Reproducing Kernel Algorithm for the Analytical-Numerical Solutions of Nonlinear Systems of Singular Periodic Boundary Value Problems

The reproducing kernel algorithm is described in order to obtain the efficient analytical-numerical solutions to nonlinear systems of two point, second-order periodic boundary value problems with finitely many singularities. The analytical-numerical solutions are obtained in the form of an infinite...

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Bibliographic Details
Published in:Mathematical problems in engineering Vol. 2015; no. 2015; pp. 1 - 13
Main Author: Abu Arqub, Omar
Format: Journal Article
Language:English
Published: Cairo, Egypt Hindawi Publishing Corporation 01.01.2015
John Wiley & Sons, Inc
Subjects:
Algorithms
Approximation
Boundary conditions
Boundary value problems
Dynamical systems
Evolutionary algorithms
Kernel functions
Kernels
Mathematical analysis
Mathematical models
Mathematical problems
Nonlinear dynamics
Nonlinear systems
Numerical methods
Series (mathematics)
Systems analysis
ISSN:1024-123X, 1563-5147
Online Access:Get full text
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Summary:The reproducing kernel algorithm is described in order to obtain the efficient analytical-numerical solutions to nonlinear systems of two point, second-order periodic boundary value problems with finitely many singularities. The analytical-numerical solutions are obtained in the form of an infinite convergent series for appropriate periodic boundary conditions in the space W230,1, whilst two smooth reproducing kernel functions are used throughout the evolution of the algorithm to obtain the required nodal values. An efficient computational algorithm is provided to guarantee the procedure and to confirm the performance of the proposed approach. The main characteristic feature of the utilized algorithm is that the global approximation can be established on the whole solution domain, in contrast with other numerical methods like onestep and multistep methods, and the convergence is uniform. Two numerical experiments are carried out to verify the mathematical results, whereas the theoretical statements for the solutions are supported by the results of numerical experiments. Our results reveal that the present algorithm is a very effective and straightforward way of formulating the analytical-numerical solutions for such nonlinear periodic singular systems.
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ISSN:1024-123X
1563-5147
DOI:10.1155/2015/518406