๐ฅ๐+2 โ ๐ฅ๐ + ๐2 ๐ฅ๐+2 โ ๐ฅ๐2 โ 1
+ ๐3 ๐ฅ๐+2 โ ๐ฅ๐3 โ 3 ๐ฅ๐+2 โ ๐ฅ๐
+๐4 ๐ฅ๐+2 โ ๐ฅ๐4 โ 8 ๐ฅ๐+2 โ ๐ฅ๐
2 + 1 = ๐ฆ๐+2, (11)
๐ฆโฒโฒ ๐ฅ๐+1 = 2๐2 + ๐36 ๐ฅ๐+1 โ ๐ฅ๐ + ๐44 3 ๐ฅ๐+1 โ ๐ฅ๐2 โ 4 = ๐๐+1, (12)
๐ฆโฒโฒ ๐ฅ๐+2 = 2๐2 + ๐36 ๐ฅ๐+2 โ ๐ฅ๐ + ๐44 3 ๐ฅ๐+2 โ ๐ฅ๐2 โ 4 = ๐๐+2, (13)
๐ฆโฒโฒโฒ ๐ฅ๐+1 = 6๐3 + ๐424 ๐ฅ๐+1 โ ๐ฅ๐ = ๐๐+1. (14)
By substituting โ = ๐ฅ๐+1 โ ๐ฅ๐ and 2โ = ๐ฅ๐+2 โ ๐ฅ๐ into equations (10)-(14), we obtain
the following:
๐ฆ๐+1 = ๐0 + โ ๐1 + ๐2 โ2 โ 1 + ๐3 โ
3 โ 3โ + ๐4 โ4 โ 8โ2 + 1 , (15)
๐ฆ๐+2 = ๐0 + 2โ ๐1 + ๐2 4โ2 โ 1 + ๐3 8โ
3 โ 6โ + ๐4 16โ4 โ 32โ2 + 1 , (16)
๐๐+1 = 2๐2 + ๐3 6โ + ๐4 12โ2 โ 16 , (17)
๐๐+2 = 2๐2 + ๐3 12โ + ๐4 48โ2 โ 16 , (18)
๐๐+1 = 6๐3 + ๐4 24โ . (19)
Rearranging equations (15)-(19) into matrix form as follows:
1 โ โ2 โ 1 โ3 โ 3โ โ4 โ 8โ2 + 11 2โ 4โ2 โ 1 8โ3 โ 6โ 16โ4 โ 16โ2 + 10 0 2โ 6โ2 12โ2 โ 160 0 2โ 12โ2 48โ2 โ 160 0 0 6โ 24โ2
๐0๐1๐2๐3๐4
=
๐ฆ๐+1๐ฆ๐+2โ๐๐+1โ๐๐+2โ๐๐+1
which can be simplified as ๐๐ด = ๐ (20)
we have ๐ด = ๐โ1๐ (21) where
๐โ1 =
2 โ110โ4 โ 7
12โ32โ4 + 6โ2 + 7
12โ3โ2โ4 + 12โ2 + 7
12โ2
โ1
โ
1
โโ13โ2 โ 12
12โ2โ5โ2 + 12
12โ23 โ2 + 2
12โ2
0 0 โ2
3โ33โ2 + 4
6โ3โ3โ2 + 2
3โ2
0 01
3โ2โ
1
3โ21
2โ
0 0 โ1
12โ31
12โ3โ
1
12โ2
Solving (21), the coefficients of ๐0,๐1, ๐2, ๐3, and ๐4 are obtained in terms of ๐ฆ๐+1, ๐ฆ๐+2 , ๐๐+1, ๐๐+2 and ๐๐+1.
