Exact analytical solution of the Cauchy problem for a linear reaction–diffusion equation with time-dependent coefficients and space–time-dependent source term

The present work derives the exact analytical solution of the Cauchy problem for a linear reaction–diffusion equation with time-dependent coefficients and space–time-dependent source term. The work also emphasizes the role of reaction–diffusion models as important particular cases of much more general equations in the kinetic theory of active particles. The analytical expression derived shows the structure of the solution and the contributions of different terms of the model to it. The result obtained enables one to solve the Cauchy problem indicated by using the exact analytical representation rather than numerical methods, which are usually time-consuming, especially when the number of spatial dimensions is greater than 2.