A NUMERICAL TECHNIQUE FOR THE SOLUTION OF GENERAL EIGHTH ORDER BOUNDARY VALUE PROBLEMS: A FINITE DIFFERENCE METHOD

In this article, we present a novel finite difference method for the numerical solution of the eighth order boundary value problems in ordinary differential equations. We have discretized the problem by using the boundary conditions in a natural way to obtain a system of equations. Then we have solved system of equations to obtain a numerical solution of the problem. Also we obtained numerical values of derivatives of solution as a byproduct of the method. The numerical experiments show that proposed method is efficient and fourth order accurate

A Numerical Technique for the Solution of General Eighth Order Boundary Value Problems: a Finite Difference Method

In this article, we present a novel finite difference method for the numerical solution of the eighth order boundary value problems in ordinary differential equations. We have discretized the problem by using the boundary conditions in a natural way to obtain a system of equations. Then we have solved system of equations to obtain a numerical solution of the problem. Also we obtained numerical values of derivatives of solution as a byproduct of the method. The numerical experiments show that proposed method is efficient and fourth order accurate.The author is grateful to the anonymous reviewers and editor for their valuable suggestions, which substantially improved the standard of the paper

Element-Free Galerkin Method Based on Block-Pulse Wavelets Integration for Solving Fourth-Order Obstacle Problem

We introduce improved element-free Galerkin method based on block pulse wavelet integration for numerical approximations to the solution of a system of fourth-order boundary-value problems associated with obstacle, unilateral, and contact problems. Moving least squares (MLS) approach is used to construct shape functions with optimized weight functions and basis. Numerical results for test problems are presented in this article to elaborate the pertinent features for the proposed technique. Comparison with existing techniques shows that our proposed method based on integration technique provides better approximation at reduced computational cost

Numerical Solution of a Class of Nonlinear System of Second-Order Boundary-Value Problems: a Fourth-Order Cubic Spline Approach

A cubic B-spline collocation approach is described and presented for the numerical solution of an extended system of linear and nonlinear second-order boundary-value problems. The system, whether regular or singularly perturbed, is tackled using a spline collocation approach constructed over uniform or non-uniform meshes. The rate of convergence is discussed theoretically and verified numerically to be of fourth-order. The efficiency and applicability of the technique are demonstrated by applying the scheme to a number of linear and nonlinear examples. The numerical solutions are contrasted with both analytical and other existing numerical solutions that exist in the literature. The numerical results demonstrate that this method is superior as it yields more accurate solutions