๐0 = 2๐ฆ๐+1 โ ๐ฆ๐+2 +1
12
10โ4 โ 7
โ2๐๐+1 +
1
12
2โ4 + 6โ2 + 7
โ2๐๐+2
โ1
12
2โ4 + 12โ2 + 7
โ2๐๐+1,
๐1 = โ๐ฆ๐+1โ
+๐ฆ๐+2โ
โ1
12
13โ2 โ 12
โ๐๐+1 โ
1
12
5โ2 + 12
โ๐๐+2
+3
4โ2 + 2 ๐๐+1,
๐2 = โ2
3
๐๐+1โ2
+1
6
3โ2 + 4
โ2๐๐+2 โ
1
3
3โ2 + 2
โ๐๐+1,
๐3 =1
3
๐๐+1โ
โ1
3
๐๐+2โ
+1
2๐๐+1,
๐4 = โ1
12
๐๐+1โ2
+1
12
๐๐+2โ2
โ1
12
๐๐+1โ
.
Substituting the coefficients into equation (6) and letting ๐ฅ = ๐ฅ๐+3, we obtain the following equation:
๐ฆ ๐ฅ๐+3 = 2๐ฆ๐+1 โ ๐ฆ๐+2 +1
12
10โ4 โ 7
โ2๐๐+1 +
1
12
2โ4 + 6โ2 + 7
โ2๐๐+2
โ1
12
2โ4 + 12โ2 + 7
โ2๐๐+1
+ โ๐ฆ๐+1โ
+๐ฆ๐+2โ
โ1
12
13โ2 โ 12
โ๐๐+1 โ
1
12
5โ2 + 12
โ๐๐+2
+3
4โ2 + 2 ๐๐+1 ๐ฅ๐+3 โ ๐ฅ๐
+ โ2
3
๐๐+1โ2
+1
6
3โ2 + 4
โ2๐๐+2 โ
1
3
3โ2 + 2
โ๐๐+1 ๐ฅ๐+3 โ ๐ฅ๐
2 โ 1
+1
3
๐๐+1โ
โ1
3
๐๐+2โ
+1
2๐๐+1 ๐ฅ๐+3 โ ๐ฅ๐
3 โ 3 ๐ฅ๐+3 โ ๐ฅ๐
+1
3
๐๐+1โ
โ1
3
๐๐+2โ
+1
2๐๐+1 ๐ฅ๐+3 โ ๐ฅ๐
4 โ 8 ๐ฅ๐+3 โ ๐ฅ๐2 + 1 = ๐ฆ๐+3.
Letting 3โ = ๐ฅ๐+3 โ ๐ฅ๐ , we obtain the discrete form of LMMC as
๐ฆ๐+3 = 2๐ฆ๐+2 โ ๐ฆ๐+1 +โ2
6โ๐๐+1 + 7๐๐+2 โ โ๐๐+1 , (22)
2.2 Order and Consistency of LMMC method
Definition 1: The linear difference operator L is defined by
๐ฟ ๐ฆ ๐ฅ ; โ = ๐ผ๐๐ฆ ๐ฅ + ๐โ โ โ2๐๐๐ ๐ฅ + ๐โ โ โ3๐๐๐ ๐ฅ + ๐โ
๐
๐=0
,
where ๐ฆ(๐ฅ) is an arbitrary function that is sufficiently differentiable on [๐, ๐]. By
expanding the test function and its first derivative as Taylor series about ๐ฅ and
collecting the terms to obtain
๐ฟ ๐ฆ ๐ฅ ; โ = ๐0๐ฆ ๐ฅ + ๐1โ๐ฆโฒ ๐ฅ +โฏ+ ๐๐โ
(๐)๐ฆ(๐) ๐ฅ + โฏ ,
where the coefficients of ๐๐ are constants independent of ๐ฆ(๐ฅ). In particular
๐0 = ๐ผ๐๐๐=0 , ๐1 = ๐๐ผ๐
๐๐=0 ,
๐2 = ๐(2)
2!๐ผ๐ โ ๐๐
๐๐=0 ,
๐3 = ๐(3)
3!๐ผ๐ โ ๐๐๐ โ ๐๐
๐
๐=0
,
โฎ
๐๐ = ๐(๐)
๐!๐ผ๐ โ
๐(๐โ2)
(๐ โ 2)!๐๐ โ
๐(๐โ3)
(๐ โ 3)!๐๐
๐
๐=0
.
