A Composite Collocation Method Based on the Fractional Chelyshkov Wavelets for Distributed-Order Fractional Mobile-Immobile Advection-Dispersion Equation

Abstract

In this study, an accurate and efficient composite collocation method based on the fractional order Chelyshkov wavelets is proposed for obtaining approximate solution of distributed-order fractional mobile-immobile advection-dispersion equation with initial and boundary conditions. Operational matrices based on the fractional Chelyshkov wavelets are constructed. The proposed method reduce the solution to a system of algebraic equations, which is solved by Newton’s iterative method. Provided examples confirm the accuracy and applicability of the proposed method in line with the studied convergence analysis and error estimation. The obtained results of demonstrated numerical schemes illustrate that this approach is very accurate and efficient.

Higher-order fractional linear multi-step methods

“…But these analytical methods have been used to solve very special types, mostly linear, of FDEs. Therefore, developing reliable and efficient numerical methods for solving general fractional differential equations, such as nonlinear types, is an attractive topic for researchers [1][2][3][4]. One can find numerical methods based on polynomial interpolation [5,6], Gauss interpolation [7], Grunwald-Letnikov approximations, and generalizations of linear multistep methods.…”

“…we can obtain fractional linear multi-step methods to solve (2). Therefore, the general fractional form of a linear multi-step method is the form…”

Numerical solution of multidimensional time‐space fractional differential equations of distributed order with Riesz derivative

“…For example, numerical solution of distributed order space-fractional diffusion equation on a two-dimensional irregular convex domain [25], one-dimensional distributed order time-fractional diffusion equation [26,27], time-space fractional reaction-diffusion equation of distributed order on two-dimensional rectangle domain [28]. Other numerical methods for solving distributed-order FDEs include implicit difference schemes [29], operational matrix scheme [30], operational matrices based on the fractional Chelyshkov wavelets [31], and using hybrid functions [32].…”