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.
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 subject to the following three types of boundary conditions, Type I: Type II: Type III:and the following type of initial condition, Type IV: where and are continuous functions defined on the interval , and is real and , , , and , , 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 and the translation parameter vary continuously, we have the following family of continuous wavelets: If we restrict the parameters and to discrete values as and , where , , and and are positive integers, we have the following family of discrete wavelets: which form a wavelet basis for . In particular, when and , then form an orthonormal basis.
The second kind Chebyshev wavelets have four arguments [1]: can assume any positive integer, , is the degree of the second kind Chebyshev polynomials, and is the normalized time. They are defined on the interval as where , and is a fixed positive integer. The coefficient in (9) is for orthonormality. Here are the second kind Chebyshev polynomials of degree which are orthogonal with respect to the weight function on the interval and satisfy the following recursive formula: Note that when dealing with the second kind Chebyshev wavelets the weight function has to be dilated and translated as
A function defined over may be expanded by the second kind Chebyshev wavelets as where in which denotes the inner production in . If the infinite series in (12) is truncated, then it can be written as where and are matrices given by
The following theorem gives the convergence and accuracy estimation of the second kind Chebyshev wavelets expansion [19].
Theorem 1. Let be a second-order derivative square-integrable function defined on with bounded second-order derivative; say for some constant ; then(i) can be expanded as an infinite sum of the second kind Chebyshev wavelets and the series converges to uniformly; that is, where .(ii)Consider where .
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 and . In this case, the six basis functions are given by on and on . Let . By integrating (18) and (19) from to and representing them in the matrix form, we obtain Thus where and . In fact, the matrix can be written as where In general, when , we have where is given in (15) and is a matrix given by here and are matrices given by in whichand in (25) is called modification item which is given by where are matrices given by . It is worthy to say that 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 where is an unknown vector which should be found and 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 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 and in type I. We introduce the following notations: Integrating (31) and using boundary conditions, we obtain the following: By means of boundary conditions, the unknown terms , , can be calculated as follows: In order to calculate unknown vector in (31), we choose the collocation points The expressions of are substituted in the given differential equation (1) and discretization is applied using the collocation points (35). Thus, a system of equations in unknowns is obtained. The unknown vector can be given by solving this system according to Newton’s iterative formula with the aid of Matlab. After finding the unknown vector , we can get the approximate solution by inserting 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 subject to the boundary conditions The exact solution is given by
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.
Example 2. Consider the linear BVP subject to the boundary conditions The exact solution is given by
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.
Example 3. We finally consider the nonlinear eighth-order BVP subject to the boundary conditions The exact solution for this problem is
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.
Example 4. Consider the linear IVP subject to the initial conditions The exact solution is given by
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 and .
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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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).