(23)
Definition 2:
The associated linear multistep method (22) is said to be of the order ๐ if
๐0 = ๐1 = โฏ = ๐๐+1 = 0 and ๐๐+2 โ 0.
Definition 3 [Consistency of the method]
The method is said to be consistence if it has order at least one.
By substituting the coefficients into equations (23), we obtain
๐0 = ๐1 = ๐2 = ๐3 = ๐4 = 0 and ๐5 =1
8.
The new method is consistent and has order ๐ = 3.
And denoted as linear multistep method with extra derivative using collocation
technique of order three (LMMC(3)).
2.2. Derivation of LMMC(4) , order and consistency of the method
In this section, we derive the LMMC of order four. For ๐ = 5, we obtain equation (4) as
๐ฆ ๐ฅ = ๐0 + ๐1 ๐ฅ โ ๐ฅ๐ + ๐2 ๐ฅ โ ๐ฅ๐2 โ 1 + ๐3 ๐ฅ โ ๐ฅ๐
3 โ 3 ๐ฅ โ ๐ฅ๐
+๐4 ๐ฅ โ ๐ฅ๐4 โ 8 ๐ฅ โ ๐ฅ๐
2 + 1
+๐5 ๐ฅ โ ๐ฅ๐5 โ 20 ๐ฅ โ ๐ฅ๐
3 + 5 ๐ฅ โ ๐ฅ๐ . (24)
Differentiating equation (24) three times gives
Differentiating equation (24) three times gives
๐ฆโฒ ๐ฅ = ๐1 + ๐22 ๐ฅ โ ๐ฅ๐ + ๐33 ๐ฅ โ ๐ฅ๐2 โ 1
+๐44 ๐ฅ โ ๐ฅ๐3 โ 4 ๐ฅ โ ๐ฅ๐ + ๐55 ๐ฅ โ ๐ฅ๐
4 โ 12 ๐ฅ โ ๐ฅ๐2 + 1 , (25)
๐ฆโฒโฒ ๐ฅ = 2๐2 + ๐36 ๐ฅ โ ๐ฅ๐ + ๐44 3 ๐ฅ โ ๐ฅ๐2 โ 4
+๐520 ๐ฅ โ ๐ฅ๐3 โ 6 ๐ฅ โ ๐ฅ๐ , (26)
๐ฆโฒโฒโฒ ๐ฅ = 6๐3 + ๐424 ๐ฅ โ ๐ฅ๐ + ๐560 ๐ฅ โ ๐ฅ๐2 โ 2 , (27)
Equations (24) and (27) are collocated at ๐ฅ = ๐ฅ๐+1, ๐ฅ๐+2 , and equation (26) at
๐ฅ = ๐ฅ๐+2, ๐ฅ๐+3 which yields
๐ฆ๐+1 = ๐0 + โ ๐1 + ๐2 โ2 โ 1 + ๐3 โ
3 โ 3โ + ๐4 โ4 โ 8โ2 + 1
+๐5 โ5 โ 20โ2 + 5โ , (28)
๐ฆ๐+2 = ๐0 + 2โ ๐1 + ๐2 4โ2 โ 1 + ๐3 8โ
3 โ 6โ + ๐4 16โ4 โ 32โ2 + 1
+๐5 32โ5 โ 160โ3 + 10โ , (29)
๐๐+2 = 2๐2 + ๐3 6โ + ๐4 12โ2 โ 16 + ๐5 20โ
3 โ 120โ , (30)
๐๐+3 = 2๐2 + ๐3 18โ + ๐4 48โ2 โ 16 + ๐5 540โ
3 โ 360โ , (31)
๐๐+1 = 6๐3 + ๐4 24โ + ๐5 60โโ2 โ 120 , (32)
๐๐+2 = 6๐3 + ๐4 48โ + ๐5 240โโ2 โ 120 . (33)
Equations (29)-(33) are written in matrix form as follows:
๐๐ด = ๐
Where
๐ =
1 โ โ2 โ 1 โ3 โ 3โ โ4 โ 8โ2 + 1 โ5 โ 20โ3 + 5โ1 2โ 4โ2 โ 1 8โ3 โ 6โ 16โ4 โ 32โ2 + 1 32โ5 โ 160โ3 + 10โ0 0 2โ 12โ2 48โ3 โ 16โ 160โ4 โ 240โ2
0 0 2โ 18โ2 108โ3 โ 16โ 540โ4 โ 360โ2
0 0 0 6โ 24โ2 60โ3 โ 120โ0 0 0 6โ 48โ2 240โ3 โ 120โ
,
๐ด = ๐0 ๐1 ๐2 ๐3 ๐4 ๐5 ๐ , and ๐ = ๐ฆ๐+1 ๐ฆ๐+2 โ๐๐+2 โ๐๐+3 โ๐๐+1 โ๐๐+2
๐.
