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Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems
Alexandria Engineering Journal (2017) xxx, xxx–xxx
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems S.C. Shiralashetti *, S. Kumbinarasaiah Department of Mathematics, Karnatak University, Dharwad 580003, India Received 5 February 2017; revised 10 July 2017; accepted 27 July 2017
KEYWORDS Hermite wavelets; Nonlinear singular initial value problems; Collocation method; Operational matrix
Abstract In this paper, new operational matrix of integration is generated by using Hermite wavelets. By aid of these matrices, Hermite wavelets operational matrix method (HWOMM) is developed for second ordered nonlinear singular initial value problems. Properties of Hermite wavelets are used to convert the differential equations into system of algebraic equations which can be efficiently solved by suitable solvers. Illustrative numerical problems are considered to demonstrate the applicability of the proposed technique and those results are comparing favourably with the exact solutions and errors. Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Wavelets are localized waves or small wave. Instead of oscillating forever, they drop to zero. Wavelets theory is a newly emerging area in mathematical research field. It has been applied in engineering disciplines; such as signal analysis for wave form representation and segmentations, harmonic analysis and time frequency analysis etc. Wavelets permit the accurate representation of a variety of functions and operators. Wavelets are assumed as basis functions wi;j ðxÞ in continuous time. Special feature of the wavelets basis is that all functions wi;j ðxÞ are constructed from a single mother wavelet wðxÞ which is a small pulse. Usually set of linearly independent
* Corresponding author. E-mail address: [email protected] (S.C. Shiralashetti). Peer review under responsibility of Faculty of Engineering, Alexandria University.
functions created by translation and dilation of mother wavelet. Many practical and physical problems in the field of science and engineering are formulated as initial value problems. The nonlinear singular initial value problems are crucial in many fields of science and engineering. For example gas dynamics, atomic structures, chemical reactions and nuclear physics [1]. In many cases, extracting the analytical solutions for singular initial value problems from analytical methods is always not possible. For such problems we disentomb numerical techniques such as, Galerkin method, Tau method, operation matrix of integration method, collocation method, series solution methods and operational matrix of differentiation etc. Operational matrix of integration with collocation method is one of the most widely used numerical schemes to solve differential equations. This approach is based on converting differential equation into integral equations using operational matrix of integration eliminating the integral operator in order to reduce the problem into system of algebraic equations.
http://dx.doi.org/10.1016/j.aej.2017.07.014 1110-0168 Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: S.C. Shiralashetti, S. Kumbinarasaiah, Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.07.014
2
S.C. Shiralashetti, S. Kumbinarasaiah
The purpose of this paper is to develop the new Hermite wavelets operational matrix of integration collocation method as an alternative method to solve nonlinear singular initial value problems of the form (1) for a given set of conditions. Here we bring all solutions of (1) under the Hermite space and present method will contribute good accuracy for linear singular problems. So that, we concentrate only nonlinear problems. That is, we are expressing solutions of nonlinear singular initial value problems in terms of Hermite wavelets basis. Let ða; bÞ R be an interval and pðxÞ; qðxÞ; rðx; yÞ : ða; bÞ ! R be continuous real valued functions. Throughout this paper we consider the arbitrary nonlinear singular second order equations given by 00
0
y ðxÞ þ pðxÞy ðxÞ þ qðxÞyðxÞ ¼ rðx; yÞ; a < x < b:
ð1Þ
subjected to following initial conditions: yð0Þ ¼ a1 ; y0 ð0Þ ¼ a2 ;
ð2Þ
where a1 ; a2 are known constants. There are some literature available on operational matrix of integration and some other techniques on singular initial value problems. Adomian decomposition method [2], Differential transformation method [3], Haar wavelet collocation method [4], New ultraspherical wavelet collocation method [5], Laguerre wavelets method [6], Physicists Hermite wavelet method [7], Hermite wavelets method for boundary value problem [8], Legendre wavelet method [9], An Accurate Numerical Algorithm [10], Finite Element Methods [11], Haar wavelet series solution [12], Chebyshev wavelet operational matrix of integration [13], The Legendre wavelets operational matrix of integration [14], A Numerical Approach for Solving A Class of The Nonlinear Lane-Emden Type Equations [15], A numerical approach for solving the high order linear singular differential-difference equations [16], An Improved Bessel Collocation Method [17], A Numerical Scheme for Solutions of a class of non-linear differential equations [18], Exponential Collocation Method For Solutions of Singularity Perturbed Delay Differential Equations [19] and also, some methods on solving partial differential equations [20–30]. There are two different approaches for solving differential equations, one approach is based on converting differential equations into integral equations then eliminate the integral operator by operational matrix of integration [31]. Another method is based on the operational matrix of derivative, here we solve the differential equations by converting into system of linear or nonlinear algebraic equations [32,33]. A vast amount of literature are available on operational matrix of integration by Legendre wavelet, Chebyshev wavelet, Laguerre wavelet but there is no literature on Hermite wavelet operational matrix of integration this impetused us to generate operational matrix of integration using Hermite wavelet. The rest of this paper is organized as follows. In Section 2, we introduce the properties of Hermite wavelets. Section 3, function approximation. Section 4, having the procedure for the operational matrix of integration. Section 5, reveals description of the proposed method. Section 6, numerical examples to test the efficiency of proposed method and conclusion is drawn in Section 7.
wavelet. When the dilation parameter a and translation parameter b varies continuously, we have the following family of continuous wavelets: xb wa;b ðxÞ ¼ jaj1=2 w ; 8 a; b 2 R; a–0: a If we restrict the parameters a and b to discrete values as k a ¼ ak 0 ; b ¼ nb0 a0 ; a0 > 1; b0 > 0: We have the following family of discrete wavelets wk;n ðxÞ ¼ jaj1=2 wðak0 x nb0 Þ;
where wk;n form a wavelet basis for L2 ðRÞ. In particular, when a0 ¼ 2 and b0 ¼ 1, then wk;n ðxÞ forms an orthonormal basis. Hermite wavelets are defined as [8], ( kþ1 2p2ffiffi n Hm ð2k x 2n þ 1Þ; 2n1 k1 6 x < k1 p 2 wn;m ðxÞ ¼ ð3Þ 0; otherwise where m ¼ 0; 1; . . . ; M 1: Here Hm ðxÞ is Hermite polynomials of degree m with respect to weight function pffiffiffiffiffiffiffiffiffiffiffiffiffi WðxÞ ¼ 1 x2 on the real line R and satisfies the following recurrence formula H0 ðxÞ ¼ 1, H1 ðxÞ ¼ 2x, Hmþ2 ðxÞ ¼ 2xHmþ1 ðxÞ 2ðm þ 1ÞHm ðxÞ; m ¼ 0; 1; 2; . . . :
Wavelets constitute a family of functions constructed from dilation and translation of a single function called mother
ð4Þ
3. Function approximation and convergence analysis We would like to bring a solution function yðxÞ under Hermite space by approximating yðxÞ by elements of Hermite wavelet basis as follows, yðxÞ ¼
1 X 1 X Cn;m wn;m ðxÞ
ð5Þ
n¼1 m¼0
where wn;m ðxÞ is given in Eq. (3). We approximate yðxÞ by truncating the series represented in Eq. (5) as, yðxÞ
