The Numerical Investigations of Non-Polynomial Spline for Solving Fractional Differential Equations

We present a crossing approach based on the new construction of non-polynomial spline function to investigate the numerical solution of the fractional differential equations. We find the accuracy of the spline method and to presenting the completion of non-polynomial spline two examples for problems are used. To clarify, we present the numerical computations that can be used to solve difficult problems while the results are found and got to be in good error estimation with comparing exact solutions

High-order methods for systems of fractional ordinary differential equations and their application to time-fractional diffusion equations

Taking into account the regularity properties of the solutions of fractional differential equations, we develop a numerical method which is able to deal, with the same accuracy, with both smooth and nonsmooth solutions of systems of fractional ordinary differential equations of the Caputo-type. We provide the error analysis of the numerical method and we illustrate its feasibility and accuracy through some numerical examples. Finally, we solve the time-fractional diffusion equation using a combination of the method of lines and the newly developed hybrid method.L.L. Ferras would like to thank FCT - Fundacao para a Ciencia e a Tecnologia, I.P. (Portuguese Foundation for Science and Technology) for financial support through the scholarship SFRH/BPD/100353/2014 and Project UID-MAT-00013/2013. M.L. Morgado aknowledges the financial support of FCT, through the Project UID/Multi/04621/2019 of CEMAT/IST-ID, Center for Computational and Stochastic Mathematics, Instituto Superior Tecnico, University of Lisbon. This work was also partially supported by FCT through the Project UID/MAT/00297/2019 (Centro de Matematica e Aplicacoes)

A nonpolynomial collocation method for fractional terminal value problems

NOTICE: this is the author’s version of a work that was accepted for publication in Journal of Computational and Applied Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Computational and Applied Mathematics, 275, February 2015, doi: 10.1016/j.cam.2014.06.013In this paper we propose a non-polynomial collocation method for solving a class of terminal (or boundary) value problems for differential equations of fractional order α, 0 < α < 1. The approach used is based on the equivalence between a problem of this type and a Fredholm integral equation of a particular form. Taking into account the asymptotic behaviour of the solution of this problem, we propose a non-polynomial collocation method on a uniform mesh. We study the order of convergence of the proposed algorithm and a result on optimal order of convergence is obtained. In order to illustrate the theoretical results and the performance of the method we present several numerical examples.The work was supported by an International Research Excellence Award funded through the Santander Universities scheme

A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations

In this paper, we develop regularized discrete least squares collocation and finite volume methods for solving two-dimensional nonlinear time-dependent partial differential equations on irregular domains. The solution is approximated using tensor product cubic spline basis functions defined on a background rectangular (interpolation) mesh, which leads to high spatial accuracy and straightforward implementation, and establishes a solid base for extending the computational framework to three-dimensional problems. A semi-implicit time-stepping method is employed to transform the nonlinear partial differential equation into a linear boundary value problem. A key finding of our study is that the newly proposed mesh-free finite volume method based on circular control volumes reduces to the collocation method as the radius limits to zero. Both methods produce a large constrained least-squares problem that must be solved at each time step in the advancement of the solution. We have found that regularization yields a relatively well-conditioned system that can be solved accurately using QR factorization. An extensive numerical investigation is performed to illustrate the effectiveness of the present methods, including the application of the new method to a coupled system of time-fractional partial differential equations having different fractional indices in different (irregularly shaped) regions of the solution domain

MKSOR iterative method with cubic b-spline approximation for solving two-point boundary value problems

In this study, two-point boundary value problems have been discretized by using cubic B-spline discretization scheme to derive the cubic B-spline approximation equations that corresponds. Then, this approximation equation is used to develop system of cubic B-spline approximation equations. To get the numerical solutions, there are three iterative methods such as Gauss-Seidel (GS), Successive Over Relaxation (SOR) and Modified Kaudd Successive Over Relaxation (MKSOR) used to solve the generated systems of linear equations. For the purpose of comparison, the GS iterative method has been designated as a control method for the SOR and MKSOR iterative methods. Three examples of problems also have been considered to test the effectiveness of these proposed iterative methods. From the numerical results, MKSOR iterative method is superior method in terms of number of iterations and computational time.Keywords: cubic B-spline approximation; two-point boundary value problem; MKSOR iteratio