Abstract
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 spaceW230,1, 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.
Highlights
Mathematical models of classical applications from physics, chemistry, and mechanics take the form of systems of singular periodic boundary value problems (BVPs) of second order which are a combination of singular differential system and periodic boundary conditions
To name but a few, computations of self-similar blow-up solutions of nonlinear partial differential equations lead to the computation of problems from this class [1, 2], the density profile equation in hydrodynamics may be reduced to a system of singular periodic BVP [3, 4], the investigation of problems in the theory of shallow membrane caps is associated with such systems [5], and in ecology, in the computation of avalanche run-up, this problem class is translated into a system of singular periodic BVP [6, 7]
Anyhow, when applied to the systems of singular periodic BVPs, standard numerical methods designed for regular BVPs suffer from a loss of accuracy or may even fail to converge [8,9,10], because of the singularity, whilst analytical methods commonly used to solve nonlinear differential equations are very restricted and numerical techniques involving discretization of the variables on the other hand give rise to rounding off errors
Summary
Mathematical models of classical applications from physics, chemistry, and mechanics take the form of systems of singular periodic boundary value problems (BVPs) of second order which are a combination of singular differential system and periodic boundary conditions. We know that except a limited number of these problems and phenomenons, most of them do not have analytical solutions These nonlinear equations should be solved using numerical methods or other analytical methods. Anyhow, when applied to the systems of singular periodic BVPs, standard numerical methods designed for regular BVPs suffer from a loss of accuracy or may even fail to converge [8,9,10], because of the singularity, whilst analytical methods commonly used to solve nonlinear differential equations are very restricted and numerical techniques involving discretization of the variables on the other hand give rise to rounding off errors. This paper ends in Appendices, with two parts about the kernel function of the space W23[0, 1]
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