New Wavelets Collocation Method for Solving Second-Order Multipoint Boundary Value Problems Using Chebyshev Polynomials of Third and Fourth Kinds

and Applied Analysis 3 The polynomials V n (x) and W n (x) are orthogonal on (−1, 1); that is,


Introduction
Spectral methods are one of the principal methods of discretization for the numerical solution of differential equations.The main advantage of these methods lies in their accuracy for a given number of unknowns (see, e.g., [1][2][3][4]).For smooth problems in simple geometries, they offer exponential rates of convergence/spectral accuracy.In contrast, finite difference and finite-element methods yield only algebraic convergence rates.The three most widely used spectral versions are the Galerkin, collocation, and tau methods.Collocation methods [5,6] have become increasingly popular for solving differential equations, also they are very useful in providing highly accurate solutions to nonlinear differential equations.
Many practical problems arising in numerous branches of science and engineering require solving high even-order and high odd-order boundary value problems.Legendre polynomials have been previously used for obtaining numerical spectral solutions for handling some of these kinds of problems (see, e.g., [7,8]).In [9], the author has constructed some algorithms by selecting suitable combinations of Legendre polynomials for solving the differentiated forms of highodd-order boundary value problems with the aid of Petrov-Galerkin method, while in the two papers [10,11], the authors handled third-and fifth-order differential equations using Jacobi tau and Jacobi collocation methods.
Multipoint boundary value problems (BVPs) arise in a variety of applied mathematics and physics.For instance, the vibrations of a guy wire of uniform cross-section composed of  parts of different densities can be set up as a multipoint BVP, as in [12]; also, many problems in the theory of elastic stability can be handled by the method of multipoint problems [13].The existence and multiplicity of solutions of multipoint boundary value problems have been studied by many authors; see [14][15][16][17] and the references therein.For two-point BVPs, there are many solution methods such as orthonormalization, invariant imbedding algorithms, finite difference, and collocation methods (see, [18][19][20]).However, there seems to be little discussion about numerical solutions of multipoint boundary value problems.
Second-order multipoint boundary value problems (BVP) arise in the mathematical modeling of deflection of cantilever beams under concentrated load [21,22], deformation in some form in any scheme of the numerical solution, and it is well known that other numerical methods do not perform well near singularities.Finally, due to their rapid convergence, Chebyshev wavelets collocation method does not suffer from the common instability problems associated with other numerical methods.
The main aim of this paper is to develop two new spectral algorithms for solving second-order multipoint BVPs based on shifted third-and fourth-kind Chebyshev wavelets.The method reduces the differential equation with its boundary conditions to a system of algebraic equations in the unknown expansion coefficients.Large systems of algebraic equations may lead to greater computational complexity and large storage requirements.However the third-and fourth-kind Chebyshev wavelets collocation method reduces drastically the computational complexity of solving the resulting algebraic system.
The structure of the paper is as follows.In Section 2, we give some relevant properties of Chebyshev polynomials of third and fourth kinds and their shifted ones.In Section 3, the third-and fourth-kind Chebyshev wavelets are constructed.Also, in this section, we ascertain the convergence of the Chebyshev wavelets series expansion.Two new shifted Chebyshev wavelets collocation methods for solving second-order linear and nonlinear multipoint boundary value problems are implemented and presented in Section 4. In Section 5, some numerical examples are presented to show the efficiency and the applicability of the presented algorithms.Some concluding remarks are given in Section 6.

Some Properties of 𝑉 𝑘 (𝑥) and 𝑊 𝑘 (𝑥)
The Chebyshev polynomials   () and   () of third and fourth kinds are polynomials of degree  in  defined, respectively, by (see [38]) where  = cos ; also they can be obtained explicitly as two particular cases of Jacobi polynomials  (,)  () for the two nonsymmetric cases correspond to  = − = ±1/2.Explicitly, we have It is readily seen that Hence, it is sufficient to establish properties and relations for   () and then deduce their corresponding properties and relations for   () (by replacing  by −).
The polynomials   () and   () are orthogonal on (−1, 1); that is, where and they may be generated by using the two recurrence relations with the initial values with the initial values The shifted Chebyshev polynomials of third and fourth kinds are defined on [0, 1], respectively, as All results of Chebyshev polynomials of third and fourth kinds can be easily transformed to give the corresponding results for their shifted ones.The orthogonality relations of  *  () and  *  () on [0, 1] are given by where

