Sinc Approximation of Solution of Heat Equation with Discontinuous Initial Condition

Abstract

We solve the one dimensional heat equation with discontinuous initial condition, the discontinuity being atx = 0, using the heat kernel and the Sine collocation method for convolution integrals. In [7], it is shown that the method is converging at an exponential rate. We compare the following formulations: the first one is based on the fact that fort > 0, the solution of the forced heat equation, the forcing term being a smooth function, is infinitely many time differentiable; the other one ignores the smoothing properties of the heat kernel. With the first approach, we apply the Sine collocation algorithm presented in [7] to the double integral for the forcing term ignoring the discontinuity in the initial condition. With the other one, we break the double integral for the forcing term into two integrals, one over (−∞,0) × [0,T] and the other over (0,∞) × [0,T], then we make the change of variable ξ = −η in the integral over (−∞,0) × [0,T] to only compute integrals over (0,∞) × [0,T]. In the first case, the conformal map is the identity; in the other one it is log(sinh(z)). Our numerical results show that for the problem considered here, the first approach is more efficient and more accurate than the second one. The method presented here is easily generalized to the case of finitely many jumps in the initial condition.