Analytical approach to solving linear diffusion–advection–reaction equations with local and nonlocal boundary conditions

Initial–boundary value problems for a linear diffusion–advection–reaction equation are considered, with general nonhomogeneous linear boundary conditions and general linear nonlocal boundary conditions. Analytical solutions are obtained using an embedding method. The solutions are expressed in terms of time-varying functions that satisfy coupled linear Volterra integral equations of the first kind. A boundary element method is applied to numerically solve the integral equations. Three examples are given to demonstrate the accuracy of the numerical solutions when compared with the analytical solutions. The embedding method is applicable to problems with bounded and unbounded spatial domains.