numerical algorithm

We investigate the existence and uniqueness of the solution and also the rate of convergence of a numerical method for a fractional differential equation in both q-calculus and (p, q)-calculus versions. We use the Banach and Schauder fixed point theorems in this study. We provide two examples, one by definition of the q-derivative and the other by (p, q)-derivative. We compare the rate of convergence of the numerical method. We like to clear some facts on (p, q)-calculus. The data from our numerical calculations show well that q-calculus works better than (p, q)-calculus in each case.

Referring to one of the recent works of the authors, presented in~\cite{differentialbpf}, for numerical solution of linear differential equations, an alternative scheme is proposed in this article to considerably improve the accuracy and efficiency. For this purpose, triangular functions as a set of orthogonal functions are used. By using a special representation of the vector forms of triangular functions and the related operational matrix of integration, solving the differential equation reduces to solve a linear system of algebraic equations. The formulation of the method is quite general, such that any arbitrary linear differential equation may be solved by it. Moreover, the algorithm does not include any integration and, instead, uses just sampling of functions, that results in a lower computational complexity. Also, the formulation of this approach needs no modification when a singularity occurs in the coefficients of differential equation. Some problems are numerically solved by the proposed method to illustrate that it is much more accurate and applicable than the prior method in~\cite{differentialbpf}.