2k1 M1 X X Cn;m wn;m ðxÞ ¼ CT wðxÞ
ð6Þ
n¼1 m¼0
where C and wðxÞ are 2k1 M 1 matrix,
h i CT ¼ C1;0 ;. .. C1;M1 ; C2;0 ;. .. C2;M1 ; ... C2k1;0 ; .. .C2k1 ;M1
ð7Þ
h i wðxÞ ¼ w1;0 ; . . . w1;M1 ; w2;0 ; . . . w2;M1 ; . . . w2k1 ;0 ; . . . w2k1;M1 ð8Þ Theorem 1. A continuous function y(x) in H2 ½0; 1Þ defined on ½0; 1Þ be bounded, then the Hermite wavelets expansion of y(x) converges to it. Proof. Let y(x) be a bounded real valued function on ½0; 1Þ. The Hermite coefficients of continuous functions y(x) is defined as, Z
2. Hermite wavelets
8a; b 2 R; a–0;
1
Cn;m ¼ Z
0
¼ I
yðxÞwn;m dx kþ1 22 n1 n k yðxÞ pffiffiffi Hm ð2 x 2n þ 1Þdx; where I ¼ k1 ; k1 p 2 2
Please cite this article in press as: S.C. Shiralashetti, S. Kumbinarasaiah, Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.07.014
Hermite wavelets operational matrix of integration for the numerical solution Z
Put 2k x 2n þ 1 ¼ z, kþ1
22 ¼ pffiffiffi p kþ1
2 2 ¼ pffiffiffi p
2 w1;3 ðxÞdx ¼ pffiffiffi ð16x4 32x3 þ 18x2 2xÞ p 1 1 1 ¼ ; 0; ; 0; ; 0 w6 ðxÞ 8 8 16
0
z 1 þ 2n Hm ðzÞ2k dz y k 2 1
Z
x
1
Z
z 1 þ 2n Hm ðzÞdz y k 2 1
Z
1
2 256 5 320 3 x 128x4 þ x 32x2 þ 2x w1;4 ðxÞdx ¼ pffiffiffi 3 p 5 1 1 1 4 ; 0; 0; ; 0; w ¼ 15 12 20 6
x
0
Using GMVT for integrals, Z 1 kþ1 2 2 w 1 þ 2n ¼ pffiffiffi y Hm ðzÞdz; p 2k 1 Put
R1 1
for some w 2 ð1; 1Þ
Z
x
0
Hm ðzÞdz ¼ h,
kþ1 2 2 w 1 þ 2n h jcn;m j ¼ pffiffiffi y p 2k Since y is bounded, therefore, n;m¼0 Cn;m is absolutely convergent. Hence the Hermite series expansion of y(x) converges uniformly. h 4. Operational matrix of integration (OMI)
2 512 6 800 3 w1;5 ðxÞdx ¼ pffiffiffi x 512x5 þ 560x4 x þ 50x2 2x 3 p 3 1 1 1 ¼ ;0; 0; 0; ; 0 w6 ðxÞ þ w1;6 ðxÞ 24 16 24
Thus
Z P1
x
6 ðxÞ w6 ðxÞdx ¼ P66 w6 ðxÞ þ w
2 w1;1 ðxÞ ¼ pffiffiffi ð4x 2Þ; p 2 w1;2 ðxÞ ¼ pffiffiffi ð16x2 16x þ 2Þ; p 2 w1;3 ðxÞ ¼ pffiffiffi ð64x3 96x2 þ 36x 2Þ; p 2 w1;4 ðxÞ ¼ pffiffiffi ð256x4 512x3 þ 320x2 64x þ 2Þ; p 2 w1;5 ðxÞ ¼ pffiffiffi ð1024x5 2560x4 þ 2240x3 800x2 þ 100x 2Þ: p Let w6 ðxÞ ¼ ððw1;0 ðxÞ; w1;1 ðxÞ;w1;2 ðxÞ; w1;3 ðxÞ; w1;4 ðxÞ;w1;5 ðxÞÞÞT . By integrating above basis functions with respect to x from 0 to x and expressing in matrix form, we obtain,
Z
x
0
Z
x
0
Z 0
x
2 1 1 w1;0 ðxÞdx ¼ pffiffiffi x ¼ ; ; 0; 0; 0; 0 w6 ðxÞ 2 4 p 2 1 1 ; 0; ; 0; 0; 0 w6 ðxÞ w1;1 ðxÞdx ¼ pffiffiffi ð2x2 2xÞ ¼ 4 8 p 2 16 3 1 1 1 x 8x2 þ 2x ¼ ; ; 0; ; 0; 0 w6 ðxÞ w1;2 ðxÞdx ¼ pffiffiffi 3 4 12 p 3
ð9Þ
0
where
2
1 2 6 1 6 4 6 1 6 6 3 6 1 6 8 6 6 1 6 15 6 1 4 24
P66 ¼
In this section, we give the structure of OMI for Hermite wavelets, particularly at k ¼ 1 and M ¼ 6. Here we consider six basis functions on ½0; 1Þ are given by, 2 w1;0 ðxÞ ¼ pffiffiffi ; p
3
Z
0
0
1 8
0 0
0 0
1 4
0
1 12
0
0 0 0
1 8
0
1 16
0 0
1 12
0
0
1 16
3 0 2 7 07 6 7 6 07 6 7 6 7 6 ðxÞ ¼ 6 0 7 and w 6 7 6 1 7 6 20 7 4 7 05
0 0 0 0 0
3 7 7 7 7 7 7 7 7 5
1 w ðxÞ 24 1;6
Again double integration of above six basis is given by, Z x 2 x2 3 1 1 ; ; ; 0; 0; 0 w6 ðxÞ ¼ w1;0 ðxÞdxdx ¼ pffiffiffi 16 8 32 p 2 0
x
0
Z
x
Z
0
Z
x
Z
x
0
Z
0
2 2 1 3 1 w1;1 ðxÞdxdx ¼ pffiffiffi x3 x2 ¼ ; ; 0; ; 0; 0 w6 ðxÞ 6 32 96 p 3 x
0
x
Z
0
Z
1 4
x
0
2 4 4 8 3 2 p ffiffiffi x x þx w1;2 ðxÞdxdx ¼ 3 p 3 3 1 1 1 ; ; ; 0; ; 0 w6 ðxÞ ¼ 32 12 24 192 2 16 5 x 8x4 þ 6x3 x2 w1;3 ðxÞdxdx ¼ pffiffiffi p 5 1 1 1 1 ; ; 0; ; 0; w ðxÞ ¼ 10 16 64 320 6
2 128 6 128 5 80 4 32 3 x x þ x x þ x2 w1;4 ðxÞdxdx ¼ pffiffiffi 5 3 3 p 15 0 0 1 1 1 1 ¼ ; ; ;0; ;0 w6 ðxÞ 24 60 96 120 x
Z 0
x
Z
x
Z
x
w1;5 ðxÞdxdx 2 512 7 256 6 200 4 50 3 ¼ pffiffiffi x x þ 112x5 x þ x x2 3 3 3 p 21 1 1 1 1 1 ¼ ; ; 0; ; 0; w ðxÞ þ w1;6 ðxÞ: 42 96 192 192 6 24 0
Please cite this article in press as: S.C. Shiralashetti, S. Kumbinarasaiah, Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.07.014
4
S.C. Shiralashetti, S. Kumbinarasaiah
Z 0
ð10Þ
0
where
P066
Z
respect to x from 0 to x together with initial conditions given in (2) and relation (9) and (10) we obtain,
Hence Z x 0 ðxÞ: w6 ðxÞdxdx ¼ P066 w6 ðxÞ þ w 6
x
2
3 16
6 1 6 6 6 3 6 6 32 6 ¼ 6 101 6 6 1 6 24 6 1 4 42
1 8 3 32 1 12 1 16 1 60 1 96
1 32
0
0
1 96
0 0
1 24
0
1 192
0
1 64
0
1 96
0
1 120
0
1 192
0
0 ðxÞÞ: yðxÞ ¼ a1 þ xa2 þ AT ðP0 wðxÞ þ w
3
2 3 0 7 7 6 7 0 7 6 7 7 6 7 7 6 7 0 7 0 1 6 7 and w6 ðxÞ ¼ 6 7 320 7 7 0 6 7 7 0 7 6 1 7 4 480 w1;6 ðxÞ 5 7 1 5 1 192 w ðxÞ 672 1;7 0 0 0
Similarly if we considered seven basis functions then x
7 ðxÞ: w7 ðxÞdx ¼ P77 w7 ðxÞ þ w
where
P77 ¼
1 2 6 1 64 6 1 63 6 61 68 6 1 6 15 6 61 6 24 6 1 4 35
1 4
0
0
1 8
0 0
1 4
0
1 12
0 0 0
0 0
1 8
0
1 16
0 0 0 0
0
1 12
0
1 20
0 0
0 0
0 0
1 16
0
0
1 20
3 2 0 07 6 7 6 7 6 07 6 7 6 7 07 6 ðxÞ ¼ and w 6 7 7 6 07 6 7 6 1 7 6 24 7 4 7 5 0
3
0
7 7 7 7 7 7 7 7 7 7 7 5
0 0 0 0 0 1 w ðxÞ 28 1;7
P077
¼
3 16 6 1 66 6 3 6 32 6 61 6 10 6 1 6 24 6 61 6 42 6 1 4 65
1 8 3 32 1 12 1 16 1 60 1 96 1 140
1 32
0
0
1 96
0 0
0 0 0
0 0 0
3
Then we can analyse the unknown vector A by solving this system through the well-known Newton iterative method using Matlab. The approximate solution can be easily recovered by inserting A into the corresponding expression of y(x). 6. Numerical experiments Test Problem 1. Consider the singular initial value problem [6], ð12Þ
yð0Þ ¼ 1; y0 ð0Þ ¼ 0; 2
3
0 7 6 7 7 0 6 7 7 1 1 6 7 7 0 24 192 6 7 7 0 6 7 7 1 0 1 0 0 6 7 7 0 64 320 0 ðxÞ ¼ and w 6 7 7 7 1 1 1 7 6 7 0 120 0 480 96 6 7 7 0 6 7 7 1 1 0 192 0 192 0 7 6 1 7 w ðxÞ 4 672 1;7 5 7 1 1 5 0 0 320 0 280 1 w ðxÞ 896 1;8
In this way we can generate the other matrix by consuming more number of basis functions. So when dealing with the problems, we need to pre-calculate the corresponding opera 0 ðxÞ for different w tional matrix of integration P; P0 and wðxÞ; values of N. 5. Description of the proposed method In this section, we will use above discussed OMI together with collocation method to solve nonlinear singular initial value problems, we assume that, y00 ¼ AT wðxÞ
Now discrete it using the following collocation points xi ¼ 2i1 ; i ¼ 1; 2; . . . ; N thus we get a system of N equations 2N with N unknowns. Here N represents size of an operational matrix of integration and corresponding equation is of the following form,
subjected to the initial conditions
0
2
AT wðxÞ þ pðxÞða2 þ AT ðPwðxÞ þ wðxÞÞÞ þ qðxÞða1 þ xa2 0 ðxÞÞÞ ¼ rðx; yÞ: þ AT ðP0 wðxÞ þ w
1 y00 ðxÞ þ y0 y3 þ 3y5 ; 0 < x < 1 x
Double integration is, Z xZ x 0 ðxÞ w6 ðxÞdxdx ¼ P077 w7 ðxÞ þ w 7 0
then we get yðxÞ; y0 ðxÞ; y00 ðxÞ are in terms of polynomial basis and their integrals. Now substitute yðxÞ; y0 ðxÞ; y00 ðxÞ are in (1) we get,
i ÞÞÞ þ qðxi Þða1 þ xi a2 AT wðxi Þ þ pðxi Þða2 þ AT ðPwðxi Þ þ wðx T 0 0 þ A ðP wðxi Þ þ w ðxi ÞÞÞ ¼ rðxi ; yðxi ÞÞ: i ¼ 1; 2; :::; N:
0
2
y0 ðxÞ ¼ a2 þ AT ðPwðxÞ þ wðxÞÞ:
ð11Þ
where A is unknown vector which should be determined and wðxÞ is the vector defined in (8), Integrate (11) twice with
ð13Þ 1 2
the exact solution is yðxÞ ¼ ð1 þ x2 Þ . Method of implementation for N ¼ 6: We assume that, y00 ¼ AT wðxÞ
ð14Þ
where wðxÞ ¼ w6 ðxÞ ¼ ððw1;0 ðxÞ;w1;1 ðxÞ; w1;2 ðxÞ; w1;3 ðxÞ;w1;4 ðxÞ; w1;5 ðxÞÞÞT & A ¼ ða0 ; a1 ;a2 ;a3 ;a4 ; a5 Þ: Integrate (14) with respect to x from 0 to x together with initial conditions given in (13), we obtain, y0 ðxÞ ¼ AT ðPwðxÞ þ wðxÞÞ: 0 ðxÞÞ: yðxÞ ¼ 1 þ AT ðP0 wðxÞ þ w Now substitute yðxÞ; y0 ðxÞ; y00 ðxÞ are in (12) and discretize it using the following collocation points xi ¼ 2i1 ; i ¼ 1; 2; . . . ; 6, 12 thus we get 6 system of equations with 6 unknowns. After solving this system by newton iteration method, we get the unknown vector A is, 3 2 a0 ¼ 0:3285 7 6 a1 ¼ 0:2908 7 6 7 6 7 6 a2 ¼ 0:0255 7 A¼6 7 6 a ¼ 0:0303 3 7 6 7 6 5 4 a4 ¼ 0:0118 a5 ¼ 2:1135 104
Please cite this article in press as: S.C. Shiralashetti, S. Kumbinarasaiah, Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.07.014
Hermite wavelets operational matrix of integration for the numerical solution Table 1
5
Comparison of absolute errors (AE) for the test problem 1, (k = 1).
