Numerical solutions for the linear and nonlinear singular boundary value problems using Laguerre wavelets
- Fengying Zhou^{1} and
- Xiaoyong Xu^{1}Email author
https://doi.org/10.1186/s13662-016-0754-1
© Zhou and Xu 2016
Received: 16 August 2015
Accepted: 11 January 2016
Published: 25 January 2016
Abstract
In this paper, a collocation method based on Laguerre wavelets is proposed for the numerical solutions of linear and nonlinear singular boundary value problems. Laguerre wavelet expansions together with operational matrix of integration are used to convert the problems into systems of algebraic equations which can be efficiently solved by suitable solvers. Illustrative examples are given to demonstrate the validity and applicability of this technique, and the results have been compared with the exact solutions.
Keywords
MSC
1 Introduction
Singular boundary value problems (BVPs) for ordinary differential equations occur frequently in the fields of engineering and science such as gas dynamics, nuclear physics, atomic structures and chemical reactions [1]. In most cases, we do not always find the exact solutions for the singular boundary values problems via analytical methods. In this case, it is very meaningful to give the high precision numerical solutions for this kind of problem by numerical methods.
Numerous research work has been invested to study the singular BVPs of the form (1)-(5). For more details, the reader is kindly recommended to see the survey in [1]. Recently, many researchers have obtained approximations for singular BVPs via various methods. For example, Kanth and Aruna applied He’s variational iteration method [3], Chang employed the Taylor series method [4], Singh and Kumar proposed a new technique based on Green’s function [5], Sahlan and Hashemizadeh used the wavelet Galerkin method [6], Arqub et al. studied a continuous genetic algorithm [7], Ebaid used the Adomian decomposition method [8], Goh et al. developed a quartic B-spline method [9], and Nasab proposed the Chebyshev finite difference method [10]. Moreover, orthogonal polynomial methods have seen significant achievements in dealing with singular boundary value problems, for example, Legendre polynomials [11], Chebyshev polynomials [2], Bernstein polynomials [12], Laguerre polynomials [13], Bessel polynomials [14], Hermite polynomials [15], and Bernoulli polynomials [16]. Note that these polynomials are supported on the whole interval. This is obviously a defect for certain analysis work, especially problems involving local functions vanishing outside a short interval. However, one advantage of wavelet analysis is the ability to perform a local analysis. This characteristic of time-frequency localization can overcome the defect and allows us to obtain very accurate numerical solutions.
There are two different approaches for solving differential equations. One approach is based on converting differential equations into integral equations through integration, approximating various signals involved in the equation by truncated orthogonal series, and using the operational matrix of integration, to eliminate the integral operations [17]. Another one is based on using operational matrix of derivatives in order to reduce the problem into solving a system of linear or nonlinear algebraic equations. There are some papers in the literature about using the operational matrix of derivatives to solve differential equations [6, 18, 19].
The rest of this paper is organized as follows. In Section 2, we introduce the Laguerre wavelets and the operational matrix of integration. The error estimation of the Laguerre wavelets expansion is also given. In Section 3, the proposed method is used to approximate solutions of the problems. Section 4 gives several examples to test the proposed method. A conclusion is drawn in Section 5.
2 Laguerre wavelets and their properties
2.1 Wavelets and Laguerre wavelets
The following theorem gives the error estimation of the Laguerre wavelets expansion.
Theorem 1
Proof
2.2 Operational matrix of integration (OMI)
3 Description of the proposed method
4 Numerical examples
In order to demonstrate the efficiency and applicability of the proposed method, several linear or nonlinear singular two-point BVPs are studied. We also compare the approximate solution with the exact solution. All computations are performed by Matlab.
Example 1
Comparison of absolute errors for Example 1 ( \(\pmb{k=2}\) )
x | Exact solution | Absolute error M = 3 | Absolute error M = 4 | Absolute error M = 5 | Absolute error M = 6 | Absolute error M = 7 |
---|---|---|---|---|---|---|
0.1 | 0.00995004165278026 | 5.05015e−5 | 1.15174e−5 | 1.67248e−7 | 1.94092e−8 | 2.04877e−10 |
0.2 | 0.0392026631136497 | 6.99671e−5 | 1.49598e−5 | 2.08568e−7 | 2.37033e−8 | 2.44321e−10 |
0.3 | 0.0859802840213046 | 8.61420e−5 | 1.65555e−5 | 2.27808e−7 | 2.59825e−8 | 2.64413e−10 |
0.4 | 0.147369759040462 | 8.52543e−5 | 1.74995e−5 | 2.42605e−7 | 2.71268e−8 | 2.74407e−10 |
0.5 | 0.219395640472593 | 8.52773e−5 | 1.84611e−5 | 2.42878e−7 | 2.80195e−8 | 2.76631e−10 |
0.6 | 0.297120821367484 | 5.87600e−5 | 1.39445e−5 | 1.76695e−7 | 2.15210e−8 | 2.10918e−10 |
0.7 | 0.374772671769399 | 3.90650e−5 | 1.02566e−5 | 1.41817e−7 | 1.58798e−8 | 1.56994e−10 |
0.8 | 0.445892293982186 | 4.85804e−5 | 6.65266e−6 | 9.25693e−8 | 1.04099e−8 | 1.06686e−10 |
0.9 | 0.503504074299238 | 2.34107e−5 | 3.09612e−6 | 5.98109e−8 | 4.99262e−9 | 5.71941e−11 |
Comparison of absolute errors for Example 1 ( \(\pmb{k=3}\) )
x | Exact solution | Absolute error M = 3 | Absolute error M = 4 | Absolute error M = 5 | Absolute error M = 6 | Absolute error M = 7 |
---|---|---|---|---|---|---|
0.1 | 0.00995004165278026 | 6.09766e−6 | 1.01327e−6 | 4.46956e−9 | 3.95315e−10 | 5.99699e−13 |
0.2 | 0.0392026631136497 | 7.11910e−6 | 1.19433e−6 | 5.22659e−9 | 4.73386e−10 | 5.69475e−13 |
0.3 | 0.0859802840213046 | 6.96418e−6 | 1.22938e−6 | 5.14196e−9 | 5.02232e−10 | 4.70207e−13 |
0.4 | 0.147369759040462 | 6.72907e−6 | 1.21170e−6 | 4.73782e−9 | 5.05799e−10 | 3.83082e−13 |
0.5 | 0.219395640472593 | 5.82617e−6 | 1.18534e−6 | 4.30528e−9 | 4.92414e−10 | 3.16524e−13 |
0.6 | 0.297120821367484 | 4.41211e−6 | 9.85413e−7 | 3.52994e−9 | 4.11916e−10 | 3.36841e−13 |
0.7 | 0.374772671769399 | 3.86910e−6 | 7.96955e−7 | 2.86759e−9 | 3.31169e−10 | 3.66096e−13 |
0.8 | 0.445892293982186 | 2.27080e−6 | 5.61629e−7 | 1.81250e−9 | 2.28235e−10 | 3.48554e−13 |
0.9 | 0.503504074299238 | 1.73623e−6 | 2.78002e−7 | 9.55841e−10 | 1.18077e−10 | 1.97952e−13 |
Example 2
Comparison of absolute errors for Example 2 ( \(\pmb{k=2}\) )
x | Exact solution | Absolute error M = 3 | Absolute error M = 4 | Absolute error M = 5 | Absolute error M = 6 | Absolute error M = 7 |
---|---|---|---|---|---|---|
0.1 | 0.990049833749168 | 4.30380e−4 | 1.19711e−5 | 4.49285e−6 | 3.92356e−8 | 3.85970e−8 |
0.2 | 0.960789439152323 | 2.87642e−4 | 9.28739e−6 | 3.38741e−6 | 3.57194e−8 | 3.70852e−8 |
0.3 | 0.913931185271228 | 1.96900e−4 | 8.49488e−6 | 3.48051e−6 | 3.46026e−8 | 3.42054e−8 |
0.4 | 0.852143788966211 | 2.44586e−4 | 8.14368e−6 | 2.98133e−6 | 2.87292e−8 | 3.09439e−8 |
0.5 | 0.778800783071405 | 2.29986e−4 | 7.24332e−6 | 2.86993e−6 | 2.88864e−8 | 2.85315e−8 |
0.6 | 0.697676326071031 | 1.02809e−4 | 6.50098e−6 | 1.33540e−6 | 2.75676e−8 | 1.36518e−8 |
0.7 | 0.612626394184416 | 5.66525e−5 | 3.52342e−6 | 7.04795e−7 | 1.58048e−8 | 6.79409e−9 |
0.8 | 0.527292424043049 | 5.30123e−5 | 2.77371e−6 | 3.32497e−7 | 9.96594e−9 | 3.16979e−9 |
0.9 | 0.444858066222941 | 1.55830e−5 | 1.84094e−6 | 1.50285e−7 | 1.09891e−8 | 1.10573e−9 |
Comparison of absolute errors for Example 2 ( \(\pmb{k=3}\) )
x | Exact solution | Absolute error M = 3 | Absolute error M = 4 | Absolute error M = 5 | Absolute error M = 6 | Absolute error M = 7 |
---|---|---|---|---|---|---|
0.1 | 0.990049833749168 | 2.08605e−4 | 5.07778e−7 | 4.15842e−7 | 2.78811e−9 | 2.71666e−9 |
0.2 | 0.960789439152323 | 2.21586e−4 | 5.87780e−7 | 4.10797e−7 | 2.80960e−9 | 2.51711e−9 |
0.3 | 0.913931185271228 | 5.48197e−5 | 1.20537e−6 | 1.38717e−7 | 4.74755e−9 | 9.86147e−10 |
0.4 | 0.852143788966211 | 1.92271e−5 | 1.45560e−6 | 2.03389e−8 | 5.53697e−9 | 1.07901e−10 |
0.5 | 0.778800783071405 | 3.85323e−5 | 1.36715e−6 | 5.24896e−8 | 5.19276e−9 | 8.98013e−11 |
0.6 | 0.697676326071031 | 2.41881e−5 | 1.19571e−6 | 4.21942e−8 | 4.19259e−9 | 4.05933e−11 |
0.7 | 0.612626394184416 | 1.59949e−5 | 1.01058e−6 | 3.36762e−8 | 3.38714e−9 | 1.56910e−11 |
0.8 | 0.527292424043049 | 9.22128e−6 | 5.95503e−7 | 2.06618e−8 | 2.03107e−9 | 3.88289e−11 |
0.9 | 0.444858066222941 | 3.14709e−6 | 2.04191e−7 | 7.61645e−9 | 7.32635e−10 | 6.69331e−11 |
Example 3
Comparison of absolute errors for Example 3 ( \(\pmb{k=2}\) )
x | Exact solution | Absolute error M = 3 | Absolute error M = 4 | Absolute error M = 5 | Absolute error M = 6 | Absolute error M = 7 |
---|---|---|---|---|---|---|
0.1 | 0.995037190209989 | 4.77401e−5 | 2.80938e−6 | 1.19035e−6 | 2.35692e−8 | 2.20068e−8 |
0.2 | 0.98058067569092 | 3.48537e−5 | 2.03669e−6 | 4.07247e−7 | 2.33623e−9 | 1.54801e−8 |
0.3 | 0.957826285221151 | 2.38207e−5 | 1.20579e−6 | 3.72277e−7 | 7.17380e−9 | 1.02347e−8 |
0.4 | 0.928476690885259 | 5.21145e−6 | 8.77799e−7 | 1.37710e−7 | 3.12745e−9 | 5.36581e−9 |
0.5 | 0.894427190999916 | 7.76968e−6 | 5.82907e−7 | 2.11945e−7 | 3.50210e−8 | 6.35488e−9 |
0.6 | 0.857492925712544 | 1.12585e−4 | 9.24255e−6 | 4.09098e−6 | 3.15261e−7 | 7.61925e−8 |
0.7 | 0.81923192051904 | 2.06616e−4 | 1.61932e−5 | 7.30904e−6 | 6.05365e−7 | 1.41539e−7 |
0.8 | 0.78086880944303 | 2.73469e−4 | 2.16091e−5 | 9.89317e−6 | 8.27822e−7 | 1.92037e−7 |
0.9 | 0.743294146247166 | 3.30810e−4 | 2.58050e−5 | 1.17167e−5 | 9.91475e−7 | 2.29895e−7 |
Comparison of absolute errors for Example 3 ( \(\pmb{k=3}\) )
x | Exact solution | Absolute error M = 3 | Absolute error M = 4 | Absolute error M = 5 | Absolute error M = 6 | Absolute error M = 7 |
