- About this Journal ·
- Abstracting and Indexing ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

Abstract and Applied Analysis

Volume 2013 (2013), Article ID 542839, 9 pages

http://dx.doi.org/10.1155/2013/542839

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

^{1}Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia^{2}Department of Mathematics, Faculty of Science, Cairo University, Giza 12613, Egypt

Received 7 August 2013; Revised 13 September 2013; Accepted 13 September 2013

Academic Editor: Soheil Salahshour

Copyright © 2013 W. M. Abd-Elhameed et al. 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.

#### Abstract

This paper is concerned with introducing two wavelets collocation algorithms for solving linear and nonlinear multipoint boundary value problems. The principal idea for obtaining spectral numerical solutions for such equations is employing third- and fourth-kind Chebyshev wavelets along with the spectral collocation method to transform the differential equation with its boundary conditions to a system of linear or nonlinear algebraic equations in the unknown expansion coefficients which can be efficiently solved. Convergence analysis and some specific numerical examples are discussed to demonstrate the validity and applicability of the proposed algorithms. The obtained numerical results are comparing favorably with the analytical known solutions.

#### 1. 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., [1–4]). 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 high-odd-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 [14–17] 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, [18–20]). 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 of beams and plate deflection theory [23], obstacle problems [24], Troesch's problem relating to the confinement of a plasma column by radiation pressure [25, 26], temperature distribution of the radiation fin of trapezoidal profile [21, 27], and a number of other engineering applications. Many authors have used numerical and approximate methods to solve second-order BVPs. The details about the related numerical methods can be found in a large number of papers (see, for instance, [21, 23, 24, 28]). The Walsh wavelets and the semiorthogonal B-spline wavelets are used in [23, 29] to construct some numerical algorithms for the solution of second-order BVPs with Dirichlet and Neumann boundary conditions. Na [21] has found the numerical solution of second-, third-, and fourth-order BVPs by converting them into initial value problems and then applying a class of methods like nonlinear shooting, method of reduced physical parameters, method of invariant imbedding, and so forth. The presented approach in this paper can be applied to both BVPs and IVPs with a slight modification, but without the transformation of BVPs into IVPs or vice versa.

Wavelets theory is a relatively new and an emerging area in mathematical research. It has been applied to a wide range of engineering disciplines; particularly, wavelets are very successfully used in signal analysis for wave form representation and segmentations, time frequency analysis, and fast algorithms for easy implementation. Wavelets permit the accurate representation of a variety of functions and operators. Moreover, wavelets establish a connection with fast numerical algorithms, (see [30, 31]).

The application of Legendre wavelets for solving differential and integral equations is thoroughly considered by many authors (see, for instance, [32, 33]). Also, Chebyshev wavelets are used for solving some fractional and integral equations (see, [34, 35]).

Chebyshev polynomials have become increasingly crucial in numerical analysis, from both theoretical and practical points of view. It is well known that there are four kinds of Chebyshev polynomials, and all of them are special cases of the more widest class of Jacobi polynomials. The first and second kinds are special cases of the symmetric Jacobi polynomials (i.e., ultraspherical polynomials), while the third and fourth kinds are special cases of the nonsymmetric Jacobi polynomials. In the literature, there is a great concentration on the first and second kinds of Chebyshev polynomials and and their various uses in numerous applications, (see, for instance, [36]). However, there are few articles that concentrate on the other two types of Chebyshev polynomials, namely, third and fourth kinds and , either from theoretical or practical point of view and their uses in various applications (see, e.g., [37]). This motivates our interest in such polynomials. We therefore intend in this work to use them in a marvelous application of multipoint BVPs arising in physics.

There are several advantages of using Chebyshev wavelets approximations based on collocation spectral method. First, unlike most numerical techniques, it is now well established that they are characterized by exponentially decaying errors. Second, approximation by wavelets handles singularities in the problem. The effect of any such singularities will appear 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.