x
Exact solution
AE at N = 6
AE at N = 7
AE at N = 8
AE at N = 9
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.99503719020 0.98058067569 0.95782628522 0.92847669088 0.89442719099 0.85749292571 0.81923192051 0.78086880944 0.74329414624
5
6
7
1:8966 108 1:0083 108 1:1329 108 5:5787 109 4:7519 109 9:1742 109 1:0085 109 3:9847 109 4:8067 108
1:8706 10 1:9982 105 1:0719 105 6:5087 106 3:8619 106 9:9962 107 4:1085 106 4:2547 106 9:8647 106
5:0872 10 3:9138 107 3:1663 107 2:2954 107 1:4662 107 6:8963 108 1:9866 109 5:8784 108 1:2314 107
2:5225 10 1:9417 106 8:0068 107 5:4375 107 4:4979 108 5:0408 107 5:1814 107 1:2857 106 7:3953 107
−5
2
x 10
AE at N=6 AE at N=7 AE at N=8 AE at N=9
1.8
Error analysis
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
Fig. 1
Physical interpretation of Absolute error analysis for the test problem 1 for different values of N.
1 Exact solution Present method
0.95
Solution
0.9 0.85 0.8 0.75 0.7 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
Fig. 2
Graphical interpretation of Exact solution and Present method solution for the test problem 1.
Substitute these values in yðxÞ, we ge the approximate solution is, yðxÞ ¼ 0:0058x7 þ 0:1344x6 0:4785x5 þ 0:6079x4 0:0566x3 0:4944x2 þ 6:8439 1021 x þ 0:9999: by applying the technique described in Section 5 with different values of N. Table 1 and Fig. 1 represents the comparison between the absolute error of approximate solution with analytical solution. Graphical representation of Exact and present method solution for the test problem 1 is drawn in Fig. 2.
Test Problem 2. Consider the following nonlinear LaneEmden equation [4]; 2 0 y00 ðxÞ þ y ðxÞ þ y5 ¼ 0; 0 < x < 1 x subjected to the initial conditions, yð0Þ ¼ 1; y0 ð0Þ ¼ 0;
Please cite this article in press as: S.C. Shiralashetti, S. Kumbinarasaiah, Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.07.014
6
S.C. Shiralashetti, S. Kumbinarasaiah Table 2
Comparison of absolute errors (AE) for the test problem 2, (k = 1).
x
Exact solution
AE at N = 6
AE at N = 7
AE at N = 8
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.99833748845 0.99339926779 0.98532927816 0.97435470369 0.96076892283 0.94491118252 0.92714554082 0.90784129900 0.88735650941
8
8
9
9:9165 10 4:5437 108 5:1208 108 5:0684 108 3:5157 108 7:0668 108 6:7098 108 2:0163 109 1:3919 107
3:6989 10 1:4473 108 2:5323 109 2:5781 109 6:6328 109 8:9403 109 6:9532 109 1:1810 108 6:5729 109
AE at N = 9 3:6640 1011 3:4148 1010 3:6797 1011 3:4785 1011 3:2490 1011 3:3358 1011 3:2694 1011 2:9498 1011 2:8965 1010
4:5191 10 2:4641 109 5:3187 1010 1:1649 1010 1:4761 1010 1:7220 1010 1:8055 1010 1:6746 1010 1:3945 1010
−7
1.4
x 10
AE at N=6 AE at N=7 AE at N=8 AE at N=9
Error analysis
1.2 1 0.8 0.6 0.4 0.2 0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
Fig. 3
Physical interpretation of Absolute error analysis for the test problem 2 for different values of N.
1 Exact solution Present method
0.98
Solution
0.96 0.94 0.92 0.9 0.88 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
Fig. 4
Graphical interpretation of Exact solution and present method solution for the test problem 2.
12 2 the exact solution is yðxÞ ¼ 1 þ x3 . We solve this equation by the HWOM method with different values of N. Table 2 and Fig. 3 reflects the comparison between the absolute error of approximate solution with analytical solution for different values of N. Graphical representation of Exact and present method solution is drawn in Fig. 4. Test Problem 3. Consider the singular initial value problem [34];
y00 ðxÞ 2e y ¼ 0; 0 < x < 1
ð15Þ
subjected to the initial conditions yð0Þ ¼ 0; y0 ð0Þ ¼ 0;
ð16Þ
the exact solution is yðxÞ ¼ 2lnðcosxÞ, Method of implementation for N ¼ 6: We assume that, y00 ¼ AT wðxÞ
ð17Þ
Please cite this article in press as: S.C. Shiralashetti, S. Kumbinarasaiah, Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.07.014
Hermite wavelets operational matrix of integration for the numerical solution
7
Comparison of absolute errors (AE) for the test problem 3, (k = 1).
Table 3 x
Exact solution
AE at N = 6
AE at N = 7
AE at N = 8
AE at N = 9
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.99833748845 0.99339926779 0.98532927816 0.97435470369 0.96076892283 0.94491118252 0.92714554082 0.90784129900 0.88735650941
5
5
6
2:2974 106 4:5037 106 6:8418 106 9:3196 106 1:2023 105 1:5040 105 1:8503 105 2:2619 105 2:7587 105
2:1447 10 4:3130 105 6:3466 105 8:6888 105 1:1148 104 1:3904 104 1:7177 104 2:0778 104 2:5754 104
6:0285 10 1:2695 104 1:8170 104 2:4653 104 3:1948 104 3:9461 104 4:8481 104 6:0067 104 7:1158 104
7:6580 10 1:5078 105 2:2661 105 3:0932 105 3:9754 105 4:9816 105 6:1115 105 7:4870 105 9:1294 105
−4
8
x 10
AE at N=6 AE at N=7 AE at N=8 AE at N=9
7
Error analysis
6 5 4 3 2 1 0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
Fig. 5
Physical interpretation of Absolute error analysis for test problem 3 for different values of N.
1.4 Exact solution Present method solution
1.2
Solution
1 0.8 0.6 0.4 0.2 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
Fig. 6
Graphical interpretation of Exact solution and present method solution for the test problem 3.
where wðxÞ ¼ w6 ðxÞ ¼ ððw1;0 ðxÞ; w1;1 ðxÞ; w1;2 ðxÞ; w1;3 ðxÞ; w1;4 ðxÞ; T
w1;5 ðxÞÞÞ and A ¼ ða0 ; a1 ; a2 ; a3 ; a4 ; a5 Þ: Integrate (17) with respect to x from 0 to x together with initial conditions given in (16) we obtain, y0 ðxÞ ¼ AT ðPwðxÞ þ wðxÞÞ:
0 ðxÞÞ: yðxÞ ¼ AT ðP0 wðxÞ þ w Now substitute yðxÞ; y0 ðxÞ; y00 ðxÞ are in (15) and discretize it ; i ¼ 1; 2; . . . ; 6 using the following collocation points xi ¼ 2i1 12 thus we get 6 system of equations with 6 unknowns. After solving this system by newton iteration method we get unknown vector A as,
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8
S.C. Shiralashetti, S. Kumbinarasaiah Table 3.1
Comparison of absolute errors (AE) for the test problem 3, (k = 2).
