The two-dimensional fractional orthogonal derivative

TL;DR

This paper extends the concept of orthogonal derivatives from one to two dimensions, incorporating fractional derivatives and utilizing biorthogonal polynomials expressed via Appell functions for integration over specific regions.

Contribution

It introduces a two-dimensional fractional orthogonal derivative framework, expanding the mathematical tools for multivariate fractional calculus using biorthogonal polynomials and Appell functions.

Findings

Extension of orthogonal derivatives to two dimensions

Use of Appell functions for integration over squares and triangles

Development of fractional orthogonal derivatives in 2D

Abstract

This paper is an edited and shortened version of Chapter 6 from the thesis of the author. First the one dimensional orthogonal derivative will be extended to the two-dimensional case. In the two-dimensional case we have to define the region of integration. In this paper we treat the integration over the square region and over the triangle region where in the last case we use biorthogonal polynomials expressed in terms of Appell functions. Next the two-dimensional orthogonal derivative will be extended to the two-dimensional fractional orthogonal derivative. The results are highly dependent on the $F_1$,\ $F_2$ and $F_3$ Appell functions.