Solving the nonlinear forced Fisher equation PDE using a proposed generalized tanh-like ansatz method

Abstract

The standard or homogeneous Fisher equation (FE) is a deterministic approximation to a stochastic model, commonly used to represent the spatial spread of gene selection and migration. It is a natural extension of the logistic growth population. This partial differential equation (PDE) can be solved using the tanh method. However, when an inhomogeneous driving function is introduced to FE, the tanh method fails. This paper proposes a new tanh-like function, the Generalized Half-Angle Tangent Hyperbolic (we called g-HATH), given as Y_p(ξ)=(1+p)[tanh⁡(ξ/2)/(1 + p⁡ tanh²(ξ/2)] as a new ansatz with variable p (0 ≤ p ≤ 1). It reduces to the tanh(ξ) when p = 1, and tanh(ξ/2) when p = 0. The new ansatz introduces a new approach to solving a broader class of nonlinear partial differential equations, allowing solutions to more complex phenomena involving inhomogeneous PDEs in natural science, social science, or physical science.