Operational matrices for solving variable order differential equations

Abstract

In this thesis, we extensively explore the role of matrices as substitutes for derivative and integral operators. By expressing an approximate solution of a partial differential in an implicit form involving polynomials, we demonstrate how to deduce novel composite operational matrices. We also show how to utilise the laws of matrix multiplication to come up with a single matrix that performs the role of differentiation and integration. In conjunction with the Garlekin technique, we apply these composite matrices to numerically solve partial differential equations. Through practical examples, we prove that these composite operational matrices are convenient in approximating the solution of partial differential equations using a computer algebra system like Mathematica.