Bernoulli-Type Spectral Numerical Scheme for Initial and Boundary Value Problems with Variable Order

Abstract

This manuscript is devoted to using Bernoulli polynomials to establish a new spectral method for computing the approximate solutions of initial and boundary value problems of variable-order fractional differential equations. With the help of the aforementioned method, some operational matrices of variable-order integration and differentiation are developed. With the aid of these operational matrices, the considered problems are converted to algebraic-type equations, which can be easily solved using computational software. Various examples are solved by applying the method described above, and their graphical presentation and accuracy performance are provided.

Keywords

• fractional-order Bernoulli polynomials
• operational matrices
• variable-order initial and boundary value problems