๐0 = 2๐ฆ๐+1 โ ๐ฆ๐+2 +3 6โ4 + 6โ2 + 7
20โ2๐๐+2 +
2โ4 โ 8โ2 โ 21
20โ2๐๐+3
โ32โ4 + 72โ2 + 49
60โ๐๐+1 โ
17โ4 โ 18โ2 โ 56
30โ๐๐+2,
๐1 = โ1
โ๐ฆ๐+1 +
1
โ๐ฆ๐+2 โ
153โ4 + 120โ2 + 110
100โ3๐๐+2
+3โ4 + 120โ2 + 100
100โ3๐๐+3 +
392โ4 + 480โ2 + 165
300โ2๐๐+1
+149โ4 โ 690โ2 โ 495
300โ2๐๐+2,
๐2 =3 3โ2 + 4
10โ2๐๐+2 โ
2 โ2 + 3
5โ2๐๐+3 โ
2 9โ2 + 7
15โ2๐๐+1
+9โ2 + 32
15โ2๐๐+2
๐3 = โ2 โ2 + 1
5โ2๐๐+2 +
2 โ2 + 1
5โ2๐๐+3 +
8โ2 + 3
15โ2๐๐+1
โ23โ2 + 18
30โ2๐๐+2,
๐4 =3
20โ2๐๐+2 โ
3
20โ2๐๐+3 โ
7
60โ๐๐+1 +
4
15โ๐๐+2,
๐5 = โ1
50โ3๐๐+2 +
1
50โ3๐๐+3 +
1
100โ2๐๐+1 โ
3
100โ2๐๐+2.
We substitute๐0,๐1, ๐2, ๐3, ๐4 and ๐5 into equation (24) and by letting ๐ฅ = ๐ฅ๐+3, and
3โ = ๐ฅ๐+3 โ ๐ฅ๐ we obtain the discrete form of LMMC as
๐ฆ๐+3 = 2๐ฆ๐+2 โ ๐ฆ๐+1 +โ2
109๐๐+2 + ๐๐+3 โ
โ3
30๐๐+1 + 2๐๐+2 , (34)
By substituting back the coefficients into equations (23), we obtain
๐0 = ๐1 = ๐2 = ๐3 = ๐4 = ๐5 = 0 and ๐6 = โ1
144.
From Definition 2, the new method is consistent and the order is ๐ = 4.
And denoted as linear multistep method with extra derivative using collocation
technique (LMMC(4)).
3. Trigonometrically-Fitting The Methods
In this section, we adapt the trigonometrically-fitting technique to
LMMC(3). By letting some of the coefficients to be unknown values of ๐ ๐,
for ๐ = 1,2,3, (22) is written as follows:
๐ฆ๐+1 = 2๐ฆ๐ โ ๐ฆ๐โ1 + โ2 ๐ 1๐๐โ1 + ๐ 2๐๐ + โ3 ๐ 3๐๐โ1 . (35)
Assuming that ๐ฆ ๐ฅ is a linear combination of the functions
{sin(๐ฃ๐ฅ), cos(๐ฃ๐ฅ)} for ๐ฃ โ โ. We obtain the following equations:
cos ๐ป = 2 โ cos ๐ป โ ๐ป2 ๐ 1 cos ๐ป + ๐ 2 + ๐ 3๐ป sin ๐ป , (36)
๐ 1 sin ๐ป = ๐ 3๐ป cos ๐ป ,
where ๐ป = ๐ฃโ, โ is the step size and๐ฃ is fitted frequency.
