Existence, uniqueness, Ulam-Hyers stability and numerical simulation of solutions for variable order fractional differential equations in fluid mechanics

In this paper, a numerical approximation is shown to solve solutions of time fractional linear differential equations of variable order in fluid mechanics where the considered fractional derivatives of variable order are in the Caputo sense. Existence, uniqueness of solutions and Ulam-Hyers stability results are displayed. To solve the considered equations a numerical approximation based on the shifted Legendre polynomials are proposed. To perform the method, an operational matrix of fractional derivative with variable-order is derived for the shifted Legendre polynomials to be applied for developing the unknown function. By substituting the aforesaid operational matrix into the considered equations and using the properties of the shifted Legendre polynomials together with the collocation points, the main equations are reduced to a system of algebraic equations. The approximate solution is calculated by solving the obtained system which is technically easier for checking. We also study the error analysis for the approximate solution yielded by the introduced method. Finally, the accuracy and performance of the proposed method are checked by some illustrative examples. The illustrative examples results establish the applicability and usefulness of the proposed method.