Analysis of Certain Nonlinear Partial Differential Equation Models Arising in Biology

Abstract

The aim of this thesis is to present and analyse mathematical models which are basically described by a system of partial differential equations. The model equations are strongly coupled partial differential equations with suitable initial and boundary conditions. Moreover, we have demonstrated the existence of classical/weak solutions for the proposed models by using, respectively, the standard regularity theory and the Faedo-Galerkin approximation method. Solutions to these model equations are approximated using the finite element method for spatial discretization and the implicit Euler method for time discretization. newlineInitially, we examine the classical solution for the Keller-Segel chemotaxis system with nonlocal diffusion. The proposed model is used to understand a large variety of fields within the life cycle of most multicellular organisms. In order to achieve boundedness, we derive the necessary and sufficient condition based on the assumption. However, the proposed system s boundedness condition is determined based on three particular situations. newlineThen, we extend the same analysis for the system of nonlinear two-species chemotaxis system with nonlocal diffusion. Based on the assumptions, the proposed system s boundness conditions are also derived for two different cases. It is shown that a globally bounded classical solution exists under the appropriate initial conditions and homogeneous Neumann boundary conditions. newlineIn contrast to earlier models, we consider a system of partial differential equations that represents the acid-mediated cancer invasion model that incorporates nonlocal diffusion. The primary goal is to utilize numerical solutions to comprehend cancer invasion and transmission within the patient. Furthermore, the Faedo-Galerkin technique and compactness principle were utilised to show the existence and uniqueness of a weak solution for the proposed model. newlineMoreover, a priori error bounds and convergence estimates for the fully discrete problem are discussed. Additionally, we conducted numerical