Analytical Solution to One-dimensional Advection-diffusion Equation with Several Point Sources through Arbitrary Time-dependent Emission Rate Patterns

Advection-diffusion equation and its related analytical solutions have gained wide applications in different areas. Compared with numerical solutions, the analytical solutions benefit from some advantages. As such, many analytical solutions have been presented for the advection-diffusion equation. The difference between these solutions is mainly in the type of boundary conditions, e.g. time patterns of the sources. Almost all the existing analytical solutions to this equation involve simple boundary conditions. Most practical problems, however, involve complex boundary conditions where it is very difficult and sometimes impossible to find the corresponding analytical solutions. In this research, first, an analytical solution of advection-diffusion equation was initially derived for a point source with a linear pulse time pattern involving constant-parameters condition (constant velocity and diffusion coefficient). Hence, using the superposition principle, the derived solution can be extended for an arbitrary time pattern involving several point sources. The given analytical solution was verified using four hypothetical test problems for a stream. Three of these test problems have analytical solutions given by previous researchers while the last one involves a complicated case of several point sources, which can only be numerically solved. The results show that the proposed analytical solution can provide an accurate estimation of the concentration; hence it is suitable for other such applications, as verifying the transport codes. Moreover, it can be applied in applications that involve optimization process where estimation of the solution in a finite number of points (e.g. as an objective function) is required. The limitations of the proposed solution are that it is valid only for constant-parameters condition, and is not computationally efficient for problems involving either a high temporal or a high spatial resolution.