A New Approach to Solve the Fractional Order Linear/Non-linear Two-Dimensional Partial Differential Equation Using Legendre Collocation Technique

Abstract

A class of boundary value problems for fractional linear/non-linear two- dimensional partial differential equations is studied. A new algorithm based on the approximation technique is proposed to solve them. To this end, the terms of the considered problems are approximated through a series expansion of triple-shifted Legendre polynomials. Then collocated these on Legendre Gauss–Lobatto points which provide an algebraic system which can be solved using the standard numerical technique. Our proposed algorithm has the exponential rate of convergence in contrast to other existing methods viz., finite difference and finite element method. Graphical presentation and tabular representations of some considered examples are illustrated to make a comparison with the existing analytical solutions which demonstrate the applicability and efficiency of the proposed algorithm. © 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Austria, part of Springer Nature.