#### 2. Some Properties of and

The Chebyshev polynomials and of third and fourth kinds are polynomials of degree in defined, respectively, by (see [38]) where ; also they can be obtained explicitly as two particular cases of Jacobi polynomials for the two nonsymmetric cases correspond to . 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 ; 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 , 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 are given by where

#### 3. 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: , is the order of the polynomial or , and is the normalized time. They are defined explicitly on the interval as

##### 3.1. Function Approximation

A function defined over may be expanded in terms of Chebyshev wavelets as where and the weights , , are given in (12).

Assume that can be approximated in terms of Chebyshev wavelets as

##### 3.2. 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 .

Theorem 1. *Assume that a function , with , 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 in (19), then we get
which in turn, and after performing integration by parts two times, yields
where
Now, we have
Finally, since , 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.

#### 4. Solution of Multipoint BVPs

In this section, we present two Chebyshev wavelets collocation methods, namely, third-kind Chebyshev wavelets collocation method (3CWCM) and fourth-kind Chebyshev wavelets collocation method (4CWCM), to numerically solve the following multipoint boundary value problem (BVP): where , , and are piecewise continuous on ; may equal zero; ; ; , , , , and are constants such that ; is a nonlinear function of , and is a nonlinear function in .

Consider an approximate solution to (25) and (26) which is given in terms of Chebyshev wavelets as then the substitution of (27) into (25) enables one to write the residual of (25) in the form Now, the application of the typical collocation method (see, e.g., [5]) gives where are the first roots of or . Moreover, the use of the boundary conditions (26) gives Equations (29) and (30) generate equations in the unknown expansion coefficients, , which can be solved with the aid of the well-known Newton's iterative method. Consequently, we get the desired approximate solution given by (27).

#### 5. 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 and . 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 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 . In Table 4, the maximum absolute error is listed for 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]):
where
The exact solution of (35) is given by . In Table 6, the maximum absolute error is listed for 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].

*Example 5. *Consider the second-order singular linear BVP:
where
and is chosen such that the exact solution of (37) is . In Table 8, the maximum absolute error is listed for and various values of , while in Figure 2, we give a comparison between the exact solution of (37) with three approximate solutions.

*Example 6. *Consider the following nonlinear second-order BVP:
with the exact solution . We solve (39) using 3CWCM for the case corresponding to and , to obtain an approximate solution of . If we make use of (27), then the approximate solution can be expanded in terms of third-kind Chebyshev wavelets as
If we set
then (42) reduces to the form
If we substitute (44) into (39), then the residual of (39) is given by
We enforce the residual to vanish at the first root of , namely, at , to get
Furthermore, the use of the boundary conditions (40) and (41) yields
The solution of the nonlinear system of (46) and (47) gives
and consequently
which is the exact solution.

*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.

#### 6. 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.

#### Acknowledgment

The authors would like to thank the referee for his valuable comments and suggestions which improved the paper in its present form.

