THEORETICAL AND NUMERICAL STUDIES OF GENERALIZED BURGERS-HUXLEY EQUATION WITH WEAKLY SINGULAR KERNELS

Abstract

The main aim of the thesis is to study the interplay between convection effects, diffusion transports, and reaction mechanisms by digging into the theoretical and numerical studies of generalized Burgers-Huxley equation (GBHE) [1] with weakly singular kernel given as ∂u ∂t + αuδ Xd i=1 ∂u ∂xi − νΔu − η Z t 0 K(t − s)Δu ds = βu(1 − uδ)(uδ − γ) + f (x, t) ∈ Ω × (0, T], u(x, 0) = u0(x), x ∈ Ω, where the domain Ω ⊂ Rd (d = 2, 3) is an open bounded simply connected convex domain and the boundary ∂Ω is Lipschitz. The kernel K(t) is a weakly singular kernel that stores the data from the previous time steps upto the time t, f is an external forcing, α, β, ν, δ, η are parameters such that α > 0 is the advection coefficient, β > 0, δ ∈ N is the retardation time, η ≥ 0 is the relaxation time, and γ ∈ (0, 1). The memory term appears in various contexts. For example, memory or time delay effects are often neglected in deriving the heat equation using Fourier’s law [68]. In population or epidemic models, the current diffusion situation heavily depends on the species past and current concentrations, represented by the memory term (see [151] and references therein). However, this effect is frequently ignored in the literature. This work takes into consideration the effect of the past history through the GBHE with weakly singular kernels making it a nonlinear partial integro-differential equation (PIDE). The GBHE comprises two models: the Burgers’ equation, which serves as a fundamental model for different complex models such as the Navier-Stokes equation (also called a toy model), and the Huxley equation, which finds various applications in nerve pulse propagation and liquid crystals [166]. These aspects highlight the broad applications of this model. Different authors in the literature propose various numerical methods for the GBHE, but there is limited research on using the finite element method for GBHE. The global solvability of the GBHE without memory (η = 0) in one dimension using conforming FEM have been studied in [112]. However, the fully discrete case has not been addressed there. In the following year, in [85], the numerical approximation using standard conforming, nonconforming, and DG approximation for the stationary counterpart in higher dimensions (R2 and R3) has been discussed under stringent conditions on parameters and given data, as stated in [85, Theorems 3.3-3.6]. This work addresses this gap by providing the well posedness results and the convergence analysis for different finite elements proposed for the numerical approximation of the solution of GBHE including memory (Weakly singular kernels), and their accuracy is validated by performing different computations in higher dimensions.