Solving nonlinear problems by Ostrowski Chun type parametric families

In this paper, by using a generalization of Ostrowski' and Chun's methods two bi-parametric families of predictor-corrector iterative schemes, with order of convergence four for solving system of nonlinear equations, are presented. The predictor of the first family is Newton's method, and the predictor of the second one is Steffensen's scheme. One of them is extended to the multidimensional case. Some numerical tests are performed to compare proposed methods with existing ones and to confirm the theoretical results. We check the obtained results by solving the molecular interaction problem.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-02 and FONDOCYT, Republica Dominicana.Cordero Barbero, A.; Maimo, J.; Torregrosa Sánchez, JR.; Vassileva, M. (2015). Solving nonlinear problems by Ostrowski Chun type parametric families. Journal of Mathematical Chemistry. 53(1):430-449. https://doi.org/10.1007/s10910-014-0432-zS430449531M.S. Petkovic̀, B. Neta, L.D. Petkovic̀, J. Dz̆unic̀, Multipoint Methods for Solving Nonlinear Equations (Academic, New York, 2013)M. Mahalakshmi, G. Hariharan, K. Kannan, The wavelet methods to linear and nonlinear reaction–diffusion model arising in mathematical chemistry. J. Math. Chem. 51(9), 2361–2385 (2013)P.G. Logrado, J.D.M. Vianna, Partitioning technique procedure revisited: Formalism and first application to atomic problems. J. Math. Chem. 22, 107–116 (1997)C.G. Jesudason, I. Numerical nonlinear analysis: differential methods and optimization applied to chemical reaction rate determination. J. Math. Chem. 49, 1384–1415 (2011)K. Maleknejad, M. Alizadeh, An efficient numerical scheme for solving hammerstein integral equation arisen in chemical phenomenon. Procedia Comput. Sci. 3, 361–364 (2011)R.C. Rach, J.S. Duan, A.M. Wazwaz, Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. J. Math. Chem. 52, 255–267 (2014)J.F. Steffensen, Remarks on iteration. Skand. Aktuar Tidskr. 16, 64–72 (1933)J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970)H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration. J. ACM 21, 643–651 (1974)J.R. Sharma, R.K. Guha, R. Sharma, An efficient fourth order weighted-Newton method for systems of nonlinear equations. Numer. Algorithms 62, 307–323 (2013)J.R. Sharma, H. Arora, On efficient weighted-Newton methods for solving systems of nonlinear equations. Appl. Math. Comput. 222, 497–506 (2013)M. Abad, A. Cordero, J.R. Torregrosa, Fourth- and fifth-order methods for solving nonlinear systems of equations: an application to the Global positioning system. Abstr. Appl. Anal.(2013) Article ID:586708. doi: 10.1155/2013/586708F. Soleymani, T. Lotfi, P. Bakhtiari, A multi-step class of iterative methods for nonlinear systems. Optim. Lett. 8, 1001–1015 (2014)M.T. Darvishi, N. Darvishi, SOR-Steffensen-Newton method to solve systems of nonlinear equations. Appl. Math. 2(2), 21–27 (2012). doi: 10.5923/j.am.20120202.05F. Awawdeh, On new iterative method for solving systems of nonlinear equations. Numer. Algorithms 5(3), 395–409 (2010)D.K.R. Babajee, A. Cordero, F. Soleymani, J.R. Torregrosa, On a novel fourth-order algorithm for solving systems of nonlinear equations. J. Appl. Math. (2012) Article ID:165452. doi: 10.1155/2012/165452A. Cordero, J.R. Torregrosa, M.P. Vassileva, Pseudocomposition: a technique to design predictor–corrector methods for systems of nonlinear equations. Appl. Math. Comput. 218(23), 1496–1504 (2012)A. Cordero, J.R. Torregrosa, M.P. Vassileva, Increasing the order of convergence of iterative schemes for solving nonlinear systems. J. Comput. Appl. Math. 252, 86–94 (2013)A.M. Ostrowski, Solution of Equations and System of Equations (Academic, New York, 1966)C. 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A new fourth-order family for solving nonlinear problems and its dynamics

In this manuscript, a new parametric class of iterative methods for solving nonlinear systems of equations is proposed. Its fourth-order of convergence is proved and a dynamical analysis on low-degree polynomials is made in order to choose those elements of the family with better conditions of stability. These results are checked by solving the nonlinear system that arises from the partial differential equation of molecular interaction.This research was supported by Ministerio de Ciencia y Tecnologia MTM2011-28636-C02-{01, 02} and Universitat Politecnica de Valencia SP20120474.Cordero Barbero, A.; Feng, L.; Magrenan, A.; Torregrosa Sánchez, JR. (2015). A new fourth-order family for solving nonlinear problems and its dynamics. Journal of Mathematical Chemistry. 53(3):893-910. https://doi.org/10.1007/s10910-014-0464-4S893910533R.C. Rach, J.S. Duan, A.M. Wazwaz, Solving coupled Lane–Emden boundary value problems in catalytic diffusion reactions by the Adomian decomposition method. J. 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Convergence of summation-by-parts finite difference methods for the wave equation

In this paper, we consider finite difference approximations of the second order wave equation. We use finite difference operators satisfying the summation-by-parts property to discretize the equation in space. Boundary conditions and grid interface conditions are imposed by the simultaneous-approximation-term technique. Typically, the truncation error is larger at the grid points near a boundary or grid interface than that in the interior. Normal mode analysis can be used to analyze how the large truncation error affects the convergence rate of the underlying stable numerical scheme. If the semi-discretized equation satisfies a determinant condition, two orders are gained from the large truncation error. However, many interesting second order equations do not satisfy the determinant condition. We then carefully analyze the solution of the boundary system to derive a sharp estimate for the error in the solution and acquire the gain in convergence rate. The result shows that stability does not automatically yield a gain of two orders in convergence rate. The accuracy analysis is verified by numerical experiments.Comment: In version 2, we have added a new section on the convergence analysis of the Neumann problem, and have improved formulations in many place

Multiderivative methods for nonlinear second order boundary value problems

Second, fourth and sixth order methods are developed and analysed for the numerical solution of non-linear second order boundary value problems. The methods arise from a two-step recurrence relation involving exponential terms, these being replaced by Padé approximants . The methods are tested on two problems from the literature