This work deals with the pseudospectral method to solve the Sturm–Liouville eigenvalue problems with Caputo fractional derivative using Chebyshev cardinal functions. The method is based on reducing the problem to a weakly singular Volterra integro-differential equation. Then, using the matrices obtained from the representation of the fractional integration operator and derivative operator based on Chebyshev cardinal functions, the equation becomes an algebraic system. To get the eigenvalues, we find the roots of the characteristics polynomial of the coefficients matrix. We have proved the convergence of the proposed method. To illustrate the ability and accuracy of the method, some numerical examples are presented.
In this paper, a new method based on combination of the natural transform method (NTM), Adomian decomposition method (ADM), and coefficient perturbation method (CPM) which is called “perturbed decomposition natural transform method” (PDNTM) is implemented for solving fractional pantograph delay differential equations with nonconstant coefficients. The fractional derivative is regarded in Caputo sense. Numerical evaluations are included to demonstrate the validity and applicability of this technique.
In this paper, we introduce a method based on Radial Basis Functions (RBFs) for the numerical approximation of Mathieu differential equation with two fractional derivatives in the Caputo sense. For this, we suggest a numerical integration method for approximating the improper integrals with a singularity point at the right end of the integration domain, which appear in the fractional computations. We study numerically the affects of characteristic parameters and damping factor on the behavior of solution for fractional Mathieu differential equation. Some examples are presented to illustrate applicability and accuracy of the proposed method. The fractional derivatives order and the parameters of the Mathieu equation are changed to study the convergency of the numerical solutions.
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