x
Exact solution
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.99833748845 0.99339926779 0.98532927816 0.97435470369 0.96076892283 0.94491118252 0.92714554082 0.90784129900 0.88735650941
Table 4
AE at N = 8 (M = 4) 3.2405 6.6123 8.9627 1.2334 1.6237 1.8461 2.6442 8.6431 3.4509
AE at N = 10 (M = 5)
5
10 105 105 104 104 104 104 104 103
3.2507 6.7550 9.4419 1.2846 1.6732 2.0306 2.5172 3.2727 2.1403
AE at N = 12 (M = 6)
5
10 105 105 104 104 104 104 104 104
6.0285 1.2695 1.8170 2.4653 3.1948 3.9461 4.8481 6.0067 7.1158
105 104 104 104 104 104 104 104 104
Comparison of absolute errors (AE) for the test problem 4 (k = 1).
x
Exact solution
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.01990066170 0.07844142630 0.17235539248 0.29684001023 0.44628710262 0.61496939949 0.79755223991 0.98939248367 1.18665369055
AE at N = 6
AE at N = 7
AE at N = 8
5
6
6
2:2300 10 1:2130 105 6:1607 106 7:8887 106 6:3569 106 1:0604 105 1:0563 105 4:2378 106 1:4494 106
1:4552 10 4:7418 108 1:0713 106 7:2831 107 1:0018 106 1:0623 106 5:1912 107 1:4039 106 1:0590 107
AE at N = 9 3:6589 107 5:4640 108 9:7692 1011 4:2075 108 5:9709 108 6:9569 108 7:4787 108 6:7446 108 1:0572 107
1:2575 10 3:7267 107 3:6764 108 5:0634 108 1:7109 107 1:8601 107 2:3240 107 2:2325 107 2:1741 107
−5
2.5
x 10
AE at N=6 AE at N=7 AE at N=8 AE at N=9
Error analysis
2
1.5
1
0.5
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
Physical interpretation of Absolute error analysis for the test problem 4 for different values of N.
Fig. 7
2 6 6 6 6 6 6 A¼6 6 6 6 6 6 4
a0 ¼ 3:0233 a1 ¼ 0:9359 a2 ¼ 0:3929 a3 ¼ 0:1169 a4 ¼ 0:0320
3 7 7 7 7 7 7 7 7 7 7 7 7 5
a5 ¼ 0:0086 Substitute these values in yðxÞ; we ge approximate solution as,
yðxÞ ¼ 0:2377x7 0:5237x6 þ 0:5892x5 0:1508x4 þ 0:0903x3 þ 0:9876x2 2:7984 1019 x þ 2:8555 1021 : by applying the technique described in Section 5 with different values of N. Table 3 and Fig. 5 represents the comparison between the absolute error of approximate solution with analytical solution. Graphical representation of Exact and present method solution for test problem 3 is drawn in Fig. 6. In the present method, for the domain wise different values of k and fixed values of M, we obtain the same accuracy in the solution. This can be observed in the Tables 3.1and 3. For
Please cite this article in press as: S.C. Shiralashetti, S. Kumbinarasaiah, Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.07.014
Hermite wavelets operational matrix of integration for the numerical solution Table 5
9
Comparison of present and other existing methods [6,34].
x
Exact solution
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.01990066170 0.07844142630 0.17235539248 0.29684001023 0.44628710262 0.61496939949 0.79755223991 0.98939248367 1.18665369055
AE in [34]
AE in [6] at k = 2 and M = 5 5
AE at N = 9 by present method 3:6589 107 5:4640 108 9:7692 1011 4:2075 108 5:9709 108 6:9569 108 7:4787 108 6:7446 108 1:0572 107
6
2:0000 10 2:4999 105 2:4999 105 1:8999 105 2:4000 105 2:9000 105 2:5999 104 1:4699 104 8:6199 103
1:1903 10 4:0724 107 3:7227 107 1:3771 107 2:1195 107 4:0909 106 7:3090 106 9:8931 106 1:1716 105
0 Exact solution BPOM solution LWM solution Present method at N=9
−0.2
Solution
−0.4 −0.6 −0.8 −1 −1.2 −1.4 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
Fig. 8 Graphical interpretation of the Exact solution, Present method solution, BPOM solution [34] and LEM solution [6] for the test problem 4.
k = 1 and M = 6(N = 6), we obtain Four decimal accuracy for the Example 3 which can be seen in the Table 3. Same accuracy can be observed in the case of k = 2 and M = 6 (N = 12) in Table 3.1. Hence, solving differential equations using any one of the continuous polynomial wavelets basis is sufficient no need to use other wavelets, which are having continuous polynomials as basis. Test Problem 4. Consider the following nonlinear LaneEmden initial value problem [6], y 2 y00 ðxÞ þ y0 ðxÞ þ 4 2e y þ e2 ; 0 < x < 1 x subjected to the initial conditions yð0Þ ¼ 0; y0 ð0Þ ¼ 0; the exact solution is yðxÞ ¼ 2lnð1 þ x2 Þ. We solve this equation by the present method with different values of N. Table 4 and Fig. 7 represents the comparison between the absolute error with exact solution for different values of N. Table 5 presents the comparison between the absolute error(AE) obtained by present method and other existed methods with analytic solutions. Fig. 8 shows graphical interpretation of solutions obtained by present method and other existed methods with analytical solution.
7. Conclusions In this paper, we attempted to introduce operational matrix of integration using Hermite wavelets for solving nonlinear singular initial value problems for different physical conditions which is important for the development of new research in the field of numerical analysis and beneficial for new researchers. The proposed scheme is tested on some examples and the results are quite satisfactory in comparison with the existing numerical solutions. Finally, we summarize the outcomes of this analysis is as follows: 1. The present method gives better accuracy in comparison with the other numerical techniques [6,34] available in literature. 2. This scheme is easy to implement in computer programs and we can extend this scheme for higher order also with slight modification in the present method. 3. This scheme is made easy by generalizing the operational matrix of integration. 4. This operational matrix of integration is newly generated using Hermite wavelets and method is also latest one. 5. This method is applicable for all type of initial value problems.
Please cite this article in press as: S.C. Shiralashetti, S. Kumbinarasaiah, Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.07.014
10 Acknowledgment It is a pleasure to thank the University Grants Commission (UGC), Govt. of India for the financial support under UGCSAP DRS-III for 2016-2021:F.510/3/DRS-III/2016(SAP-I) Dated: 29th Feb. 2016. References [1] M. Kumar, N. Singh, A collection of computational techniques for solving singular boundary-value problems, Adv. Eng. Softw. 40 (2009) 288–297. [2] A.M. Wazwaz, A new method for solving singular initial value problem in the second order ordinary differential equations, Appl. Math. Comput. 128 (2002) 45–57. [3] V.S. Erturk, Differential transformation method for solving differential equations of Lanr-Emden type, Math. Comput. Appl. 12 (3) (2007) 135–139. [4] S.C. Shiralashetti, A.B. Deshi, P.B. Mutalik Desai, Haar wavelet collocation method for the numerical solution of singular initial value problems, Ain Shams Eng. J. 7 (2016) 663–670. [5] E.H. Doha, W.M. Abd-Elhameed, Y.H. Youssri, New ultraspherical wavelet collocation method for solving 2nth order initial and boundary value problem, J. Egypt. Math. Soc. 24 (2016) 319–327. [6] F. Zhou, X. Xu, Numerical solutions for linear and nonlinear Boundary Value problems using Laguerre wavelets, Adv. Diff. Eq. 1–15 (2016). [7] M. Usman, S.T. Mohyud-Din, Physicists Hermites wavelets method for singular differential equation, Int. J. Adv. Appl. Math. Mech. 1 (2) (2013) 16–29. [8] A. Ayyaz, A.I. Muhammad, T.M. Syed, Hermite wavelets method for boundary value problems, Int. J. Mod. Appl. Phys. 3 (1) (2013) 38–47. [9] S.G. Venkatesh, S.K. Ayaswamy, S.R. Balachandar, Legendre wavelet method for solving initial problem of Bratu-type, Comput. Math. Appl. 63 (2012) 1287–1295. [10] A.K. Nasab, Z.P. Atabakan, A.I. Ismail, An accurate numerical algorithm for solving singular and nonsingular system of initial value problems on large interval, Mediterr. J. Math. 3 (6) (2016) 5033–5051. [11] S.C. Shiralashetti, P.B. Mutalik Desai, A.B. Deshi, Comparison of haar wavelet collocation and finite element methods for solving the typical ordinary differential equations, Int. J. Basic Sci. Appl. Comput. 1 (3) (2015) 1–11. [12] N.M. Bujurke, S.C. Shiralashetti, C.S. Salimath, An application of single-term Haar wavelet series in the solution of nonlinear oscillator equations, J. Comput. Appl. Math. 227 (2009) 234– 244. [13] E. Babolian, F. Fattahzadeh, Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration, Appl. Math. Comput. 188 (2007) 417–426. [14] M. Razzaghi, S. Yousefi, The Legendre wavelets operational matrix of integration, Int. J. Syst. Sci. 32 (4) (2001) 495–502. [15] Y. Suayip, A numerical approach for solving a class of the nonlinear lane-emden type equations arising in astrophysics, Math. Methods Appl. Sci. 34 (2011) 2218–2230. [16] Y. Suayip, A numerical approach for solving the high-order linear singular differential-difference equations, Comput. Appl. 62 (2011) 2289–2303.
