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.

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A fourth-order spline collocation approach for the numerical solution of a generalized system of second-order boundary-value problems

10.1063/1.4756508 ◽

2012 ◽

Author(s):

Ali Sayfy ◽

S. Khoury

Keyword(s):

Numerical Solution ◽

Boundary Value Problems ◽

Fourth Order ◽

Boundary Value ◽

Second Order ◽

Spline Collocation ◽

Generalized System ◽

Collocation Approach ◽

Order Spline

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A Fourth-Order Collocation Scheme for Two-Point Interface Boundary Value Problems

Advances in Numerical Analysis ◽

10.1155/2014/875013 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8

Author(s):

Rakhim Aitbayev ◽

Nazgul Yergaliyeva

Keyword(s):

Numerical Solution ◽

Boundary Value Problems ◽

Fourth Order ◽

Boundary Value ◽

Nodal Point ◽

Optimal Order ◽

Spline Collocation ◽

Order Error ◽

Jump Discontinuities ◽

Collocation Scheme

A fourth-order accurate orthogonal spline collocation scheme is formulated to approximate linear two-point boundary value problems with interface conditions. The coefficients of the differential operator may have jump discontinuities at the interface point, a nodal point of the scheme. Existence and uniqueness of the numerical solution are proved. Optimal order error estimates in the maximum norm are obtained, and a superconvergence property of the numerical solution in the maximal nodal norm is proved. Numerical results are presented confirming the theoretical estimates.

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Numerical Solutions of General Fourth Order Two Point Boundary Value Problems by Galerkin Method with Legendre Polynomials

Dhaka University Journal of Science ◽

10.3329/dujs.v62i2.21973 ◽

2015 ◽

Vol 62 (2) ◽

pp. 103-108 ◽

Cited By ~ 1

Author(s):

Md Bellal Hossain ◽

Md Shafiqul Islam

Keyword(s):

Differential Equations ◽

Boundary Conditions ◽

Boundary Value Problems ◽

Fourth Order ◽

Legendre Polynomials ◽

Numerical Solutions ◽

Nonlinear Differential Equations ◽

Boundary Value ◽

Equivalent Form ◽

Linear And Nonlinear

In this paper, Galerkin weighted residual method is presented to find the numerical solutions of the general fourth order linear and nonlinear differential equations with essential boundary conditions. For this, the given differential equations and the boundary conditions over arbitrary finite domain [a, b] are converted into its equivalent form over the interval [0, 1]. Here the Legendre polynomials, over the interval [0, 1], are chosen as trial functions satisfying the corresponding homogeneous form of the Dirichlet boundary conditions. Details matrix formulations are derived for solving linear and nonlinear boundary value problems (BVPs). Numerical examples for both linear and nonlinear BVPs are considered to verify the proposed formulation and the results obtained are compared. DOI: http://dx.doi.org/10.3329/dujs.v62i2.21973 Dhaka Univ. J. Sci. 62(2): 103-108, 2014 (July)

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Nature Inspired Computational Technique for the Numerical Solution of Nonlinear Singular Boundary Value Problems Arising in Physiology

The Scientific World JOURNAL ◽

10.1155/2014/837021 ◽

2014 ◽

Vol 2014 ◽

pp. 1-10

Author(s):

Suheel Abdullah Malik ◽

Ijaz Mansoor Qureshi ◽

Muhammad Amir ◽

Ihsanul Haq

Keyword(s):

Numerical Solution ◽

Boundary Value Problems ◽

Numerical Solutions ◽

Fitness Function ◽

Boundary Value ◽

Computational Technique ◽

Singular Boundary ◽

Interior Point Algorithm ◽

Singular Boundary Value Problems ◽

Hybrid Heuristic

We present a hybrid heuristic computing method for the numerical solution of nonlinear singular boundary value problems arising in physiology. The approximate solution is deduced as a linear combination of some log sigmoid basis functions. A fitness function representing the sum of the mean square error of the given nonlinear ordinary differential equation (ODE) and its boundary conditions is formulated. The optimization of the unknown adjustable parameters contained in the fitness function is performed by the hybrid heuristic computation algorithm based on genetic algorithm (GA), interior point algorithm (IPA), and active set algorithm (ASA). The efficiency and the viability of the proposed method are confirmed by solving three examples from physiology. The obtained approximate solutions are found in excellent agreement with the exact solutions as well as some conventional numerical solutions.

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