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

A collocation method based on the second kind Chebyshev wavelets is proposed for the numerical solution of eighth-order twopoint 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.Theuniform convergence analysis and error estimation for the proposedmethod 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.

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.

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  =  − 0 and  =  0  − 0 , where  0 > 1,  0 > 0, and  and  are positive integers, we have the following family of discrete wavelets: which form a wavelet basis for  2 (R).In particular, when  0 = 2 and  0 = 1, then  , () form an orthonormal basis.
The second kind Chebyshev wavelets  , () = (, , , ) have four arguments [1]:  can assume any positive integer,  = 1, 2, 3, . . ., 2 −1 ,  is the degree of the second kind Chebyshev polynomials, and  is the normalized time.They are defined on the interval [0, 1) as Advances in Mathematical Physics 3 where = 0, 1, 2, . . .,  − 1, 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 () = √ 1 −  2 on the interval [−1, 1] 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 () ∈  2 (R) defined over [0, 1) may be expanded by the second kind Chebyshev wavelets as where in which ⟨⋅, ⋅⟩  2  [0,1) denotes the inner production in  2  [0, 1).If the infinite series in (12) is truncated, then it can be written as where  and Ψ() are 2 −1  × 1 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 squareintegrable function defined on [0, 1) with bounded secondorder 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 where  ,, = (∫

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  = 2 and  = 3.In this case, the six basis functions are given by on [0, 1/2) and . By integrating ( 18) and ( 19) from 0 to  and representing them in the matrix form, we obtain Thus where and Ψ6 () = (1/24) (0 0  1,3 () 0 0  2,3 ())  .In fact, the matrix  6×6 can be written as where In general, when  ≥ 4, we have where Ψ() is given in (15) and  is a 2 −1  × 2 −1  matrix given by here  and  are  ×  matrices given by ( in which and Ψ() in (25) is called modification item which is given by where   are 1 ×  matrices given by = 1, 2, 3, . . ., 2 −1 .It is worthy to say that Ψ() in ( 25) is often omitted in many literatures for simplicity when performing numerical calculations [15][16][17].

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 (0) = 0, = 0,  (1) (1) = − ,  (2) (1) = − 4, 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 Bspline 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  (8) (1) (0) = 0, (1) (1) = − ,  (2) (1) = − 2. ( 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  (8) subject to the boundary conditions =  (2) (1) =  (4) (1) =  (6) (1) = . ( 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.

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.

Table 2 :
Comparison of numerical results for Example 1.

Table 3 :
Comparison of numerical results for Example 2.

Table 4 :
Comparison of numerical results for Example 3.

Table 5 :
Comparison of numerical results for Example 4.