Bivariate Jacobi polynomials depending on four parameters and their effect on solutions of time-fractional Burgers’ equations

Abstract

The utilization of time-fractional Burgers’ equations is widespread, employed in modeling various phenomena such as heat conduction, acoustic wave propagation, gas turbulence, and the propagation of chaos in non-linear Markov processes. This study introduces a novel pseudo-operational collocation method, leveraging two-variable Jacobi polynomials. These polynomials are obtained through the Kronecker product of their one-variable counterparts, concerning both spatial ($x$) and temporal ($t$) domains. The study explores the impact of four parameters ($\theta ,\vartheta ,\sigma ,\varsigma >-1$) on the accuracy of resulting approximate solutions, marking the first examination of such influence. Collocation nodes in a tensor approach are constructed employing the roots of one-variable Jacobi polynomials of varying degrees in $x$ and $t$. The study delves into analyzing how the distribution of these roots affects the outcomes. Consequently, pseudo-operational matrices are devised to integrate both integer and fractional orders, presenting a novel methodological advancement. By employing these matrices and appropriate approximations, the governing equations transform into an algebraic system, facilitating computational analysis. Furthermore, the existence and uniqueness of the equations under study are investigated and the study estimates error bounds within a Jacobi-weighted space for the obtained approximate solutions. Numerical simulations underscore the simplicity, applicability, and efficiency of the proposed matrix spectral scheme.