First integrals, conserved vectors of nonlinear partial difference equations

Abstract

We perform a symmetry analysis of some nonlinear partial difference equations (nP$\triangle$Es), where the discrete version is obtained using some discretization approach. The discrete versions of the wave, diffusion, Fisher and Huxley equations are the subject of this research. At first, the initial invariance approach is the Lie symmetry approach. The first integrals technique that Hydon introduced to be used with discrete ordinary difference equations (O$\triangle$Es) serves as our inspiration in this situation. We develop a similar technique for generating the first integral vectors of the nP$\triangle$Es without recourse to symmetry generators.

Invariant analysis of the two-cell tumor growth model in the brain

“…Importantly, although not explicitly stated, each cell population might have the ability to diffuse more rapidly in regions of white matter. Lie group methods [8][9][10][11][12][13][14][15][16][17] possess a versatile toolkit with broad applications in the realms of science and mathematics. They provide a systematic approach to understanding symmetries, solving differential equations, and gaining insights into complex physical and mathematical systems.…”