Abstract
In this article, a new reproducing kernel approach is developed for obtaining a numerical solution of multi-order fractional nonlinear three-point boundary value problems. This approach is based on a reproducing kernel, which is constructed by shifted Legendre polynomials (L-RKM). In the considered problem, fractional derivatives with respect to α and β are defined in the Caputo sense. This method has been applied to some examples that have exact solutions. In order to show the robustness of the proposed method, some examples are solved and numerical results are given in tabulated forms.
Highlights
A new iterative reproducing kernel approach will be constructed for obtaining the numerical solution of a multi-order fractional nonlinear three-point boundary value problem as follows: a2(ξ) cDαz(ξ) + a1(ξ) cDβz(ξ) + a0(ξ)z(ξ) = g(ξ, z(ξ), z (ξ))
Many important phenomena have been concerned in engineering and applied science, such as dynamical systems, fluid mechanics, control theory, oil industries, and heat conduction, and can be well-turned by fractional differential equations [8,9,10]
The main motivation of this paper is to extend the Legendre reproducing kernel approach for solving multi-order fractional nonlinear three-point boundary value problems with a Caputo derivative
Summary
A new iterative reproducing kernel approach will be constructed for obtaining the numerical solution of a multi-order fractional nonlinear three-point boundary value problem as follows: a2(ξ) cDαz(ξ) + a1(ξ) cDβz(ξ) + a0(ξ)z(ξ) = g(ξ, z(ξ), z (ξ)). In 1908, Zaremba firstly introduced the reproducing kernel concept [33] His researches regarded boundary value problems, which include the Dirichlet condition. The Legendre reproducing kernel method is proposed for the fractional two-point boundary value problem of Bratu type equations [43]. The main motivation of this paper is to extend the Legendre reproducing kernel approach for solving multi-order fractional nonlinear three-point boundary value problems with a Caputo derivative.
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