In this article, we propose a one-dimensional heat conduction model for a double-pane window with a temperature-jump boundary condition and a thermal lagging interfacial effect condition between layers. We construct a second-order accurate finite difference scheme to solve the heat conduction problem. The designed scheme is mainly based on approximations satisfying the facts that all inner grid points has second-order temporal and spatial truncation errors, while at the boundary points and at inter-facial points has second-order temporal truncation error and first-order spatial truncation error, respectively. We prove that the finite difference scheme introduced is unconditionally stable, convergent, and has a rate of convergence two in space and time for the $L_{\infty }$-norm. Moreover, we give a numerical example to confirm our theoretical results.

In this paper, we consider the finite difference approximation for a one-dimensional mathematical model of heat conduction in a three-layered solid with interfacial conditions for temperature and heat flux between the layers. The finite difference scheme is unconditionally stable, convergent, and equivalent to the solution of two linear algebraic systems. We evaluate various methods for solving the involved linear systems by analyzing direct and iterative solvers, including GPU-accelerated approaches using CuPy and PyCUDA. We evaluate performance and scalability and contribute to advancing computational techniques for modeling complex physical processes accurately and efficiently.

We introduce new necessary conditions for the existence and uniqueness of stationary weak solutions and the existence of the weak solutions for the evolution problem in the system arising from the modeling of the bioconvective flow problem. Our analysis is based on the application of the Galerkin method, and the system considered consists of three equations: the nonlinear Navier–Stokes equation, the incompressibility equation, and a parabolic conservation equation, where the unknowns are the fluid velocity, the hydrostatic pressure, and the concentration of microorganisms. The boundary conditions are homogeneous and of zero-flux-type, for the cases of fluid velocity and microorganism concentration, respectively.

In this work, we investigate the calibration of diffusion coefficients in a reaction-diffusion system arising in the modelling of the spatial-temporal dynamics of an epidemic. The model assumes a population partitioned into two compartments: susceptible and infected individuals. It captures a scenario with no lasting immunity, allowing recovered individuals to return to the susceptible class. We consider that the diffusion is given by a diagonal matrix with unknown space-dependent entries that should be determined by the knowledge of susceptible and infected populations at some fixed time. The identification of the diffusion parameter is posed as an optimal control problem, where the objective functional is the comparison between the solution of the reaction-diffusion system and the observed data in the L 2-norm, supplemented by a regularization term in L 2-norm of the diffusion coefficient. The analysis of the optimal control formulation is carried out using the Dubovitskii-Milyutin Methodology. We obtain the following main results: the existence of strictly positive solutions for the state equations, demonstrating the existence of optimal solutions, and the necessary optimality conditions of first order.

In this note we prove the existence and uniqueness of weak solutions for the boundary value problem modelling the stationary case of the bioconvective flow problem introduced by Tuval et. al. (2005, {\it PNAS} 102, 2277--2282). We derive some appropriate a priori estimates for the weak solution, which implies the existence, by application of Gossez theorem, and the uniqueness by standard methodology of comparison of two arbitrary solutions.