Numerical Solutions for the Eighth-Order Initial and Boundary Value Problems Using the Second Kind Chebyshev Wavelets

Abstract

A collocation method based on the second kind Chebyshev wavelets is proposed for the numerical solution of eighth-order two-point boundary value problems (BVPs) and initial value problems (IVPs) in ordinary differential equations. The second kind Chebyshev wavelets operational matrix of integration is derived and used to transform the problem to a system of algebraic equations. The uniform convergence analysis and error estimation for the proposed method are given. Accuracy and efficiency of the suggested method are established through comparing with the existing quintic B-spline collocation method, homotopy asymptotic method, and modified decomposition method. Numerical results obtained by the present method are in good agreement with the exact solutions available in the literatures.

Academic Editor:Ricardo Weder

School of Science, East China Institute of Technology, Nanchang 330013, China

Received 20 March 2015; Accepted 1 July 2015; 14 July 2015

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Eighth-order differential equations govern the physics of some hydrodynamic stability problems. Chandrasekhar reported that when an infinite horizontal layer of fluid is heated from below and under the action of rotation, instability sets in [1]. When the instability sets in as overstability, the problem is modeled by an eighth-order ordinary differential equation for which the existence and uniqueness of the solution can be found in the book [2]. In this paper, we consider linear and nonlinear eighth-order BVPs of the form [figure omitted; refer to PDF] subject to the following three types of boundary conditions,

: Type I: [figure omitted; refer to PDF]

: Type II: [figure omitted; refer to PDF]

: Type III: [figure omitted; refer to PDF]

and the following type of initial condition,

: Type IV: [figure omitted; refer to PDF]

where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are continuous functions defined on the interval [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is real and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , are finite real numbers. Such differential equations can be solved numerically by finite difference method [3], homotopy asymptotic method [4, 5], the use of octic and nonic polynomial splines [6, 7], nonpolynomial splines [8], modified decomposition method [9], variational iteration decomposition method [10], and quintic B-spline collocation method [11]. In recent years, wavelets have received considerable attention in the field of numerical approximations [12, 13]. Different types of wavelets and approximating functions have been used in the numerical solution of boundary value problems [14]. Chebyshev wavelets are widely used in solving nonlinear integrodifferential equations and partial differential equations [15-18]. The motivation of this research is to find a simple and accurate method based on the second kind Chebyshev wavelets for the numerical solution of BVPs given in (1) under the assumption of unique solution for the problem. Orthogonal polynomial methods have seen significant achievements in dealing with various numerical problems, for example, Legendre polynomials, Chebyshev polynomials, Laguerre polynomials, and Hermite polynomials. However, 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. But according to the definition of wavelets, wavelets window shape can be arbitrarily changed by the dilations and the time localization can be moved through the translations. This characteristic of time-frequency localization can overcome the defect and allows us to obtain very accurate numerical solutions.

The rest of this paper is organized as follows. Section 2 introduces the second kind Chebyshev wavelets and their properties. The uniform convergence analysis and error estimation of the second kind Chebyshev wavelets expansion are also given. In Section 3, Chebyshev wavelets operational matrix of integration is derived. In Section 4, the proposed method is applied to approximate solution of the problem. Section 5 gives some examples to test the proposed method. A conclusion is drawn in Section 6.

2. Properties of the Second Kind Chebyshev Wavelets

Wavelets constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet. When the dilation parameter [figure omitted; refer to PDF] and the translation parameter [figure omitted; refer to PDF] vary continuously, we have the following family of continuous wavelets: [figure omitted; refer to PDF] If we restrict the parameters [figure omitted; refer to PDF] and [figure omitted; refer to PDF] to discrete values as [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are positive integers, we have the following family of discrete wavelets: [figure omitted; refer to PDF] which form a wavelet basis for [figure omitted; refer to PDF] . In particular, when [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] form an orthonormal basis.

The second kind Chebyshev wavelets [figure omitted; refer to PDF] have four arguments [1]: [figure omitted; refer to PDF] can assume any positive integer, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is the degree of the second kind Chebyshev polynomials, and [figure omitted; refer to PDF] is the normalized time. They are defined on the interval [figure omitted; refer to PDF] as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] is a fixed positive integer. The coefficient in (9) is for orthonormality. Here [figure omitted; refer to PDF] are the second kind Chebyshev polynomials of degree [figure omitted; refer to PDF] which are orthogonal with respect to the weight function [figure omitted; refer to PDF] on the interval [figure omitted; refer to PDF] and satisfy the following recursive formula: [figure omitted; refer to PDF] Note that when dealing with the second kind Chebyshev wavelets the weight function has to be dilated and translated as [figure omitted; refer to PDF]

A function [figure omitted; refer to PDF] defined over [figure omitted; refer to PDF] may be expanded by the second kind Chebyshev wavelets as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] in which [figure omitted; refer to PDF] denotes the inner production in [figure omitted; refer to PDF] . If the infinite series in (12) is truncated, then it can be written as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are [figure omitted; refer to PDF] matrices given by [figure omitted; refer to PDF]

The following theorem gives the convergence and accuracy estimation of the second kind Chebyshev wavelets expansion [19].

Theorem 1.

Let [figure omitted; refer to PDF] be a second-order derivative square-integrable function defined on [figure omitted; refer to PDF] with bounded second-order derivative; say [figure omitted; refer to PDF] for some constant [figure omitted; refer to PDF] ; then

(i) [figure omitted; refer to PDF] can be expanded as an infinite sum of the second kind Chebyshev wavelets and the series converges to [figure omitted; refer to PDF] uniformly; that is, [figure omitted; refer to PDF]

: where [figure omitted; refer to PDF] .

(ii) Consider [figure omitted; refer to PDF]

: where [figure omitted; refer to PDF] .

3. The Second Kind Chebyshev Wavelets Operational Matrix of Integration

In this section, we will derive precise integral of the second kind Chebyshev wavelet functions which play a great role in dealing with differential equations. First, we figure out the precise integral of the second kind Chebyshev wavelet functions with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . In this case, the six basis functions are given by [figure omitted; refer to PDF] on [figure omitted; refer to PDF] and [figure omitted; refer to PDF] on [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] . By integrating (18) and (19) from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] and representing them in the matrix form, we obtain [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . In fact, the matrix [figure omitted; refer to PDF] can be written as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] In general, when [figure omitted; refer to PDF] , we have [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is given in (15) and [figure omitted; refer to PDF] is a [figure omitted; refer to PDF] matrix given by [figure omitted; refer to PDF] here [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are [figure omitted; refer to PDF] matrices given by [figure omitted; refer to PDF] in which [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in (25) is called modification item which is given by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] are [figure omitted; refer to PDF] matrices given by [figure omitted; refer to PDF] [figure omitted; refer to PDF] . It is worthy to say that [figure omitted; refer to PDF] in (25) is often omitted in many literatures for simplicity when performing numerical calculations [15-17].

4. Description of the Proposed Method

In this section, we will use the second kind Chebyshev wavelets operational matrix of integration for solving eighth-order initial and boundary value problems. We assume that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is an unknown vector which should be found and [figure omitted; refer to PDF] is the vector defined in (15). Equation (31) is integrated repeatedly with appropriate limits of integration based on the boundary conditions. In this way, the solution [figure omitted; refer to PDF] and its eight derivatives are expressed in terms of Chebyshev wavelet functions and integrals. We take type I boundary conditions as an example to show the proposed method. The other types of boundary conditions can also be manipulated in a similar way. For simplicity, we take [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in type I. We introduce the following notations: [figure omitted; refer to PDF] Integrating (31) and using boundary conditions, we obtain the following: [figure omitted; refer to PDF] By means of boundary conditions, the unknown terms [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , can be calculated as follows: [figure omitted; refer to PDF] In order to calculate unknown vector [figure omitted; refer to PDF] in (31), we choose the collocation points [figure omitted; refer to PDF] The expressions of [figure omitted; refer to PDF] are substituted in the given differential equation (1) and discretization is applied using the collocation points (35). Thus, a system of [figure omitted; refer to PDF] equations in [figure omitted; refer to PDF] unknowns is obtained. The unknown vector [figure omitted; refer to PDF] can be given by solving this system according to Newton's iterative formula with the aid of Matlab. After finding the unknown vector [figure omitted; refer to PDF] , we can get the approximate solution by inserting [figure omitted; refer to PDF] into (33).

5. Numerical Examples

In order to illustrate the applicability and effectiveness of the proposed method, we apply it on several numerical examples with different types of boundary conditions. For the sake of comparison, we take problems from [5, 9, 11, 20]. Double precision arithmetic is used to reduce the round-off errors to minimum.

Example 1.

Consider the linear BVP [figure omitted; refer to PDF] subject to the boundary conditions [figure omitted; refer to PDF] The exact solution is given by [figure omitted; refer to PDF]

From Table 1, we can see that the approximate solutions obtained by adding modification item in (25) are more accurate than the case where it is omitted. So the following examples are all the case where the modification item in (25) is not omitted. We have compared our results with quintic B-spline collocation method [11]. It is clear from Table 2 that our scheme produces stable results and performs better when the number of points is increased.

Table 1: Comparison of approximate solution and exact solution of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]	Exact solution	Absolute error without modification item [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	Absolute error without modification item [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	Absolute error without modification item [figure omitted; refer to PDF] , [figure omitted; refer to PDF]
0.1	0.0994653826268083	1.10305e - 7	3.64211e - 9	1.11685e - 10
0.2	0.195424441305627	3.48090e - 7	3.09490e - 9	5.62525e - 11
0.3	0.283470349590961	3.00007e - 7	1.31128e - 8	1.44296e - 10
0.4	0.358037927433905	7.80905e - 7	3.00872e - 8	1.92387e - 10
0.5	0.412180317675032	4.11782e - 5	6.45118e - 7	1.15242e - 9
0.6	0.437308512093722	1.21810e - 5	1.21577e - 7	5.91031e - 10
0.7	0.4228880685688	3.52869e - 6	2.39439e - 7	1.11384e - 9
0.8	0.356086548558795	7.07550e - 6	3.91677e - 7	1.32972e - 9
0.9	0.221364280004125	3.65371e - 5	4.91306e - 7	1.28918e - 9

Table 2: Comparison of numerical results for Example 1.

[figure omitted; refer to PDF]	Exact solution	Absolute error with modification item [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	Absolute error with modification item [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	Absolute error with modification item [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	Absolute error reported in [11]
0.1	0.0994653826268083	1.88066e - 12	1.19071e - 14	1.63757e - 15	2.458692e - 7
0.2	0.195424441305627	1.95242e - 11	1.46077e - 13	1.98729e - 14	8.195639e - 7
0.3	0.283470349590961	5.97782e - 11	5.52058e - 13	7.41628e - 14	1.996756e - 6
0.4	0.358037927433905	1.04164e - 10	1.26049e - 12	1.67477e - 13	4.291534e - 6
0.5	0.412180317675032	1.23344e - 10	2.13401e - 12	2.82995e - 13	6.198883e - 6
0.6	0.437308512093722	1.03258e - 10	2.91611e - 12	3.92352e - 13	7.182360e - 6
0.7	0.4228880685688	5.82228e - 11	3.34476e - 12	4.70345e - 13	7.033348e - 6
0.8	0.356086548558795	1.74023e - 11	3.28548e - 12	5.08759e - 13	5.066395e - 6
0.9	0.221364280004125	1.03128e - 12	2.81263e - 12	5.21582e - 13	2.413988e - 6

Example 2.

Consider the linear BVP [figure omitted; refer to PDF] subject to the boundary conditions [figure omitted; refer to PDF] The exact solution is given by [figure omitted; refer to PDF]

A comparison of the absolute errors in some different points between the present method and optimal homotopy asymptotic method [5] is presented in Table 3. It is evident from Table 3 that the present method yields more accurate results.

Table 3: Comparison of numerical results for Example 2.

[figure omitted; refer to PDF]	Exact solution	Absolute error [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	Absolute error [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	Absolute error [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	Absolute error reported in [5]
0.1	0.994653826268083	3.44169e - 15	7.77156e - 16	1.11022e - 16	2.55072e - 9
0.2	0.977122206528136	1.97619e - 13	4.16333e - 14	1.11022e - 16	2.83738e - 9
0.3	0.944901165303202	2.00095e - 12	4.30100e - 13	0	3.12018e - 9
0.4	0.895094818584762	9.89686e - 12	2.17337e - 12	0	3.39829e - 9
0.5	0.824360635350064	3.28992e - 11	7.36377e - 12	0	3.67096e - 9
0.6	0.728847520156204	8.43374e - 11	1.92198e - 11	0	3.93756e - 9
0.7	0.604125812241143	1.78625e - 10	4.14863e - 11	1.11022e - 16	4.19756e - 9
0.8	0.445108185698493	3.22716e - 10	7.68903e - 11	3.88578e - 16	4.45065e - 9
0.9	0.245960311115695	4.97527e - 10	1.24352e - 10	7.77156e - 16	4.69673e - 9

Example 3.