Solving equation (36) and letting, ๐ 1 = โ1/6, the value of the remaining coefficients
is obtain in terms of ๐ป.
๐ 2 =7
6+
3
80๐ป4 +
851
60480๐ป6 +
20777
3628800๐ป8 +
7939
3421440๐ป10 + ๐ ๐ป12 ,
๐ 3 = โ1
6โ
1
18๐ป2 โ
1
45๐ป4 โ
17
1890๐ป6 โ
31
8505๐ป8 โ
691
467775๐ป10 + ๐ ๐ป12 .
This new method is denoted as Trigonometrically-fitted linear multistep method
with extra derivative using collocation technique of order three (TF-LMMC(3)).
Then we apply the trigonometrically-fitting technique to LMMC(4). By letting
some of the coefficients to be unknown values of ๐ ๐, for ๐ = 1,2,3,4, rewrite the
formula in (34) as
๐ฆ๐+1 = 2๐ฆ๐ โ ๐ฆ๐โ1 + โ2 ๐ 1๐๐ + ๐ 2๐๐+1 + โ3 ๐ 3๐๐โ1 + ๐ 4๐๐ . (37)
Assuming that ๐ฆ ๐ฅ is the linear combination of the function {sin(๐ฃโ), cos(๐ฃโ)}
for ๐ฃ โ โ. Therefore, the following equations are obtained.
cos ๐ป = 2 โ cos ๐ป โ ๐ป2 ๐ 1 + ๐ 2 cos ๐ป + ๐ 3๐ป sin ๐ป ,
(38)
๐ 2 sin ๐ป = โ๐ป ๐ 3 cos ๐ป +๐ 4 ,
where ๐ป = ๐ฃโ, โ is the step size and๐ฃ is fitted frequency.
Solving equations in (38) simultaneously by letting ๐ 1 = 9/10 and ๐ 3 = โ1/30
the value of the remaining coefficients is obtained in terms of ๐ป as follows:
๐ 2 =1
10โ
1
144๐ป4 โ
313
100800๐ป6 โ
923
725760๐ป8 โ
6437
12474000๐ป10 + ๐ ๐ป12 ,
๐ 4 = โ1
15+
3
400๐ป4 +
83
432000๐ป6 +
983
1209600๐ป8 + ๐ ๐ป10 .
This new method is denoted as Trigonometrically-fitted linear multistep method
with extra derivative using collocation technique of order four (TF-LMMC(4)).
The other coefficients of the method remain the same. This method is
4. Numerical Results and Discussion
In this section, the new methods LMMC(3), LMMC(4), TF-LMMC(3) and TF-
LMMC(4) are tested for problems .1-5 . Comparisons are made with other
existing methods.
The following are the notation used in figures 1-10:
TF-LMMC(3) Trigonometrically-fitted linear multistep method with collocation
method of order three developed in this paper.
LMMC(3) A linear multistep method with collocation method of order three
developed in this paper.
TF-LMMC(4) Trigonometrically-fitted linear multistep method with collocation
method of order four developed in this paper.
LMMC(4) A linear multistep method with collocation method of order four
developed in this paper.
EHM3(4) Explicit three-stage fourth-order hybrid method derived by Franco
[12]
RKN3(4) Explicit three-stage fourth-order RKN method by Hairer et al.[14].
PFRKN4(4) Explicit four-stage fourth-order Phase-fitted RKN method by
Papadopoulos et al.[15].
DIRKN(HS) Diagonally implicit three-stage fourth-order RKN method derived in
Sommeijer [16]
DIRKN3(4) Diagonally implicit three-stage fourth-order RKN method derived in
Senu et al. [17]
SIHM3(4) Semi-implicit three-stage fourth-order hybrid method developed in
Ahmad et al. [18]
Problem 1(An almost Periodic Orbit problem studied by Stiefel and Bettis [19]) ๐ฆ1" ๐ฅ + ๐ฆ1 ๐ฅ = 0.001cos ๐ฅ , ๐ฆ1 0 = 1,๐ฆ1โฒ 0 = 0, ๐ฆ2" ๐ฅ + ๐ฆ2 ๐ฅ = 0.001sin ๐ฅ , ๐ฆ2 0 = 0,๐ฆ2โฒ 0 = 0.9995, Exact solution is ๐ฆ1 = cos ๐ฅ + 0.0005๐ฅsin ๐ฅ , and ๐ฆ2 = sin ๐ฅ โ0.0005๐ฅcos ๐ฅ . The fitted frequency is ๐ฃ = 1.
Figure 1: The efficiency curve for TF-LMMC(3) for
Problem 1 with ๐๐๐๐ = 104 and โ =0.9
2๐ for ๐ = 1,โฆ , 5.
The following are efficiency curves TF-LMMC(3) Method.
Problem 2 (Inhomogeneous system
by Lambert and Watson [20])
๐2๐ฆ1 ๐ฅ
๐๐ก2= โ๐ฃ2๐ฆ1 ๐ฅ + ๐ฃ2๐ ๐ฅ + ๐" ๐ฅ ,
๐ฆ1 0 = ๐ + ๐ 0 , ๐ฆ1โฒ 0 = ๐โฒ 0 ,
๐2๐ฆ2 ๐ฅ
๐๐ก2= โ๐ฃ2๐ฆ2 ๐ฅ + ๐ฃ2๐ ๐ฅ + ๐" ๐ฅ
๐ฆ2 0 = ๐ 0 , ๐ฆ2โฒ 0 = ๐ฃ๐ + ๐โฒ 0
Exact solution is ๐ฆ1 ๐ฅ = ๐cos ๐ฃ๐ฅ +
๐ ๐ฅ ,๐ฆ2 ๐ฅ = ๐sin ๐ฃ๐ฅ + ๐ ๐ฅ , ๐ ๐ฅ is chosen
to be ๐โ0.05๐ฅ and parameters ๐ฃ and ๐
are 20 and 0.1 respectively.
Figure 2: The efficiency curve for TF-LMMC(3) for
Problem 2 with ๐๐๐๐ = 104 and โ =0.125
2๐ for
๐ = 2,โฆ , 6.
Problem 3 (Inhomogeneous system studied by Franco [12])
๐ฆ" ๐ฅ =1012
โ992
โ992
1012
๐ฆ ๐ฅ = ๐ฟ932cos(2๐ฅ) โ99
2sin 2๐ฅ
932sin(2๐ฅ) โ
992cos(2๐ฅ)
๐ฆ 0 =โ1 + ๐ฟ
1, ๐ฆโฒ 0 =
โ1010 + 2๐ฟ
for ๐ฟ = 10โ3. Exact solution ๐ฆ ๐ก =โcos 10๐ฅ โ sin 10๐ฅ + ๐ฟcos(2๐ฅ)
cos 10๐ฅ + sin 10๐ฅ + ๐ฟsin(2๐ฅ).
The Eigen-value of the problem are ๐ฃ = 10 and ๐ฃ = 1. The fitted frequency is chosen to be ๐ฃ = 10 because it is dominant than ๐ฃ = 1.
Figure 3: The efficiency curve for TF-LMMC(3) for
Problem 3 with ๐๐๐๐ = 104 and โ =0.125
2๐ for
๐ = 1,โฆ , 5.
Problem 4 (Homogenous given
in Attili et.al [2])
๐ฆ" ๐ฅ = โ64๐ฆ ๐ฅ ,๐ฆ 0 = 14 ,
๐ฆโฒ 0 = โ 12 .
