Analytical Solution to One-dimensional Advection-di ffusion Equation with Several Point Sources through Arbitra ry Time-dependent Emission Rate Patterns

Abstract

Advection-diffusion equation and its related analyt ical solutions have gained wide applications in different areas. Compared with nume rical solutions, the analytical solutions benefit from some advantages. As such, ma ny 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. tim e patterns of the sources. Almost all the existing analytical solutions to this equation invo lve simple boundary conditions. Most practical problems, however, involve complex bounda ry conditions where it is very difficult and sometimes impossible to find the corr esponding analytical solutions. In this research, first, an analytical solution of advectio n-diffusion equation was initially derived for a point source with a linear pulse time pattern involving constant-parameters condition (constant velocity and diffusion coeffici ent). Hence, using the superposition principle, the derived solution can be extended for an arbitrary time pattern involving several point sources. The given analytical solutio n was verified using four hypothetical test problems for a stream. Three of these test pro blems 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 estimat ion 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 p rocess 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 involvin g either a high temporal or a high spatial resolution. Keywords: Advection-diffusion equation, Analytical solution, Laplace transformation, Point source, Solute transport.