#### References

- E. H. Doha, W. M. Abd-Elhameed, and Y. H. Youssri, “Efficient spectral-Petrov-Galerkin methods for the integrated forms of third- and fifth-order elliptic differential equations using general parameters generalized Jacobi polynomials,”
*Applied Mathematics and Computation*, vol. 218, no. 15, pp. 7727–7740, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. M. Abd-Elhameed, E. H. Doha, and Y. H. Youssri, “Efficient spectral-Petrov-Galerkin methods for third- and fifth-order differential equations using general parameters generalized Jacobi polynomials,”
*Quaestiones Mathematicae*, vol. 36, no. 1, pp. 15–38, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - E. H. Doha, W. M. Abd-Elhameed, and A. H. Bhrawy, “New spectral-Galerkin algorithms for direct solution of high even-order differential equations using symmetric generalized Jacobi polynomials,”
*Collectanea Mathematica*, vol. 64, no. 3, pp. 373–394, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - E. H. Doha, A. H. Bhrawy, and R. M. Hafez, “On shifted Jacobi spectral method for high-order multi-point boundary value problems,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 10, pp. 3802–3810, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang,
*Spectral Methods*, Springer, Berlin, Germany, 2006. View at MathSciNet - Siraj-ul-Islam, I. Aziz, and B. Šarler, “The numerical solution of second-order boundary-value problems by collocation method with the Haar wavelets,”
*Mathematical and Computer Modelling*, vol. 52, no. 9-10, pp. 1577–1590, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. H. Bhrawy, A. S. Alofi, and S. I. El-Soubhy, “An extension of the Legendre-Galerkin method for solving sixth-order differential equations with variable polynomial coefficients,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 896575, 13 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. H. Doha and W. M. Abd-Elhameed, “Efficient solutions of multidimensional sixth-order boundary value problems using symmetric generalized Jacobi-Galerkin method,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 749370, 19 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. M. Abd-Elhameed, “Efficient spectral Legendre dual-Petrov-Galerkin algorithms for the direct solution of $(2n+1)$th-order linear differential equations,”
*Journal of the Egyptian Mathematical Society*, vol. 17, no. 2, pp. 189–211, 2009. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. H. Bhrawy and W. M. Abd-Elhameed, “New algorithm for the numerical solutions of nonlinear third-order differential equations using Jacobi-Gauss collocation method,”
*Mathematical Problems in Engineering*, vol. 2011, Article ID 837218, 14 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. H. Bhrawy, A. S. Alofi, and S. I. El-Soubhy, “Spectral shifted Jacobi tau and collocation methods for solving fifth-order boundary value problems,”
*Abstract and Applied Analysis*, vol. 2011, Article ID 823273, 14 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Moshiinsky, “Sobre los problemas de condiciones a la frontiera en una dimension de caracteristicas discontinuas,”
*Boletin de La Sociedad Matematica Mexicana*, vol. 7, article 125, 1950. View at Google Scholar - S. P. Timoshenko,
*Theory of Elastic Stability*, McGraw-Hill Book, New York, NY, USA, 2nd edition, 1961. View at MathSciNet - R. P. Agarwal and I. Kiguradze, “On multi-point boundary value problems for linear ordinary differential equations with singularities,”
*Journal of Mathematical Analysis and Applications*, vol. 297, no. 1, pp. 131–151, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Du, “Solvability of functional differential equations with multi-point boundary value problems at resonance,”
*Computers & Mathematics with Applications*, vol. 55, no. 11, pp. 2653–2661, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Feng and J. R. L. Webb, “Solvability of $m$-point boundary value problems with nonlinear growth,”
*Journal of Mathematical Analysis and Applications*, vol. 212, no. 2, pp. 467–480, 1997. View at Publisher · View at Google Scholar · View at MathSciNet - H. B. Thompson and C. Tisdell, “Three-point boundary value problems for second-order, ordinary, differential equations,”
*Mathematical and Computer Modelling*, vol. 34, no. 3-4, pp. 311–318, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. R. Scott and H. A. Watts, “SUPORT—a computer code for two-point boundary-value problems via orthonormalization,”
*Sandia Labs Report*75-0198, Sandia Laboratories, Albuquerque, NM, USA, 1975. View at Google Scholar - M. R. Scott and H. A. Watts, “Computational solution of linear two-point boundary value problems via orthonormalization,”
*SIAM Journal on Numerical Analysis*, vol. 14, no. 1, pp. 40–70, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. R. Scott and W. H. Vandevender, “A comparison of several invariant imbedding algorithms for the solution of two-point boundary-value problems,”