We finally consider the nonlinear eighth-order BVP [figure omitted; refer to PDF] subject to the boundary conditions [figure omitted; refer to PDF] The exact solution for this problem is [figure omitted; refer to PDF]

Table 4 gives absolute errors for different points in the case of present method and methods in [9, 20]. It is clear from Table 4 that our scheme produces stable results and performs better when the number of points increases.

Table 4: Comparison of numerical results for Example 3.

[figure omitted; refer to PDF]	Exact solution	Absolute error [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	Absolute error [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	Absolute error [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	Absolute error reported in [9]	Absolute error reported in [20]
0.1	1.10517091807565	2.31710e - 10	2.63833e - 12	2.66453e - 15	1.9130e - 7	1.613e - 8
0.2	1.22140275816017	4.41112e - 10	5.26023e - 12	5.55111e - 15	1.2537e - 7	3.070e - 8
0.3	1.349858807576	6.08021e - 10	7.85194e - 12	8.21565e - 15	7.2467e - 8	4.227e - 8
0.4	1.49182469764127	7.16342e - 10	1.04021e - 11	1.11022e - 14	4.8491e - 8	4.972e - 8
0.5	1.64872127070013	7.55685e - 10	1.29052e - 11	1.37667e - 14	2.9101e - 7	5.231e - 8
0.6	1.82211880039051	7.22459e - 10	1.53608e - 11	1.68753e - 14	7.7974e - 8	4.978e - 8
0.7	2.01375270747048	6.20262e - 10	1.77764e - 11	1.95399e - 14	1.1064e - 7	4.237e - 8
0.8	2.22554092849247	4.59459e - 10	2.01625e - 11	2.26485e - 14	1.7131e - 7	3.080e - 8
0.9	2.45960311115695	2.56072e - 10	2.25375e - 11	2.57571e - 14	7.9265e - 8	1.619e - 8

Example 4.

Consider the linear IVP [figure omitted; refer to PDF] subject to the initial conditions [figure omitted; refer to PDF] The exact solution is given by [figure omitted; refer to PDF]

Table 5 gives absolute errors for different points in the case of the present method and optimal homotopy asymptotic method [5]. It is clear from Table 5 that our scheme gives better results. Actually, we can get more accurate results by increasing the numbers of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Table 5: Comparison of numerical results for Example 4.

[figure omitted; refer to PDF]	Exact solution	Absolute error [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	Absolute error [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	Absolute error [figure omitted; refer to PDF] , [figure omitted; refer to PDF]	Absolute error reported in [5]
0.1	0.994653826268083	1.11022e - 16	1.11022e - 16	1.11022e - 16	1.11022e - 16
0.2	0.977122206528136	3.26405e - 14	1.11022e - 16	1.11022e - 16	1.11022e - 16
0.3	0.944901165303202	6.47371e - 13	6.66133e - 16	0	3.33067e - 16
0.4	0.895094818584762	5.12467e - 12	5.21804e - 15	0	1.11022e - 15
0.5	0.824360635350064	2.49454e - 11	2.50910e - 14	0	1.11022e - 16
0.6	0.728847520156204	9.00054e - 11	9.07052e - 14	1.11022e - 16	2.22045e - 16
0.7	0.604125812241143	2.65085e - 10	2.68451e - 13	1.11022e - 16	1.11022e - 16
0.8	0.445108185698493	6.75052e - 10	6.86395e - 13	2.77555e - 16	1.60982e - 15
0.9	0.245960311115695	1.54368e - 9	1.57462e - 12	6.66133e - 16	7.49401e - 16

6. Conclusion

In this paper, the second kind Chebyshev wavelets and operational matrix of integration are used to find solutions of linear and nonlinear boundary value problems with different types for eighth-order differential equations. The numerical results obtained by the proposed method are in good agreement with the exact solutions available in the literatures. The uniform convergence analysis and error estimation for the second kind Chebyshev wavelets expansion are given. The method is computationally efficient and the algorithm can easily be implemented on a computer. Numerical comparison shows that our method is very reliable and accurate.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant no. 11326089), the Education Department Youth Science Foundation of Jiangxi Province (Grant no. GJJ14492), the Youth Science Foundation of Jiangxi Province (Grant no. 20151BAB211004), and PhD Research Startup Foundation of East China Institute of Technology (Grant no. DHBK2012205).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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