The Neumann series as a fundamental solution of the two-dimensional convection–diffusion equation with variable velocity

Abstract

The fundamental solution of the two-dimensional convection–diffusion equation with variable coefficients and its adjoint equation are obtained in complex form in terms of the unknown density of two equivalent uniquely solvable Volterra integral equations of the second kind whose analytical solutions are given explicitly as convergent Neumann series. The Volterra integral equations are obtained by integrating the complex form of the original differential equations, without additional change of variables as proposed by previously authors. In the numerical examples, cases corresponding to non-self-adjoint operators are considered. As a validation, the proposed approach is used to derive the fundamental solution of the adjoint to the convection–diffusion equation with constant velocity. In this case, the series solution can be evaluated analytically. For more general velocity fields, the recursive terms of the series can be evaluated by symbolic computation or numerical integration.