Shifted Third-and Fourth-Kind Chebyshev Wavelets
Wavelets constitute of a family of functions constructed from dilation and translation of single function called the mother wavelet.When the dilation parameter  and the translation parameter  vary continuously, then we have the following family of continuous wavelets: Each of the third-and fourth-kind Chebyshev wavelets   () = (, , , ) has four arguments: ,  ∈ N,  is the order of the polynomial  *  () or  *  (), and  is the normalized time.They are defined explicitly on the interval [0, 1] as

Function Approximation.
A function () defined over [0, 1] may be expanded in terms of Chebyshev wavelets as where and the weights  *  ,  = 1, 2, are given in (12).Assume that () can be approximated in terms of Chebyshev wavelets as

Convergence Analysis.
In this section, we state and prove a theorem to ascertain that the third-and fourth-kind Chebyshev wavelets expansion of a function (), with bounded second derivative, converges uniformly to ().
can be expanded as an infinite series of third-kind Chebyshev wavelets; then this series converges uniformly to ().Explicitly, the expansion coefficients in (16) satisfy the following inequality: Proof.From ( 16), it follows that If we make use of the substitution 2   −  = cos  in ( 19), then we get which in turn, and after performing integration by parts two times, yields where Now, we have Finally, since  ⩽ 2  − 1, we have Remark 2. The estimation in ( 18) is also valid for the coefficients of fourth-kind Chebyshev wavelets expansion.The proof is similar to the proof of Theorem 1.

Numerical Examples
In this section, the presented algorithms in Section 4 are applied to solve both of linear and nonlinear multipoint BVPs.Some examples are considered to illustrate the efficiency and applicability of the two proposed algorithms.
Example 1.Consider the second-order nonlinear BVP (see [6,28]): The two proposed methods are applied to the problem for the case corresponding to  = 0 and  = 8.The numerical solutions are shown in Table 1.Due to nonavailability of the exact solution, we compare our results with Haar wavelets method [6], ADM solution [28] and ODEs Solver from Mathematica which is carried out by using Runge-Kutta method.This comparison is also shown in Table 1.
Example 2. Consider the second-order linear BVP (see, [39,40]): ) . ( The exact solution of problem ( 32) is given by In Table 2, the maximum absolute error  is listed for  = 1 and various values of , while in Table 3, we give a comparison between the best errors resulted from the application of various methods for Example 2, while in Figure 1, we give a comparison between the exact solution of (32) with three approximate solutions.
Example 3. Consider the second-order singular nonlinear BVP (see [40,41]): ) , with the exact solution () = sinh .In Table 4, the maximum absolute error  is listed for  = 0 and various values of , while in Table 5 we give a comparison between the best errors resulted from the application of various methods for Example 3.This table shows that our two algorithms are more accurate if compared with the two methods developed in [40,41].
Example 4. Consider the second-order nonlinear BVP (see [42]): ) − 0.0286634, ) − 0.0401287, where The exact solution of ( 35) is given by () = (1/3) sin( −  2 ).In Table 6, the maximum absolute error  is listed for  = 2 and various values of , and in Table 7 we give a comparison between best errors resulted from the application of various methods for Example 4. This table shows that our two algorithms are more accurate if compared with the method developed in [42]. where and () is chosen such that the exact solution of ( 37) is () = (1 − )  .In Table 8, the maximum absolute error  is listed for  = 1 and various values of , while in Figure 2, we give a comparison between the exact solution of (37) with three approximate solutions.
Remark 3. It is worth noting here that the obtained numerical results in the previous solved six examples are very accurate, although the number of retained modes in the spectral expansion is very few, and again the numerical results are comparing favorably with the known analytical solutions.

Concluding Remarks
In this paper, two algorithms for obtaining numerical spectral wavelets solutions for second-order multipoint linear and nonlinear boundary value problems are analyzed and discussed.Chebyshev polynomials of third and fourth kinds are used.One of the advantages of the developed algorithms is their availability for application on singular boundary value problems.Another advantage is that high accurate approximate solutions are achieved using a few number of terms of the approximate expansion.The obtained numerical results are comparing favorably with the analytical ones.

Table 1 :
Comparison between different solutions for Example 1.

Table 2 :
The maximum absolute error  for Example 2.

Table 3 :
The best errors for Example 2.

Table 4 :
The maximum absolute error  for Example 3.

Table 5 :
The best errors for Example 3.

Table 6 :
The maximum absolute error  for Example 4.

Table 7 :
The best errors for Example 4.

Table 8 :
The maximum absolute error  for Example 5.