Approximation of inverse Laplace transform solution to heat transport in a stream system

The Laplace transform is a powerful tool used in solving partial differential equations. However, due to considerable difficulty associated with inverting analytical solutions from Laplace space to the original time variable, this final step is often performed by numerical techniques. Unfortunately, this does not deliver a closed form solution to the original model. Here we illustrate a technique for approximating a closed form inversion to the Laplace transformed solution of a heat transport model for a natural river system. We show the approximation method provides good results when compared to the analytical solution that is dependent upon numerical techniques for the inversion of the Laplace transform. Our approximation results in a simple and concise expression in terms of the model parameters without relying on difficult numerical computations. We focus on the contribution to downstream temperatures from upstream boundary conditions, illustrating how boundary condition temperature decays in terms of model parameters.