Computational Numerical Solution Algorithm for Fractional Differential Equations

This study focused on three main problems, firstly, a study on the existence of the solution for a coupled system of fractional differential equations with integral boundary conditions. The solution process for the existence and uniqueness of solutions for the proposed problem was obtained by using the contraction mapping principle, and then by using Leray–Schauder’s alternative method. Secondly, investigation and approximation of solutions of Caputo type fractional differential equation with nonlinear boundary conditions has been solved by using an appropriate parameterization technique, where nonlinear boundary conditions were transformed to linear boundary conditions by using vector parameters. To solve the transformed problem, a numerical-analytic scheme was constructed to find the relation between different type’s two-point and multipoint linear boundary condition and nonlinear boundary conditions. Finally, efficient numerical - analytical computational algorithm for solving systems of fractional differential equations (SFDEs) Nonlinear Point Boundary-Value Problem with Nonlinear Boundary Conditions were considered. The fractional derivative was described in the Caputo sense. The method is based on numerical approximations of systems of fractional differential equations, where the properties of this method were utilized to reduce SFDEs to the system of algebraic equations. Special attention is given to study the convergence and estimate the error of the presented method. The methods introduce a promising tool for solving many systems of non-linear fractional differential equations. Numerical examples were presented to illustrate the validity and the great potential of both proposed techniques.