S.C. Shiralashetti, S. Kumbinarasaiah [17] Y. Suayip, T.S. Mehme, An improved Bessel collocation method with a residual error function to solve a class of lane-emden differential equations. Math. Comput. Model. [18] Y. Suayip, A numerical scheme for solutions of a class of nonlinear differential equations, J. Taibah Univ. Sci. (2013). [19] Y. Suayip, Exponential collocation method for solutions of singularly perturbed delay differential equations, Abstr. Appl. Anal. (2013), ID: 493204. [20] T.M. Syed, Y. Ahmet, S. Selin, Numerical soliton solution of the Kaup-Kupershmidt equation, Int. J. Numer. Methods Heat Fluid Flow, Emerald 21 (3) (2011) 272–281. [21] T.M. Syed, Y. Ahmet, A.S. Sefa, Numericalsoliton solutions of the improved Boussinesq equation, Int. J. Numer. Meth. Heat Fluid Flow 21 (7) (2011) 822–827. [22] T.M. Syed, K. Yasir, F. Naeem, Y. Ahmet, Exp-function method for solitary and periodic solutions of Fitzhugh Nagumo equations, Int. J. Numer. Meth. Heat Fluid Flow 22 (3) (2012) 335–341. [23] T.M. Syed, N. Elham, U. Muhammad, A meshless numerical solution of the family of generalized fifth-order Korteweg-de Vries equations, Int. J. Numer. Meth. Heat Fluid Flow 22 (5) (2012) 641–658. [24] M.N. Aslam, M.A. Akbar, T.M. Syed, A novel -expansion method and its application to the Boussinesq equation, Chin. Phys. B 23 (2) (2014). [25] T.M. Syed, A.N. Muhammad, I.N. Khalida, Traveling wave solutions of seventh-order generalized KdV equations using He’s polynomials, Int. J. Nonlinear Sci. Numer. Simul. 10 (2) (2009) 223–229. [26] T.M. Syed, A.N. Muhammad, K.I. Noor, M.M. Hosseini, Variational iteration method for reformulated partial differential equations, Int. J. Nonlinear Sci. Numer. Simul. 11 (2) (2010) 87–92. [27] T.M. Syed, A.N. Muhammad, V. Asif, A.A. Eisa, Exp-function method for traveling wave solutions of nonlinear evolution equations, Appl. Math. Comput. 216 (2010) 477–483. [28] T.M. Syed, A.N. Muhammad, I.N. Khalida, Some relatively new techniques for nonlinear problems, Math. Probl. Eng. (2009) Article ID-234849. [29] T.M. Syed, A.N. Muhammad, V. Asif, Exp-function method for generalized traveling solutions of master partial differential equations, Acta Appl. Math. 104 (2) (2008) 131–137. [30] T.M. Syed, A.N. Muhammad, V. Asif, Exp-function method for generalized traveling solutions of Calogero-Degasperis-Fokas equation, Zeitschrift fu¨r Naturforschung A J. Phys. Sci. 65 (2010) 78–84. [31] S. Yousefi, Legendre wavelet method for solving differential equation of Lane-Emden type, Appl. Math. Comput. 181 (2006) 1417–11422. [32] d.S. Panagiotis, G.M. Spyridon, The operational matrix of differentiation for orthogonal polynomial series, Int. J. Control 44 (2007) 1–15. [33] N.K. Rajeev, A. Raigar, Numerical solution based on operational matrix of differentiation of shifted second kind Chebyshev wavelets for a Stefan problem, Int. J. Math., Comput., Phys., Electr. Comput. Eng. 9 (7) (2015) 374–377. [34] B. Basirat, M.A. Shahdadi, Application of the BPOM for numerical solution of the isothermal gas spheres equations, Ijrras 18 (1) (2014).
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H O S T E D BY
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ORIGINAL ARTICLE
Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems S.C. Shiralashetti *, S. Kumbinarasaiah Department of Mathematics, Karnatak University, Dharwad 580003, India Received 5 February 2017; revised 10 July 2017; accepted 27 July 2017
KEYWORDS Hermite wavelets; Nonlinear singular initial value problems; Collocation method; Operational matrix
Abstract In this paper, new operational matrix of integration is generated by using Hermite wavelets. By aid of these matrices, Hermite wavelets operational matrix method (HWOMM) is developed for second ordered nonlinear singular initial value problems. Properties of Hermite wavelets are used to convert the differential equations into system of algebraic equations which can be efficiently solved by suitable solvers. Illustrative numerical problems are considered to demonstrate the applicability of the proposed technique and those results are comparing favourably with the exact solutions and errors. Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction Wavelets are localized waves or small wave. Instead of oscillating forever, they drop to zero. Wavelets theory is a newly emerging area in mathematical research field. It has been applied in engineering disciplines; such as signal analysis for wave form representation and segmentations, harmonic analysis and time frequency analysis etc. Wavelets permit the accurate representation of a variety of functions and operators. Wavelets are assumed as basis functions wi;j ðxÞ in continuous time. Special feature of the wavelets basis is that all functions wi;j ðxÞ are constructed from a single mother wavelet wðxÞ which is a small pulse. Usually set of linearly independent
* Corresponding author. E-mail address: [email protected] (S.C. Shiralashetti). Peer review under responsibility of Faculty of Engineering, Alexandria University.
functions created by translation and dilation of mother wavelet. Many practical and physical problems in the field of science and engineering are formulated as initial value problems. The nonlinear singular initial value problems are crucial in many fields of science and engineering. For example gas dynamics, atomic structures, chemical reactions and nuclear physics [1]. In many cases, extracting the analytical solutions for singular initial value problems from analytical methods is always not possible. For such problems we disentomb numerical techniques such as, Galerkin method, Tau method, operation matrix of integration method, collocation method, series solution methods and operational matrix of differentiation etc. Operational matrix of integration with collocation method is one of the most widely used numerical schemes to solve differential equations. This approach is based on converting differential equation into integral equations using operational matrix of integration eliminating the integral operator in order to reduce the problem into system of algebraic equations.
http://dx.doi.org/10.1016/j.aej.2017.07.014 1110-0168 Ó 2017 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: S.C. Shiralashetti, S. Kumbinarasaiah, Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.07.014
2
S.C. Shiralashetti, S. Kumbinarasaiah
The purpose of this paper is to develop the new Hermite wavelets operational matrix of integration collocation method as an alternative method to solve nonlinear singular initial value problems of the form (1) for a given set of conditions. Here we bring all solutions of (1) under the Hermite space and present method will contribute good accuracy for linear singular problems. So that, we concentrate only nonlinear problems. That is, we are expressing solutions of nonlinear singular initial value problems in terms of Hermite wavelets basis. Let ða; bÞ R be an interval and pðxÞ; qðxÞ; rðx; yÞ : ða; bÞ ! R be continuous real valued functions. Throughout this paper we consider the arbitrary nonlinear singular second order equations given by 00
0
y ðxÞ þ pðxÞy ðxÞ þ qðxÞyðxÞ ¼ rðx; yÞ; a < x < b:
ð1Þ
subjected to following initial conditions: yð0Þ ¼ a1 ; y0 ð0Þ ¼ a2 ;
ð2Þ
where a1 ; a2 are known constants. There are some literature available on operational matrix of integration and some other techniques on singular initial value problems. Adomian decomposition method [2], Differential transformation method [3], Haar wavelet collocation method [4], New ultraspherical wavelet collocation method [5], Laguerre wavelets method [6], Physicists Hermite wavelet method [7], Hermite wavelets method for boundary value problem [8], Legendre wavelet method [9], An Accurate Numerical Algorithm [10], Finite Element Methods [11], Haar wavelet series solution [12], Chebyshev wavelet operational matrix of integration [13], The Legendre wavelets operational matrix of integration [14], A Numerical Approach for Solving A Class of The Nonlinear Lane-Emden Type Equations [15], A numerical approach for solving the high order linear singular differential-difference equations [16], An Improved Bessel Collocation Method [17], A Numerical Scheme for Solutions of a class of non-linear differential equations [18], Exponential Collocation Method For Solutions of Singularity Perturbed Delay Differential Equations [19] and also, some methods on solving partial differential equations [20–30]. There are two different approaches for solving differential equations, one approach is based on converting differential equations into integral equations then eliminate the integral operator by operational matrix of integration [31]. Another method is based on the operational matrix of derivative, here we solve the differential equations by converting into system of linear or nonlinear algebraic equations [32,33]. A vast amount of literature are available on operational matrix of integration by Legendre wavelet, Chebyshev wavelet, Laguerre wavelet but there is no literature on Hermite wavelet operational matrix of integration this impetused us to generate operational matrix of integration using Hermite wavelet. The rest of this paper is organized as follows. In Section 2, we introduce the properties of Hermite wavelets. Section 3, function approximation. Section 4, having the procedure for the operational matrix of integration. Section 5, reveals description of the proposed method. Section 6, numerical examples to test the efficiency of proposed method and conclusion is drawn in Section 7.
