Stepwise superposition approximation approach for analytical solutions with non-zero initial concentration using existing solutions of zero initial concentration in contaminate transport

Analytical solutions for contaminant transport are widely used for both theoretical and practical purposes. However, many existing solutions are obtained subject to an initial condition of zero concentration, which is often unrealistic in many practical cases. This article proposed a stepwise superposition approximation approach to solve the non-zero initial concentration problem for first-type and third-type boundary conditions by using the existing zero initial concentration solution. Theoretical examples showed that the approach was highly efficient if a proper superposition scheme with relative concentration increments was constructed. The key parameter that controlled the convergence speed was the time increment (Δt) multiplied by the rate constant (λ). The approach served also as an alternative way to make a convenient concentration calculation even if the non-zero initial concentration solution of a problem was known.