Numerical solution of two-dimensional fractional-order partial differential equations using hybrid functions

Fractional differential equations fit perfectly into nature modeling, requiring the finding of efficient numerical methods of solving them. This paper aims to propose a method for solving two-dimensional fractional order equations by constructing a new set of fractional functions called fractional-order hybrid of block-pulse functions and Bernoulli polynomials. The functions are obtained by replacing t with xα in the hybrid of block-pulse functions and Bernoulli polynomials. One of the main reasons why the proposed method is better than other methods is that, in this method, the Riemann–Liouville fractional integral operator is calculated exactly. The proposed numerical method determines a system of algebraic equations that can be solved by the collocation technique. Four illustrative examples show the effectiveness and validity of the proposed technique. The simplicity of the method makes it easy to use in order to examine the behavior of fractional differential equations with two variables, being efficient even for a relatively small base size.