Exact solution is ๐ฆ = 1716 sin 8๐ฅ + ๐ ,
๐ = ๐ โ tanโ1 4 .
The fitted frequency is ๐ฃ = 8.
Figure 4: The efficiency curve for TF-LMMC(3) for
Problem 5 with ๐๐๐๐ = 104 and โ =0.1
2๐ for
๐ = 3,โฆ , 7.
Problem 5 (Inhomogeneous equation
studied by Papadopoulos et.al [15])
๐ฆโฒโฒ(๐ฅ) = โ๐2๐ฆ(๐ฅ) + (๐2 โ 1)sin(๐ฅ),
๐ฆ(0) = 1, ๐ฆโ(0) = ๐ + 1.
Exact solution is y(๐ฅ) = cos(๐๐ฅ) +
sin(๐๐ฅ) + sin(๐ฅ).
The fitted frequency is ๐ = 10.
Figure 6: The efficiency curve for TF-LMMC(3) for Problem
6 with ๐๐๐๐ = 104 and โ =0.125
2๐ for ๐ = 3,โฆ , 7.
The efficiency curves are shown in Figures 1-5, where problems 1-5 are
tested for a very large interval ๐๐๐๐ = 10000.
It is observed that TF-LMMC(3) lies below all of the other methods efficiency
graphs. It prove that TF-LMMC(3) is superior compared to the other existing
methods in the literature.
LMMC(3) is as competitive as the other methods for solving oscillatory
problems, though LMMC(3) is method of order three which is one order less
compared to the other methods in comparison.
Although explicit methods such as EHM3(4) and RKN3(4) needs less time to
do the computation, the methods are less efficient compared to the other
implicit and fitted methods.
Implicit methods: DIRKN(HS), DIRKN3(4), SIHM3(4) and phase-fitted
method: PFRKN4(4), need more time to do the computation, thus less
efficient compared to TF-LMMC(3). For all of the problems tested,
Problem 1(An almost Periodic Orbit problem studied by Stiefel and Bettis [19]) ๐ฆ1" ๐ฅ + ๐ฆ1 ๐ฅ = 0.001cos ๐ฅ , ๐ฆ1 0 = 1,๐ฆ1โฒ 0 = 0, ๐ฆ2" ๐ฅ + ๐ฆ2 ๐ฅ = 0.001sin ๐ฅ , ๐ฆ2 0 = 0,๐ฆ2โฒ 0 = 0.9995, Exact solution is ๐ฆ1 = cos ๐ฅ + 0.0005๐ฅsin ๐ฅ , and ๐ฆ2 = sin ๐ฅ โ0.0005๐ฅcos ๐ฅ . The fitted frequency is ๐ฃ = 1.
Figure 6: The efficiency curve for TF-LMMC(4) for Problem 1
with ๐๐๐๐ = 104 and โ =0.9
2๐ for ๐ = 1,โฆ , 5.
The following are efficiency curves TF-LMMC(4) Method.
Problem 2 (Inhomogeneous system
by Lambert and Watson [20])
๐2๐ฆ1 ๐ฅ
๐๐ก2= โ๐ฃ2๐ฆ1 ๐ฅ + ๐ฃ2๐ ๐ฅ + ๐" ๐ฅ ,
๐ฆ1 0 = ๐ + ๐ 0 , ๐ฆ1โฒ 0 = ๐โฒ 0 ,
๐2๐ฆ2 ๐ฅ
๐๐ก2= โ๐ฃ2๐ฆ2 ๐ฅ + ๐ฃ2๐ ๐ฅ + ๐" ๐ฅ
๐ฆ2 0 = ๐ 0 , ๐ฆ2โฒ 0 = ๐ฃ๐ + ๐โฒ 0
Exact solution is ๐ฆ1 ๐ฅ = ๐cos ๐ฃ๐ฅ +
๐ ๐ฅ ,๐ฆ2 ๐ฅ = ๐sin ๐ฃ๐ฅ + ๐ ๐ฅ , ๐ ๐ฅ is chosen
to be ๐โ0.05๐ฅ and parameters ๐ฃ and ๐
are 20 and 0.1 respectively.