---|---|---|---|---|---|---|
0.1 | 0.995037190209989 | 7.56440e−7 | 5.71715e−8 | 2.72876e−9 | 3.20637e−10 | 3.77395e−11 |
0.2 | 0.98058067569092 | 7.76980e−8 | 2.63566e−8 | 3.70661e−10 | 2.55319e−10 | 1.92960e−11 |
0.3 | 0.957826285221151 | 6.00860e−8 | 2.26763e−8 | 1.66112e−8 | 6.70535e−10 | 3.12874e−10 |
0.4 | 0.928476690885259 | 2.29220e−6 | 1.91073e−7 | 4.98419e−8 | 1.35278e−9 | 8.25356e−10 |
0.5 | 0.894427190999916 | 2.52812e−6 | 3.30368e−7 | 7.10777e−8 | 1.72866e−9 | 1.16419e−9 |
0.6 | 0.857492925712544 | 6.98583e−6 | 1.99788e−7 | 1.14244e−7 | 5.91857e−9 | 1.36581e−9 |
0.7 | 0.81923192051904 | 1.05020e−5 | 6.25059e−7 | 1.43766e−7 | 9.14354e−9 | 1.45891e−9 |
0.8 | 0.78086880944303 | 1.36172e−5 | 1.04964e−6 | 1.59582e−7 | 1.10759e−8 | 1.41417e−9 |
0.9 | 0.743294146247166 | 1.60925e−5 | 1.51021e−6 | 1.60847e−7 | 1.20134e−8 | 1.26230e−9 |
Example 4
Comparison of absolute errors for Example 4 ( \(\pmb{k=2}\) )
x | Exact solution | Absolute error M = 3 | Absolute error M = 4 | Absolute error M = 5 | Absolute error M = 6 | Absolute error M = 7 |
---|---|---|---|---|---|---|
0.1 | −0.0199006617063362 | 7.12293e−5 | 4.60723e−6 | 1.03881e−6 | 1.97866e−9 | 9.02739e−9 |
0.2 | −0.0784414263065627 | 1.66987e−6 | 6.19572e−7 | 6.99131e−7 | 1.51682e−8 | 1.68836e−9 |
0.3 | −0.172355392482105 | 1.21751e−4 | 1.76275e−6 | 3.42269e−7 | 2.01945e−8 | 6.31737e−9 |
0.4 | −0.296840010236547 | 2.74878e−5 | 1.49126e−6 | 1.06726e−6 | 5.07048e−9 | 9.34269e−9 |
0.5 | −0.44628710262842 | 1.06032e−5 | 2.81301e−6 | 4.23783e−8 | 4.20504e−8 | 1.76553e−10 |
0.6 | −0.614969399495921 | 2.72115e−4 | 1.79133e−5 | 8.97718e−6 | 5.75857e−7 | 1.78242e−7 |
0.7 | −0.797552239914736 | 4.58685e−4 | 2.48304e−5 | 1.45806e−5 | 9.99011e−7 | 2.95038e−7 |
0.8 | −0.989392483672214 | 5.79446e−4 | 2.95584e−5 | 1.83900e−5 | 1.27201e−6 | 3.69959e−7 |
0.9 | −1.18665369055547 | 6.57413e−4 | 3.25099e−5 | 2.04445e−5 | 1.43231e−6 | 4.14540e−7 |
Comparison of absolute errors for Example 4 ( \(\pmb{k=3}\) )
x | Exact solution | Absolute error M = 3 | Absolute error M = 4 | Absolute error M = 5 | Absolute error M = 6 | Absolute error M = 7 |
---|---|---|---|---|---|---|
0.1 | −0.0199006617063362 | 1.64143e−8 | 7.15543e−8 | 4.63072e−9 | 1.09733e−10 | 4.51913e−12 |
0.2 | −0.0784414263065627 | 4.26643e−7 | 7.99963e−8 | 7.54245e−9 | 8.38618e−11 | 2.46411e−11 |
0.3 | −0.172355392482105 | 1.45496e−6 | 1.90341e−7 | 3.52023e−8 | 4.66119e−10 | 5.31878e−10 |
0.4 | −0.296840010236547 | 7.54378e−6 | 7.77384e−7 | 8.52194e−8 | 1.55276e−10 | 1.16498e−9 |
0.5 | −0.44628710262842 | 7.18512e−6 | 1.12532e−6 | 1.07965e−7 | 6.28860e−10 | 1.47615e−9 |
0.6 | −0.614969399495921 | 1.65622e−5 | 1.03010e−7 | 1.82404e−7 | 7.06173e−9 | 1.72201e−9 |
