A Numerical Method for Partial Differential Algebraic Equations Based on Differential Transform Method

Abstract

We have considered linear partial differential algebraic equations (LPDAEs) of the form $A{u}_t(t,x)+B{u}_{xx}(t,x)+Cu(t,x)=f(t,x)$, which has at least one singular matrix of $A,B\in {ℝ}^{n\times n}$. We have first introduced a uniform differential time index and a differential space index. The initial conditions and boundary conditions of the given system cannot be prescribed for all components of the solution vector $u$ here. To overcome this, we introduced these indexes. Furthermore, differential transform method has been given to solve LPDAEs. We have applied this method to a test problem, and numerical solution of the problem has been compared with analytical solution.