Miklos Vajta

Approximate solution of a nonlinear partial differential equation

Nonlinear partial differential equations (PDE) are notorious to solve. In only a limited number of cases can we find an analytic solution. In most cases, we can only apply some numerical scheme to simulate the process described by a nonlinear PDE. Therefore, approximate solutions are important for they may provide more insight about the process and its properties (stability, sensitivity etc.). The paper investigates the transient solution of a second order, nonlinear parabolic partial differential equation with given boundary-and initial conditions. The PDE may describe various physical processes, but we interpret it as a thermal process with exponential source term. We develop an analytical approximation, which describes the inverse solution. Accuracy and feasibility will be demonstrated. We also provide an expression for the time-derivative of the transient at time zero. The results can be extended for other boundary conditions as well.