Numerical study of 1D and 2D advection-diffusion-reaction equations using Lucas and Fibonacci polynomials

Ihteram Ali,Sirajul Haq,Shams Ul Arifeen,Kottakkaran Sooppy Nisar

doi:10.1007/s40065-021-00330-4

Abstract

In this work, a numerical scheme based on combined Lucas and Fibonacci polynomials is proposed for one- and two-dimensional nonlinear advection–diffusion–reaction equations. Initially, the given partial differential equation (PDE) reduces to discrete form using finite difference method and theta - weighted scheme. Thereafter, the unknown functions have been approximated by Lucas polynomial while their derivatives by Fibonacci polynomials. With the help of these approximations, the nonlinear PDE transforms into a system of algebraic equations which can be solved easily. Convergence of the method has been investigated theoretically as well as numerically. Performance of the proposed method has been verified with the help of some test problems. Efficiency of the technique is examined in terms of root mean square (RMS), L_2 and L_infty error norms. The obtained results are then compared with those available in the literature.

Highlights

Partial differential equations (PDEs) have dominant applications in various physical, chemical, and biological processes. These processes are mostly modeled in the form of heat, Laplace, and wave-type differential equations
Ali et al [8,15,17] applied Lucas polynomials coupled with finite differences and obtained accurate solution of various classes of one- and two-dimensional PDEs
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Summary

Introduction

Partial differential equations (PDEs) have dominant applications in various physical, chemical, and biological processes. In the form of advection-diffusion equation it describes heat transfer in a solid body and dissipation of salt in ground water In image processing it handles image at different scales [28]. The higher order derivative of unknown functions can be approximated via Lucas and Fibonacci Polynomials It is straightforward and produces better accuracy for less number of nodal points. Omer Oruc [31,32] applied a combined Lucas and Fibonacci polynomials approach for numerical solution of evolutionary equation for the first time. Ali et al [8,15,17] applied Lucas polynomials coupled with finite differences and obtained accurate solution of various classes of one- and two-dimensional PDEs. In this paper, we implement the proposed method to one- and two-dimensional Burger and heat equations.

Basic concepts and definition

Solution methodology

Truncation error estimate

Numerical examples

Conclusion