Analytical solution of nonlinear partial differential equations of physics

摘要

A general method is proposed to approximate the analytical solution of any time-dependent partial differential equation with boundary conditions defined on the four sides of a rectangle. To ensure that the approximant satisfies the boundary conditions problem the differential operator is modified with one additional term which takes into account the effect of boundary conditions. Then the new problem can be directly integrated in the same way as an ordinary differential equation. In this work Adomian's decomposition method with analytic extension is used to obtain the first-order approximant to the solution of a test case. The result is an analytic approximation to the solution which is compatible with both the exact boundary conditions and the accuracy imposed in the whole domain. The solution obtained is compared with the analytic approximation obtained with a Tau-Legendre spectral method.