wavelet. When the dilation parameter a and translation parameter b varies continuously, we have the following family of continuous wavelets: xb wa;b ðxÞ ¼ jaj1=2 w ; 8 a; b 2 R; a–0: a If we restrict the parameters a and b to discrete values as k a ¼ ak 0 ; b ¼ nb0 a0 ; a0 > 1; b0 > 0: We have the following family of discrete wavelets wk;n ðxÞ ¼ jaj1=2 wðak0 x nb0 Þ;
where wk;n form a wavelet basis for L2 ðRÞ. In particular, when a0 ¼ 2 and b0 ¼ 1, then wk;n ðxÞ forms an orthonormal basis. Hermite wavelets are defined as [8], ( kþ1 2p2ffiffi n Hm ð2k x 2n þ 1Þ; 2n1 k1 6 x < k1 p 2 wn;m ðxÞ ¼ ð3Þ 0; otherwise where m ¼ 0; 1; . . . ; M 1: Here Hm ðxÞ is Hermite polynomials of degree m with respect to weight function pffiffiffiffiffiffiffiffiffiffiffiffiffi WðxÞ ¼ 1 x2 on the real line R and satisfies the following recurrence formula H0 ðxÞ ¼ 1, H1 ðxÞ ¼ 2x, Hmþ2 ðxÞ ¼ 2xHmþ1 ðxÞ 2ðm þ 1ÞHm ðxÞ; m ¼ 0; 1; 2; . . . :
Wavelets constitute a family of functions constructed from dilation and translation of a single function called mother
ð4Þ
3. Function approximation and convergence analysis We would like to bring a solution function yðxÞ under Hermite space by approximating yðxÞ by elements of Hermite wavelet basis as follows, yðxÞ ¼
1 X 1 X Cn;m wn;m ðxÞ
ð5Þ
n¼1 m¼0
where wn;m ðxÞ is given in Eq. (3). We approximate yðxÞ by truncating the series represented in Eq. (5) as, yðxÞ
2k1 M1 X X Cn;m wn;m ðxÞ ¼ CT wðxÞ
ð6Þ
n¼1 m¼0
where C and wðxÞ are 2k1 M 1 matrix,
h i CT ¼ C1;0 ;. .. C1;M1 ; C2;0 ;. .. C2;M1 ; ... C2k1;0 ; .. .C2k1 ;M1
ð7Þ
h i wðxÞ ¼ w1;0 ; . . . w1;M1 ; w2;0 ; . . . w2;M1 ; . . . w2k1 ;0 ; . . . w2k1;M1 ð8Þ Theorem 1. A continuous function y(x) in H2 ½0; 1Þ defined on ½0; 1Þ be bounded, then the Hermite wavelets expansion of y(x) converges to it. Proof. Let y(x) be a bounded real valued function on ½0; 1Þ. The Hermite coefficients of continuous functions y(x) is defined as, Z
2. Hermite wavelets
8a; b 2 R; a–0;
1
Cn;m ¼ Z
0
¼ I
yðxÞwn;m dx kþ1 22 n1 n k yðxÞ pffiffiffi Hm ð2 x 2n þ 1Þdx; where I ¼ k1 ; k1 p 2 2
Please cite this article in press as: S.C. Shiralashetti, S. Kumbinarasaiah, Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.07.014
Hermite wavelets operational matrix of integration for the numerical solution Z
Put 2k x 2n þ 1 ¼ z, kþ1
22 ¼ pffiffiffi p kþ1
2 2 ¼ pffiffiffi p
2 w1;3 ðxÞdx ¼ pffiffiffi ð16x4 32x3 þ 18x2 2xÞ p 1 1 1 ¼ ; 0; ; 0; ; 0 w6 ðxÞ 8 8 16
0
z 1 þ 2n Hm ðzÞ2k dz y k 2 1
Z
x
1
Z
z 1 þ 2n Hm ðzÞdz y k 2 1
Z
1
2 256 5 320 3 x 128x4 þ x 32x2 þ 2x w1;4 ðxÞdx ¼ pffiffiffi 3 p 5 1 1 1 4 ; 0; 0; ; 0; w ¼ 15 12 20 6
x
0
Using GMVT for integrals, Z 1 kþ1 2 2 w 1 þ 2n ¼ pffiffiffi y Hm ðzÞdz; p 2k 1 Put
R1 1
for some w 2 ð1; 1Þ
Z
x
0
Hm ðzÞdz ¼ h,
kþ1 2 2 w 1 þ 2n h jcn;m j ¼ pffiffiffi y p 2k Since y is bounded, therefore, n;m¼0 Cn;m is absolutely convergent. Hence the Hermite series expansion of y(x) converges uniformly. h 4. Operational matrix of integration (OMI)
2 512 6 800 3 w1;5 ðxÞdx ¼ pffiffiffi x 512x5 þ 560x4 x þ 50x2 2x 3 p 3 1 1 1 ¼ ;0; 0; 0; ; 0 w6 ðxÞ þ w1;6 ðxÞ 24 16 24
Thus
Z P1
x
6 ðxÞ w6 ðxÞdx ¼ P66 w6 ðxÞ þ w
2 w1;1 ðxÞ ¼ pffiffiffi ð4x 2Þ; p 2 w1;2 ðxÞ ¼ pffiffiffi ð16x2 16x þ 2Þ; p 2 w1;3 ðxÞ ¼ pffiffiffi ð64x3 96x2 þ 36x 2Þ; p 2 w1;4 ðxÞ ¼ pffiffiffi ð256x4 512x3 þ 320x2 64x þ 2Þ; p 2 w1;5 ðxÞ ¼ pffiffiffi ð1024x5 2560x4 þ 2240x3 800x2 þ 100x 2Þ: p Let w6 ðxÞ ¼ ððw1;0 ðxÞ; w1;1 ðxÞ;w1;2 ðxÞ; w1;3 ðxÞ; w1;4 ðxÞ;w1;5 ðxÞÞÞT . By integrating above basis functions with respect to x from 0 to x and expressing in matrix form, we obtain,
Z
x
0
Z
x
0
Z 0
x
2 1 1 w1;0 ðxÞdx ¼ pffiffiffi x ¼ ; ; 0; 0; 0; 0 w6 ðxÞ 2 4 p 2 1 1 ; 0; ; 0; 0; 0 w6 ðxÞ w1;1 ðxÞdx ¼ pffiffiffi ð2x2 2xÞ ¼ 4 8 p 2 16 3 1 1 1 x 8x2 þ 2x ¼ ; ; 0; ; 0; 0 w6 ðxÞ w1;2 ðxÞdx ¼ pffiffiffi 3 4 12 p 3
ð9Þ
0
where
2
1 2 6 1 6 4 6 1 6 6 3 6 1 6 8 6 6 1 6 15 6 1 4 24
P66 ¼
In this section, we give the structure of OMI for Hermite wavelets, particularly at k ¼ 1 and M ¼ 6. Here we consider six basis functions on ½0; 1Þ are given by, 2 w1;0 ðxÞ ¼ pffiffiffi ; p
3
Z
0
0
1 8
0 0
0 0
1 4
0
1 12
0
0 0 0
1 8
0
1 16
0 0
1 12
0
0
1 16
3 0 2 7 07 6 7 6 07 6 7 6 7 6 ðxÞ ¼ 6 0 7 and w 6 7 6 1 7 6 20 7 4 7 05
0 0 0 0 0
3 7 7 7 7 7 7 7 7 5
1 w ðxÞ 24 1;6
Again double integration of above six basis is given by, Z x 2 x2 3 1 1 ; ; ; 0; 0; 0 w6 ðxÞ ¼ w1;0 ðxÞdxdx ¼ pffiffiffi 16 8 32 p 2 0
x
0
Z
x
Z
0
Z
x
Z
x
0
Z
0
2 2 1 3 1 w1;1 ðxÞdxdx ¼ pffiffiffi x3 x2 ¼ ; ; 0; ; 0; 0 w6 ðxÞ 6 32 96 p 3 x
0
x
Z
0
Z
1 4
x
0
2 4 4 8 3 2 p ffiffiffi x x þx w1;2 ðxÞdxdx ¼ 3 p 3 3 1 1 1 ; ; ; 0; ; 0 w6 ðxÞ ¼ 32 12 24 192 2 16 5 x 8x4 þ 6x3 x2 w1;3 ðxÞdxdx ¼ pffiffiffi p 5 1 1 1 1 ; ; 0; ; 0; w ðxÞ ¼ 10 16 64 320 6
2 128 6 128 5 80 4 32 3 x x þ x x þ x2 w1;4 ðxÞdxdx ¼ pffiffiffi 5 3 3 p 15 0 0 1 1 1 1 ¼ ; ; ;0; ;0 w6 ðxÞ 24 60 96 120 x
Z 0
x
Z
x
Z
x
w1;5 ðxÞdxdx 2 512 7 256 6 200 4 50 3 ¼ pffiffiffi x x þ 112x5 x þ x x2 3 3 3 p 21 1 1 1 1 1 ¼ ; ; 0; ; 0; w ðxÞ þ w1;6 ðxÞ: 42 96 192 192 6 24 0
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4
S.C. Shiralashetti, S. Kumbinarasaiah
Z 0
ð10Þ
0
where
P066
Z
respect to x from 0 to x together with initial conditions given in (2) and relation (9) and (10) we obtain,
Hence Z x 0 ðxÞ: w6 ðxÞdxdx ¼ P066 w6 ðxÞ þ w 6
x
2
3 16
6 1 6 6 6 3 6 6 32 6 ¼ 6 101 6 6 1 6 24 6 1 4 42
1 8 3 32 1 12 1 16 1 60 1 96
1 32
0
0
1 96
0 0
1 24
0
1 192
0
1 64
0
1 96
0
1 120
0
1 192
0
0 ðxÞÞ: yðxÞ ¼ a1 þ xa2 þ AT ðP0 wðxÞ þ w
3
2 3 0 7 7 6 7 0 7 6 7 7 6 7 7 6 7 0 7 0 1 6 7 and w6 ðxÞ ¼ 6 7 320 7 7 0 6 7 7 0 7 6 1 7 4 480 w1;6 ðxÞ 5 7 1 5 1 192 w ðxÞ 672 1;7 0 0 0
Similarly if we considered seven basis functions then x
7 ðxÞ: w7 ðxÞdx ¼ P77 w7 ðxÞ þ w
where
P77 ¼
1 2 6 1 64 6 1 63 6 61 68 6 1 6 15 6 61 6 24 6 1 4 35
1 4
0
0
1 8
0 0
1 4
0
1 12
0 0 0
0 0
1 8
0
1 16
0 0 0 0
0
1 12
0
1 20
0 0
0 0
0 0
1 16
0
0
1 20
3 2 0 07 6 7 6 7 6 07 6 7 6 7 07 6 ðxÞ ¼ and w 6 7 7 6 07 6 7 6 1 7 6 24 7 4 7 5 0
3
0
7 7 7 7 7 7 7 7 7 7 7 5
0 0 0 0 0 1 w ðxÞ 28 1;7
P077
¼
3 16 6 1 66 6 3 6 32 6 61 6 10 6 1 6 24 6 61 6 42 6 1 4 65
1 8 3 32 1 12 1 16 1 60 1 96 1 140
1 32
0
0
1 96
0 0
0 0 0
0 0 0
3
Then we can analyse the unknown vector A by solving this system through the well-known Newton iterative method using Matlab. The approximate solution can be easily recovered by inserting A into the corresponding expression of y(x). 6. Numerical experiments Test Problem 1. Consider the singular initial value problem [6], ð12Þ
yð0Þ ¼ 1; y0 ð0Þ ¼ 0; 2
3
0 7 6 7 7 0 6 7 7 1 1 6 7 7 0 24 192 6 7 7 0 6 7 7 1 0 1 0 0 6 7 7 0 64 320 0 ðxÞ ¼ and w 6 7 7 7 1 1 1 7 6 7 0 120 0 480 96 6 7 7 0 6 7 7 1 1 0 192 0 192 0 7 6 1 7 w ðxÞ 4 672 1;7 5 7 1 1 5 0 0 320 0 280 1 w ðxÞ 896 1;8
In this way we can generate the other matrix by consuming more number of basis functions. So when dealing with the problems, we need to pre-calculate the corresponding opera 0 ðxÞ for different w tional matrix of integration P; P0 and wðxÞ; values of N. 5. Description of the proposed method In this section, we will use above discussed OMI together with collocation method to solve nonlinear singular initial value problems, we assume that, y00 ¼ AT wðxÞ
Now discrete it using the following collocation points xi ¼ 2i1 ; i ¼ 1; 2; . . . ; N thus we get a system of N equations 2N with N unknowns. Here N represents size of an operational matrix of integration and corresponding equation is of the following form,
subjected to the initial conditions
0
2
AT wðxÞ þ pðxÞða2 þ AT ðPwðxÞ þ wðxÞÞÞ þ qðxÞða1 þ xa2 0 ðxÞÞÞ ¼ rðx; yÞ: þ AT ðP0 wðxÞ þ w
1 y00 ðxÞ þ y0 y3 þ 3y5 ; 0 < x < 1 x
Double integration is, Z xZ x 0 ðxÞ w6 ðxÞdxdx ¼ P077 w7 ðxÞ þ w 7 0
then we get yðxÞ; y0 ðxÞ; y00 ðxÞ are in terms of polynomial basis and their integrals. Now substitute yðxÞ; y0 ðxÞ; y00 ðxÞ are in (1) we get,
i ÞÞÞ þ qðxi Þða1 þ xi a2 AT wðxi Þ þ pðxi Þða2 þ AT ðPwðxi Þ þ wðx T 0 0 þ A ðP wðxi Þ þ w ðxi ÞÞÞ ¼ rðxi ; yðxi ÞÞ: i ¼ 1; 2; :::; N:
0
2
y0 ðxÞ ¼ a2 þ AT ðPwðxÞ þ wðxÞÞ:
ð11Þ
where A is unknown vector which should be determined and wðxÞ is the vector defined in (8), Integrate (11) twice with
ð13Þ 1 2
the exact solution is yðxÞ ¼ ð1 þ x2 Þ . Method of implementation for N ¼ 6: We assume that, y00 ¼ AT wðxÞ
ð14Þ
where wðxÞ ¼ w6 ðxÞ ¼ ððw1;0 ðxÞ;w1;1 ðxÞ; w1;2 ðxÞ; w1;3 ðxÞ;w1;4 ðxÞ; w1;5 ðxÞÞÞT & A ¼ ða0 ; a1 ;a2 ;a3 ;a4 ; a5 Þ: Integrate (14) with respect to x from 0 to x together with initial conditions given in (13), we obtain, y0 ðxÞ ¼ AT ðPwðxÞ þ wðxÞÞ: 0 ðxÞÞ: yðxÞ ¼ 1 þ AT ðP0 wðxÞ þ w Now substitute yðxÞ; y0 ðxÞ; y00 ðxÞ are in (12) and discretize it using the following collocation points xi ¼ 2i1 ; i ¼ 1; 2; . . . ; 6, 12 thus we get 6 system of equations with 6 unknowns. After solving this system by newton iteration method, we get the unknown vector A is, 3 2 a0 ¼ 0:3285 7 6 a1 ¼ 0:2908 7 6 7 6 7 6 a2 ¼ 0:0255 7 A¼6 7 6 a ¼ 0:0303 3 7 6 7 6 5 4 a4 ¼ 0:0118 a5 ¼ 2:1135 104
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Hermite wavelets operational matrix of integration for the numerical solution Table 1
5
Comparison of absolute errors (AE) for the test problem 1, (k = 1).
x
Exact solution
AE at N = 6
AE at N = 7
AE at N = 8
AE at N = 9
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.99503719020 0.98058067569 0.95782628522 0.92847669088 0.89442719099 0.85749292571 0.81923192051 0.78086880944 0.74329414624
5
6
7
1:8966 108 1:0083 108 1:1329 108 5:5787 109 4:7519 109 9:1742 109 1:0085 109 3:9847 109 4:8067 108
1:8706 10 1:9982 105 1:0719 105 6:5087 106 3:8619 106 9:9962 107 4:1085 106 4:2547 106 9:8647 106
5:0872 10 3:9138 107 3:1663 107 2:2954 107 1:4662 107 6:8963 108 1:9866 109 5:8784 108 1:2314 107
2:5225 10 1:9417 106 8:0068 107 5:4375 107 4:4979 108 5:0408 107 5:1814 107 1:2857 106 7:3953 107
−5
2
x 10
AE at N=6 AE at N=7 AE at N=8 AE at N=9
1.8
Error analysis
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
Fig. 1
Physical interpretation of Absolute error analysis for the test problem 1 for different values of N.
1 Exact solution Present method
0.95
Solution
0.9 0.85 0.8 0.75 0.7 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
Fig. 2
Graphical interpretation of Exact solution and Present method solution for the test problem 1.
Substitute these values in yðxÞ, we ge the approximate solution is, yðxÞ ¼ 0:0058x7 þ 0:1344x6 0:4785x5 þ 0:6079x4 0:0566x3 0:4944x2 þ 6:8439 1021 x þ 0:9999: by applying the technique described in Section 5 with different values of N. Table 1 and Fig. 1 represents the comparison between the absolute error of approximate solution with analytical solution. Graphical representation of Exact and present method solution for the test problem 1 is drawn in Fig. 2.
Test Problem 2. Consider the following nonlinear LaneEmden equation [4]; 2 0 y00 ðxÞ þ y ðxÞ þ y5 ¼ 0; 0 < x < 1 x subjected to the initial conditions, yð0Þ ¼ 1; y0 ð0Þ ¼ 0;
Please cite this article in press as: S.C. Shiralashetti, S. Kumbinarasaiah, Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.07.014
6
S.C. Shiralashetti, S. Kumbinarasaiah Table 2
Comparison of absolute errors (AE) for the test problem 2, (k = 1).
x
Exact solution
AE at N = 6
AE at N = 7
AE at N = 8
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.99833748845 0.99339926779 0.98532927816 0.97435470369 0.96076892283 0.94491118252 0.92714554082 0.90784129900 0.88735650941
8
8
9
9:9165 10 4:5437 108 5:1208 108 5:0684 108 3:5157 108 7:0668 108 6:7098 108 2:0163 109 1:3919 107
3:6989 10 1:4473 108 2:5323 109 2:5781 109 6:6328 109 8:9403 109 6:9532 109 1:1810 108 6:5729 109
AE at N = 9 3:6640 1011 3:4148 1010 3:6797 1011 3:4785 1011 3:2490 1011 3:3358 1011 3:2694 1011 2:9498 1011 2:8965 1010
4:5191 10 2:4641 109 5:3187 1010 1:1649 1010 1:4761 1010 1:7220 1010 1:8055 1010 1:6746 1010 1:3945 1010
−7
1.4
x 10
AE at N=6 AE at N=7 AE at N=8 AE at N=9
Error analysis
1.2 1 0.8 0.6 0.4 0.2 0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
Fig. 3
Physical interpretation of Absolute error analysis for the test problem 2 for different values of N.
1 Exact solution Present method
0.98
Solution
0.96 0.94 0.92 0.9 0.88 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
Fig. 4
Graphical interpretation of Exact solution and present method solution for the test problem 2.
12 2 the exact solution is yðxÞ ¼ 1 þ x3 . We solve this equation by the HWOM method with different values of N. Table 2 and Fig. 3 reflects the comparison between the absolute error of approximate solution with analytical solution for different values of N. Graphical representation of Exact and present method solution is drawn in Fig. 4. Test Problem 3. Consider the singular initial value problem [34];
y00 ðxÞ 2e y ¼ 0; 0 < x < 1
ð15Þ
subjected to the initial conditions yð0Þ ¼ 0; y0 ð0Þ ¼ 0;
ð16Þ
the exact solution is yðxÞ ¼ 2lnðcosxÞ, Method of implementation for N ¼ 6: We assume that, y00 ¼ AT wðxÞ
ð17Þ
Please cite this article in press as: S.C. Shiralashetti, S. Kumbinarasaiah, Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.07.014
Hermite wavelets operational matrix of integration for the numerical solution
7
Comparison of absolute errors (AE) for the test problem 3, (k = 1).
Table 3 x
Exact solution
AE at N = 6
AE at N = 7
AE at N = 8
AE at N = 9
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.99833748845 0.99339926779 0.98532927816 0.97435470369 0.96076892283 0.94491118252 0.92714554082 0.90784129900 0.88735650941
5
5
6
2:2974 106 4:5037 106 6:8418 106 9:3196 106 1:2023 105 1:5040 105 1:8503 105 2:2619 105 2:7587 105
2:1447 10 4:3130 105 6:3466 105 8:6888 105 1:1148 104 1:3904 104 1:7177 104 2:0778 104 2:5754 104
6:0285 10 1:2695 104 1:8170 104 2:4653 104 3:1948 104 3:9461 104 4:8481 104 6:0067 104 7:1158 104
7:6580 10 1:5078 105 2:2661 105 3:0932 105 3:9754 105 4:9816 105 6:1115 105 7:4870 105 9:1294 105
−4
8
x 10
AE at N=6 AE at N=7 AE at N=8 AE at N=9
7
Error analysis
6 5 4 3 2 1 0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
Fig. 5
Physical interpretation of Absolute error analysis for test problem 3 for different values of N.
1.4 Exact solution Present method solution
1.2
Solution
1 0.8 0.6 0.4 0.2 0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
Fig. 6
Graphical interpretation of Exact solution and present method solution for the test problem 3.
where wðxÞ ¼ w6 ðxÞ ¼ ððw1;0 ðxÞ; w1;1 ðxÞ; w1;2 ðxÞ; w1;3 ðxÞ; w1;4 ðxÞ; T
w1;5 ðxÞÞÞ and A ¼ ða0 ; a1 ; a2 ; a3 ; a4 ; a5 Þ: Integrate (17) with respect to x from 0 to x together with initial conditions given in (16) we obtain, y0 ðxÞ ¼ AT ðPwðxÞ þ wðxÞÞ:
0 ðxÞÞ: yðxÞ ¼ AT ðP0 wðxÞ þ w Now substitute yðxÞ; y0 ðxÞ; y00 ðxÞ are in (15) and discretize it ; i ¼ 1; 2; . . . ; 6 using the following collocation points xi ¼ 2i1 12 thus we get 6 system of equations with 6 unknowns. After solving this system by newton iteration method we get unknown vector A as,
Please cite this article in press as: S.C. Shiralashetti, S. Kumbinarasaiah, Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.07.014
8
S.C. Shiralashetti, S. Kumbinarasaiah Table 3.1
Comparison of absolute errors (AE) for the test problem 3, (k = 2).
x
Exact solution
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.99833748845 0.99339926779 0.98532927816 0.97435470369 0.96076892283 0.94491118252 0.92714554082 0.90784129900 0.88735650941
Table 4
AE at N = 8 (M = 4) 3.2405 6.6123 8.9627 1.2334 1.6237 1.8461 2.6442 8.6431 3.4509
AE at N = 10 (M = 5)
5
10 105 105 104 104 104 104 104 103
3.2507 6.7550 9.4419 1.2846 1.6732 2.0306 2.5172 3.2727 2.1403
AE at N = 12 (M = 6)
5
10 105 105 104 104 104 104 104 104
6.0285 1.2695 1.8170 2.4653 3.1948 3.9461 4.8481 6.0067 7.1158
105 104 104 104 104 104 104 104 104
Comparison of absolute errors (AE) for the test problem 4 (k = 1).
x
Exact solution
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.01990066170 0.07844142630 0.17235539248 0.29684001023 0.44628710262 0.61496939949 0.79755223991 0.98939248367 1.18665369055
AE at N = 6
AE at N = 7
AE at N = 8
5
6
6
2:2300 10 1:2130 105 6:1607 106 7:8887 106 6:3569 106 1:0604 105 1:0563 105 4:2378 106 1:4494 106
1:4552 10 4:7418 108 1:0713 106 7:2831 107 1:0018 106 1:0623 106 5:1912 107 1:4039 106 1:0590 107
AE at N = 9 3:6589 107 5:4640 108 9:7692 1011 4:2075 108 5:9709 108 6:9569 108 7:4787 108 6:7446 108 1:0572 107
1:2575 10 3:7267 107 3:6764 108 5:0634 108 1:7109 107 1:8601 107 2:3240 107 2:2325 107 2:1741 107
−5
2.5
x 10
AE at N=6 AE at N=7 AE at N=8 AE at N=9
Error analysis
2
1.5
1
0.5
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
Physical interpretation of Absolute error analysis for the test problem 4 for different values of N.
Fig. 7
2 6 6 6 6 6 6 A¼6 6 6 6 6 6 4
a0 ¼ 3:0233 a1 ¼ 0:9359 a2 ¼ 0:3929 a3 ¼ 0:1169 a4 ¼ 0:0320
3 7 7 7 7 7 7 7 7 7 7 7 7 5
a5 ¼ 0:0086 Substitute these values in yðxÞ; we ge approximate solution as,
yðxÞ ¼ 0:2377x7 0:5237x6 þ 0:5892x5 0:1508x4 þ 0:0903x3 þ 0:9876x2 2:7984 1019 x þ 2:8555 1021 : by applying the technique described in Section 5 with different values of N. Table 3 and Fig. 5 represents the comparison between the absolute error of approximate solution with analytical solution. Graphical representation of Exact and present method solution for test problem 3 is drawn in Fig. 6. In the present method, for the domain wise different values of k and fixed values of M, we obtain the same accuracy in the solution. This can be observed in the Tables 3.1and 3. For
Please cite this article in press as: S.C. Shiralashetti, S. Kumbinarasaiah, Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.07.014
Hermite wavelets operational matrix of integration for the numerical solution Table 5
9
Comparison of present and other existing methods [6,34].
x
Exact solution
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.01990066170 0.07844142630 0.17235539248 0.29684001023 0.44628710262 0.61496939949 0.79755223991 0.98939248367 1.18665369055
AE in [34]
AE in [6] at k = 2 and M = 5 5
AE at N = 9 by present method 3:6589 107 5:4640 108 9:7692 1011 4:2075 108 5:9709 108 6:9569 108 7:4787 108 6:7446 108 1:0572 107
6
2:0000 10 2:4999 105 2:4999 105 1:8999 105 2:4000 105 2:9000 105 2:5999 104 1:4699 104 8:6199 103
1:1903 10 4:0724 107 3:7227 107 1:3771 107 2:1195 107 4:0909 106 7:3090 106 9:8931 106 1:1716 105
0 Exact solution BPOM solution LWM solution Present method at N=9
−0.2
Solution
−0.4 −0.6 −0.8 −1 −1.2 −1.4 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
x
Fig. 8 Graphical interpretation of the Exact solution, Present method solution, BPOM solution [34] and LEM solution [6] for the test problem 4.
k = 1 and M = 6(N = 6), we obtain Four decimal accuracy for the Example 3 which can be seen in the Table 3. Same accuracy can be observed in the case of k = 2 and M = 6 (N = 12) in Table 3.1. Hence, solving differential equations using any one of the continuous polynomial wavelets basis is sufficient no need to use other wavelets, which are having continuous polynomials as basis. Test Problem 4. Consider the following nonlinear LaneEmden initial value problem [6], y 2 y00 ðxÞ þ y0 ðxÞ þ 4 2e y þ e2 ; 0 < x < 1 x subjected to the initial conditions yð0Þ ¼ 0; y0 ð0Þ ¼ 0; the exact solution is yðxÞ ¼ 2lnð1 þ x2 Þ. We solve this equation by the present method with different values of N. Table 4 and Fig. 7 represents the comparison between the absolute error with exact solution for different values of N. Table 5 presents the comparison between the absolute error(AE) obtained by present method and other existed methods with analytic solutions. Fig. 8 shows graphical interpretation of solutions obtained by present method and other existed methods with analytical solution.
7. Conclusions In this paper, we attempted to introduce operational matrix of integration using Hermite wavelets for solving nonlinear singular initial value problems for different physical conditions which is important for the development of new research in the field of numerical analysis and beneficial for new researchers. The proposed scheme is tested on some examples and the results are quite satisfactory in comparison with the existing numerical solutions. Finally, we summarize the outcomes of this analysis is as follows: 1. The present method gives better accuracy in comparison with the other numerical techniques [6,34] available in literature. 2. This scheme is easy to implement in computer programs and we can extend this scheme for higher order also with slight modification in the present method. 3. This scheme is made easy by generalizing the operational matrix of integration. 4. This operational matrix of integration is newly generated using Hermite wavelets and method is also latest one. 5. This method is applicable for all type of initial value problems.
Please cite this article in press as: S.C. Shiralashetti, S. Kumbinarasaiah, Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.07.014
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Please cite this article in press as: S.C. Shiralashetti, S. Kumbinarasaiah, Hermite wavelets operational matrix of integration for the numerical solution of nonlinear singular initial value problems, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.07.014