Inverse problem for the linear bi-flux diffusion equation using homotopy analysis method

Abstract The linear bi-flux diffusion equation extends the classical diffusion process by incorporating a bimodal flux distribution. In this paper, we address an inverse problem associated with this model. Specifically, we consider the case where the coefficient of the lowest order term in u ( x , t ) is unknown and focus on recovering this coefficient function alongside the solution. To solve this problem, we propose a homotopy analysis method. By reformulating the inverse problem as an integro-differential equation, we solve it using the homotopy analysis method, which transforms the problem into an infinite series of simpler iterative equations. The numerical solution is then obtained as the sum of these functions. To illustrate the simplicity and effectiveness of our approach, we present three test examples. In one instance, where the analytical solution is too complex to derive, we utilize a numerical method to solve the iterative equations. The results demonstrate that even a few terms from the series yield an accurate approximation of the solution.