Figure8: The efficiency curve for TF-LMMC(4) for
Problem 2 with ๐๐๐๐ = 104 and โ =0.125
2๐ for
๐ = 2,โฆ , 6.
Problem 3 (Inhomogeneous system studied by Franco [12])
๐ฆ" ๐ฅ =1012
โ992
โ992
1012
๐ฆ ๐ฅ = ๐ฟ932cos(2๐ฅ) โ99
2sin 2๐ฅ
932sin(2๐ฅ) โ
992cos(2๐ฅ)
๐ฆ 0 =โ1 + ๐ฟ
1, ๐ฆโฒ 0 =
โ1010 + 2๐ฟ
for ๐ฟ = 10โ3. Exact solution ๐ฆ ๐ก =โcos 10๐ฅ โ sin 10๐ฅ + ๐ฟcos(2๐ฅ)
cos 10๐ฅ + sin 10๐ฅ + ๐ฟsin(2๐ฅ).
The Eigen-value of the problem are ๐ฃ = 10 and ๐ฃ = 1. The fitted frequency is chosen to be ๐ฃ = 10 because it is dominant than ๐ฃ = 1.
Figure 9: The efficiency curve for TF-LMMC(4) for
Problem 3 with ๐๐๐๐ = 104 and โ =0.125
2๐ for
๐ = 1,โฆ , 5.
Problem 4 (Homogenous given
in Attili et.al [2])
๐ฆ" ๐ฅ = โ64๐ฆ ๐ฅ ,๐ฆ 0 = 14 ,
๐ฆโฒ 0 = โ 12 .
Exact solution is ๐ฆ = 1716 sin 8๐ฅ + ๐ ,
๐ = ๐ โ tanโ1 4 .
The fitted frequency is ๐ฃ = 8.
Figure 9: The efficiency curve for TF-LMMC(4) for
Problem 4 with ๐๐๐๐ = 104 and โ =0.1
2๐ for
๐ = 3,โฆ , 7.
Problem 5 (Inhomogeneous equation
studied by Papadopoulos et.al [15])
๐ฆโฒโฒ(๐ฅ) = โ๐2๐ฆ(๐ฅ) +(๐2 โ
1)sin(๐ฅ),
๐ฆ(0) = 1, ๐ฆโ(0) = ๐ + 1.
Exact solution is y(๐ฅ) = cos(๐๐ฅ) +
sin(๐๐ฅ) + sin(๐ฅ).
The fitted frequency is ๐ = 10.
Figure 10: The efficiency curve for TF-LMMC(4) for
Problem 5 with ๐๐๐๐ = 104 and โ =0.125
2๐ for ๐ = 3, โฆ , 7.
From the efficiency curves in Figures 6-10, we observed that methods with fitting
properties have smaller error.
DIRKN methods have more functions evaluation than LMMs and SIHMs, hence more
computational time is required to implement DIRKN methods.
It is shown that TF-LMMC(4) have better performance compared to the original
method LMMC(4), and superior compared to all the existing methods in comparisons.
Conclusion
โข In this research, we developed linear multistep methods with extra
derivatives using collocation technique of order three (LMMC(3)) and four
(LMMC(4))
โข Trigonometrically-Fitted the Linear Multistep Method With Collocation
technique of Order Three (TF-LMMC(3)) and four (TF-LMMC(4)) respectively.
โข Numerical results for LMMC(3) which has order three is as comparable as other
existing methods which are of order four
โข Numerical results for LMMC(4) which is order four is slightly better than other
existing methods in comparisons.
โข Hence having extra derivtives in the multistep method do improved the accuracy
of the methods.
โข TF-LMMC(3) and TF-LMMC(4) are clearly superior in solving special second order
ODEs with oscillatory solutions since it involves lesser computational time and
better accuracy
โข We can conclude that Trigonometrically-fitting the methods improved the effiency
of the methods for integrating oscillatary problems.
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