*Applied Mathematics and Computation*, vol. 1, no. 3, pp. 187–218, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Y. Na,
*Computational Methods in Engineering Boundary Value Problems*, vol. 145, Academic Press, New York, NY, USA, 1979. View at MathSciNet - K. E. Bisshopp and D. C. Drucker, “Large deflection of cantilever beams,”
*Quarterly of Applied Mathematics*, vol. 3, pp. 272–275, 1945. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Glabisz, “The use of Walsh-wavelet packets in linear boundary value problems,”
*Computers & Structures*, vol. 82, no. 2-3, pp. 131–141, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - Siraj-ul-Islam, M. A. Noor, I. A. Tirmizi, and M. A. Khan, “Quadratic non-polynomial spline approach to the solution of a system of second-order boundary-value problems,”
*Applied Mathematics and Computation*, vol. 179, no. 1, pp. 153–160, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Robert and J. Shipman, “Solution of Troesch's two-point boundary value problems by shooting techniques,”
*Journal of Computational Physics*, vol. 10, pp. 232–241, 1972. View at Publisher · View at Google Scholar - E. Wiebel,
*Confinement of a Plasma Column by Radiation Pressure in the Plasma in a Magnetic Field*, Stanford University Press, Stanfor, Calif, USA, 1958. - H. H. Keller and E. S. Holdrege, “Radiation heat transfer for annular fins of trapezoidal profile,”
*International Journal of High Performance Computing Applications*, vol. 92, pp. 113–116, 1970. View at Google Scholar - M. Tatari and M. Dehgan, “The use of the Adomian decomposition method for solving multipoint boundary value problems,”
*Physica Scripta*, vol. 73, pp. 672–676, 2006. View at Google Scholar - M. Lakestani and M. Dehghan, “The solution of a second-order nonlinear differential equation with Neumann boundary conditions using semi-orthogonal B-spline wavelets,”
*International Journal of Computer Mathematics*, vol. 83, no. 8-9, pp. 685–694, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Constantmldes,
*Applied Numerical Methods with Personal Computers*, McGraw-Hill, New York, NY, USA, 1987. - D. E. Newland,
*An Introduction to Random Vibrations, Spectral and Wavelet Analysis*, Longman Scientific and Technical, New York, NY, USA, 1993. - M. Razzaghi and S. Yousefi, “Legendre wavelets method for the solution of nonlinear problems in the calculus of variations,”
*Mathematical and Computer Modelling*, vol. 34, no. 1-2, pp. 45–54, 2001. View at Publisher · View at Google Scholar · View at MathSciNet - M. Razzaghi and S. Yousefi, “Legendre wavelets method for constrained optimal control problems,”
*Mathematical Methods in the Applied Sciences*, vol. 25, no. 7, pp. 529–539, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. Babolian and F. Fattahzadeh, “Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration,”
*Applied Mathematics and Computation*, vol. 188, no. 1, pp. 417–426, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Zhu and Q. Fan, “Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelet,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 6, pp. 2333–2341, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - E. H. Doha, W. M. Abd- Elhameed, and Y. H. Youssri, “Second kind Chebyshev operational matrix algorithm for solving differential equations of Lane-Emden type,”
*New Astronomy*, vol. 23-24, pp. 113–117, 2013. View at Google Scholar - E. H. Doha, W. M. Abd-Elhameed, and M. A. Bassuony, “New algorithms for solving high even-order differential equations using third and fourth Chebyshev-Galerkin methods,”
*Journal of Computational Physics*, vol. 236, pp. 563–579, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - J. C. Mason and D. C. Handscomb,
*Chebyshev Polynomials*, Chapman & Hall, New York, NY, USA, 2003. View at MathSciNet - Y. Lin, J. Niu, and M. Cui, “A numerical solution to nonlinear second order three-point boundary value problems in the reproducing kernel space,”
*Applied Mathematics and Computation*, vol. 218, no. 14, pp. 7362–7368, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Geng, “Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method,”
*Applied Mathematics and Computation*, vol. 215, no. 6, pp. 2095–2102, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Z. Geng, “A numerical algorithm for nonlinear multi-point boundary value problems,”
*Journal of Computational and Applied Mathematics*, vol. 236, no. 7, pp. 1789–1794, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Saadatmandi and M. Dehghan, “The use of sinc-collocation method for solving multi-point boundary value problems,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 2, pp. 593–601, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet