Abstract
In this paper, we investigate a spatially two-dimensional Burgers–Huxley equation that depicts the interaction between convection effects, diffusion transport, reaction gadget, nerve proliferation in neurophysics, as well as motion in liquid crystals. We have used the Lie symmetry method to study the vector fields, optimal systems of first order, symmetry reductions, and exact solutions. Furthermore, using the power series method, a set of series solutions are obtained. Finally, conservation laws are derived using optimal systems.
Highlights
Lie symmetry analysis is a powerful and influential tool for mathematically analyzing partial differential equations. It can be used in securing analytic solutions or in switching PDEs into solvable ordinary differential equations (ODEs)
Each member of this system is used in lessening independent variables of the system until analytic solutions are obtained or PDEs are switched to solvable ODEs [3,4,5,6,7]
The one-parameter Lie symmetry groups generated by infinitesimals X1, X2, and X3 are given by r1 : ( x, y, t, u) −→ ( x + e1, y, t, u), r2 : ( x, y, t, u) −→ ( x, y + e2, t, u), r3 : ( x, y, t, u) −→ ( x, y, t + e3, u), where e1, e2, and e3 are group parameters
Summary
We will study the Lie symmetries and optimal systems of the Burgers–Huxley equation. Substituting extended transformations into the obtained compatible conditions and making the coefficients of several monomials in partial derivatives and numerous powers of u equal, we get the following over determining system of PDEs: ηy − ξ x = 0, ξ xx + ξ yy − ξ t − 2φ xu − φ = 0, ηt − ηxx − ηyy + 2φyu + φ = 0, ηxu + ξ yu = 0, ζ xu = 0, ηu = 0, ξ u = 0, ζ u = 0, ζ xx + ζ yy − ζ t + 2ηy = 0, φuu − 2ξ xu = 0, ξ uu = 0, ζ uu = 0, ηuu = 0, φuu − 2ηyu = 0, ζ x + ζ y + kζ u = 0, ξ x + ξ y + kξ u − 2ηy = 0, ηy − ηx = 0, ηx + ξ y = 0, ζ x = 0, ζ y = 0,.
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