Convergence–Focused Numerical Solutions for Nonlinear Moving Boundary Models of Rheumatoid Arthritis

In this paper, we present a novel numerical framework to address a nonlinear moving boundary mathematical model of rheumatoid arthritis. This complex model integrates two hyperbolic equations, one elliptic equation, and one ordinary differential equation, all intricately coupled to capture the dynamics of the disease. Our approach begins with the implementation of the front-fixing transformation, which effectively stabilizes the free boundary and transforms the problem into a more manageable form. By leveraging a hybrid scheme that combines finite difference methods with collocation techniques, we rigorously establish the convergence of the proposed method. To validate our approach, we provide a series of numerical experiments, showcasing its accuracy and efficiency in resolving the model’s intricacies. The innovative proof of numerical stability and convergence underscores the robustness of this method in tackling challenging biomedical problems.