Numerical solutions of systems of first-order, two-point BVPs based on the reproducing kernel algorithm

The aim of the present analysis is to implement a relatively recent computational algorithm, reproducing kernel Hilbert space, for obtaining the solutions of systems of first-order, two-point boundary value problems for ordinary differential equations. The reproducing kernel Hilbert space is constru...

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Bibliographic Details
Published in:Calcolo Vol. 55; no. 3; pp. 1 - 28
Main Author: Abu Arqub, Omar
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.09.2018
Springer Nature B.V
Subjects:
Algorithms
Boundary value problems
Computation
Computer simulation
Differential equations
Evolutionary algorithms
Hilbert space
Kernel functions
Mathematics
Mathematics and Statistics
Numerical Analysis
Numerical methods
Ordinary differential equations
Simulated annealing
Theory of Computation
Viability
Boundary value problems
34B15
Gram–Schmidt process
Reproducing kernel algorithm
34K28
47B32
Initial–final conditions
ISSN:0008-0624, 1126-5434
Online Access:Get full text
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Summary:The aim of the present analysis is to implement a relatively recent computational algorithm, reproducing kernel Hilbert space, for obtaining the solutions of systems of first-order, two-point boundary value problems for ordinary differential equations. The reproducing kernel Hilbert space is constructed in which the initial–final conditions of the systems are satisfied. Whilst, three smooth kernel functions are used throughout the evolution of the algorithm in order to obtain the required grid points. An efficient construction is given to obtain the numerical solutions for the systems together with an existence proof of the exact solutions based upon the reproducing kernel theory. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed. Finally, the utilized results show that the present algorithm and simulated annealing provide a good scheduling methodology to such systems compared with other numerical methods.
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ISSN:0008-0624
1126-5434
DOI:10.1007/s10092-018-0274-3