0.7 | −0.797552239914736 | 2.32175e−5 | 6.06844e−7 | 2.25375e−7 | 1.22023e−8 | 1.80556e−9 |
0.8 | −0.989392483672214 | 2.86341e−5 | 1.40638e−6 | 2.44832e−7 | 1.51659e−8 | 1.67463e−9 |
0.9 | −1.18665369055547 | 3.30731e−5 | 2.31416e−6 | 2.41118e−7 | 1.67655e−8 | 1.39453e−9 |
Example 5
Comparison of absolute errors for Example 5 ( \(\pmb{k=2, 3}\) )
x | Exact solution | Absolute error k = 2, M = 3 | Absolute error k = 2, M = 4 | Absolute error k = 3, M = 3 | Absolute error k = 3, M = 4 |
---|---|---|---|---|---|
0.1 | 0.009 | 8.96289e−10 | 3.92161e−10 | 5.95982e−12 | 7.96377e−12 |
0.2 | 0.032 | 8.72558e−10 | 3.32835e−10 | 2.50069e−11 | 1.72426e−11 |
0.3 | 0.063 | 8.44767e−10 | 3.09044e−10 | 3.99208e−11 | 2.53607e−11 |
0.4 | 0.096 | 8.01978e−10 | 3.03655e−10 | 6.62202e−11 | 1.77049e−11 |
0.5 | 0.125 | 7.15163e−10 | 2.80179e−10 | 9.72653e−11 | 1.43694e−11 |
0.6 | 0.144 | 7.48198e−10 | 3.93943e−12 | 9.33972e−11 | 2.52972e−11 |
0.7 | 0.147 | 9.12601e−10 | 1.80271e−10 | 8.55649e−11 | 4.49204e−11 |
0.8 | 0.128 | 9.68271e−10 | 6.99664e−11 | 1.43124e−10 | 4.37169e−11 |
0.9 | 0.081 | 7.16510e−10 | 3.93030e−11 | 3.19812e−10 | 4.91018e−11 |
Example 6
Comparison of absolute errors for Example 6 ( \(\pmb{k=2}\) )
x | Exact solution | Absolute error M = 3 | Absolute error M = 4 | Absolute error M = 5 | Absolute error M = 6 | Absolute error M = 7 |
---|---|---|---|---|---|---|
0.1 | 0.313265850498063 | 2.55607e−7 | 8.14189e−7 | 1.81121e−8 | 1.21233e−9 | 3.14035e−10 |
0.2 | 0.3030154228323 | 4.26548e−7 | 8.60558e−7 | 1.57268e−8 | 1.29968e−9 | 3.06900e−10 |
0.3 | 0.286047265304854 | 1.31980e−6 | 8.60956e−7 | 1.57118e−8 | 1.31094e−9 | 2.98647e−10 |
0.4 | 0.262531127456033 | 8.54592e−7 | 8.45643e−7 | 1.38501e−8 | 1.26470e−9 | 2.88410e−10 |
0.5 | 0.232696783873834 | 5.95794e−7 | 8.74915e−7 | 1.51067e−8 | 1.33441e−9 | 2.82368e−10 |
0.6 | 0.196826805692954 | 4.48716e−8 | 6.66434e−7 | 1.27714e−8 | 1.07644e−9 | 2.13563e−10 |
0.7 | 0.155248106682756 | 2.34595e−7 | 4.72070e−7 | 7.86159e−9 | 7.56736e−10 | 1.50629e−10 |
0.8 | 0.108322763444465 | 9.46220e−7 | 3.02305e−7 | 5.09872e−9 | 4.93439e−10 | 9.44874e−11 |
0.9 | 0.0564386024692362 | 4.47318e−7 | 1.49726e−7 | 1.05906e−9 | 2.86694e−10 | 4.39479e−11 |
Comparison of absolute errors for Example 6 ( \(\pmb{k=3}\) )
x | Exact solution | Absolute error M = 3 | Absolute error M = 4 | Absolute error M = 5 | Absolute error M = 6 | Absolute error M = 7 |
---|---|---|---|---|---|---|
0.1 | 0.313265850498063 | 1.12155e−7 | 7.47073e−8 | 7.77016e−11 | 2.25000e−11 | 1.12567e−11 |
0.2 | 0.3030154228323 | 1.18254e−7 | 7.42881e−8 | 7.21018e−11 | 2.28459e−11 | 1.06202e−11 |
0.3 | 0.286047265304854 | 8.26092e−8 | 6.72829e−8 | 1.32635e−10 | 1.64894e−11 | 9.71828e−12 |
0.4 | 0.262531127456033 | 7.92919e−8 | 5.59685e−8 | 1.80870e−10 | 5.77443e−12 | 8.57608e−12 |
0.5 | 0.232696783873834 | 3.19442e−8 | 4.68132e−8 | 2.27523e−10 | 2.24772e−12 | 7.28159e−12 |
0.6 | 0.196826805692954 | 1.34284e−8 | 3.36344e−8 | 2.33153e−10 | 8.22350e−12 | 5.78329e−12 |
0.7 | 0.155248106682756 | 6.97215e−9 | 2.21505e−8 | 2.24769e−10 | 1.30745e−11 | 4.31280e−12 |
0.8 | 0.108322763444465 | 3.25668e−8 | 1.34976e−8 | 1.93600e−10 | 1.16738e−11 | 2.82023e−12 |
0.9 | 0.0564386024692362 | 8.78766e−9 | 6.50385e−9 | 8.87945e−11 | 5.50390e−12 | 1.33990e−12 |
x | Exact solution | N = 10 | N = 14 |
---|---|---|---|
0.1 | 0.313265850498063 | 1.05e−7 | 6.69e−8 |
0.2 | 0.3030154228323 | 6.33e−9 | 7.87e−9 |
0.3 | 0.286047265304854 | 5.91e−8 | 6.92e−9 |
0.4 | 0.262531127456033 | 2.12e−7 | 2.87e−8 |
0.5 | 0.232696783873834 | 1.00e−8 | 7.40e−10 |
0.6 | 0.196826805692954 | 5.36e−7 | 6.32e−8 |
0.7 | 0.155248106682756 | 4.25e−8 | 6.95e−8 |
0.8 | 0.108322763444465 | 8.32e−7 | 3.38e−9 |
0.9 | 0.0564386024692362 | 4.67e−8 | 7.85e−8 |
Example 7
The numerical results of Example 7 with \(\pmb{k=3}\)
x | M = 3 | M = 4 | M = 5 | M = 6 | Method in [16] with n = 14 |
---|---|---|---|---|---|
0.1 | 0.829706090093794 | 0.82970609213969 | 0.829706092330806 | 0.829706092433877 | 0.82970609243390 |
0.2 | 0.833374731260013 | 0.833374733298822 | 0.833374733492775 | 0.833374733591078 | 0.83337473359110 |
0.3 | 0.839489911535405 | 0.839489913690883 | 0.8394899138631 | 0.83948991395376 | 0.83948991395381 |
0.4 | 0.848052782678684 | 0.848052784769831 | 0.848052784915137 | 0.848052784996097 | 0.84805278499617 |
0.5 | 0.85906492472985 | 0.859064926965802 | 0.859064927099445 | 0.85906492716925 | 0.85906492716933 |
0.6 | 0.872528317441265 | 0.872528319803658 | 0.872528319900291 | 0.87252831995828 | 0.87252831995828 |
0.7 | 0.888445303225948 | 0.888445305504435 | 0.888445305576983 | 0.888445305623171 | 0.88844530562329 |
0.8 | 0.906818545614923 | 0.906818547978651 | 0.906818548031817 | 0.906818548066776 | 0.90681854806690 |
0.9 | 0.927650986181403 | 0.927650988306455 | 0.927650988340659 | 0.927650988365551 | 0.92765098836568 |
Example 8
Comparison of absolute errors for Example 8 ( \(\pmb{k=2}\) )
x | Exact solution | Absolute error M = 3 | Absolute error M = 4 | Absolute error M = 5 | Absolute error M = 6 | Absolute error M = 7 |
---|---|---|---|---|---|---|
0.1 | 0.100665339016074 | 1.99905e−4 | 2.28846e−5 | 1.72044e−6 | 4.80911e−7 | 5.02872e−8 |
0.2 | 0.205291382243876 | 1.91458e−4 | 2.34563e−5 | 1.63517e−6 | 4.79635e−7 | 4.89559e−8 |
0.3 | 0.317687905980875 | 1.67910e−4 | 2.28553e−5 | 1.57308e−6 | 4.65931e−7 | 4.71142e−8 |
0.4 | 0.441387397450343 | 1.61539e−4 | 2.17240e−5 | 1.47290e−6 | 4.39266e−7 | 4.45112e−8 |
0.5 | 0.579559511251008 | 1.51975e−4 | 2.13789e−5 | 1.36356e−6 | 4.18131e−7 | 4.14292e−8 |
0.6 | 0.734970520367994 | 9.83797e−5 | 1.29024e−5 | 9.77157e−7 | 2.95283e−7 | 2.86342e−8 |
0.7 | 0.909981686939921 | 4.84761e−5 | 5.13638e−6 | 6.43969e−7 | 1.71463e−7 | 1.62533e−8 |
0.8 | 1.10657514524663 | 7.88980e−6 | 1.98660e−6 | 2.77744e−7 | 5.15655e−8 | 4.25484e−9 |
0.9 | 1.32639533423358 | 3.47573e−5 | 7.91713e−6 | 5.42440e−8 | 5.87647e−8 | 6.74598e−9 |
Comparison of absolute errors for Example 8 ( \(\pmb{k=3}\) )
x | Exact solution | Absolute error M = 3 | Absolute error M = 4 | Absolute error M = 5 | Absolute error M = 6 | Absolute error M = 7 |
---|---|---|---|---|---|---|
0.1 | 0.100665339016074 | 2.00943e−5 | 2.49095e−6 | 7.39087e−8 | 8.51470e−9 | 9.01484e−12 |
0.2 | 0.205291382243876 | 1.91219e−5 | 2.43795e−6 | 7.17544e−8 | 8.33840e−9 | 3.18067e−12 |
0.3 | 0.317687905980875 | 1.75163e−5 | 2.21871e−6 | 6.28094e−8 | 7.40476e−9 | 3.37537e−11 |
0.4 | 0.441387397450343 | 1.46803e−5 | 1.89551e−6 | 4.88112e−8 | 5.95511e−9 | 9.98910e−11 |
0.5 | 0.579559511251008 | 1.21171e−5 | 1.59113e−6 | 3.61323e−8 | 4.63996e−9 | 1.48223e−10 |
0.6 | 0.734970520367994 | 8.42905e−6 | 1.08465e−6 | 2.44077e−8 | 3.18752e−9 | 1.28915e−10 |
0.7 | 0.909981686939921 | 5.01234e−6 | 5.98191e−7 | 1.33754e−8 | 1.77942e−9 | 1.03023e−10 |
0.8 | 1.10657514524663 | 1.47814e−6 | 1.20797e−7 | 4.04928e−9 | 6.02002e−10 | 6.26907e−11 |
0.9 | 1.32639533423358 | 1.54737e−6 | 3.27982e−7 | 2.46634e−9 | 3.25944e−10 | 2.24953e−12 |
5 Conclusion
The main goal of this paper is to develop an efficient and accurate method to solve linear or nonlinear singular boundary value problems with four different types’ initial boundary conditions and mixed boundary conditions. The Laguerre wavelets operational matrix of integration together with the collocation method is utilized to reduce the problem to the solution of linear or nonlinear algebraic equations. One of the main advantages of the developed algorithm is that it does not require any modification while switching from the linear case to the nonlinear case. Another one is that high accuracy approximate solutions are achieved using very small values of k and M. Illustrative examples are included to demonstrate the validity and applicability of the proposed method.
Declarations
Acknowledgements
This work is supported by the Education Department Youth Science Foundation of Jiangxi Province (Grant No. GJJ14492), the Youth Science Foundation of Jiangxi Province (Grant No. 20151BAB211004), and the Ph.D. Research Startup Foundation of East China University of Technology (Grant No. DHBK2012205).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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