An iterative technique for solving singularly perturbed parabolic PDE

Abstract

In this paper, we examine the applicability of a variant of iterative Tikhonov regularization for solving parabolic PDE with its highest order space derivative multiplied by a small parameter $\epsilon$. The solution of the operator equation $\frac{\partial u }{\partial t}-\epsilon \frac{\partial ^2 u}{\partial x^2}+a(x,t)=f(x,t)$ is not uniformly convergent to the solution of the operator equation $\frac{\partial u }{\partial t} +a(x,t) =f(x,t)$, when $\epsilon \rightarrow 0$. Although many numerical techniques are employed in practice to tackle the problem, the discretization of the PDE often leads to ill-conditioned system and hence the perturbed parabolic operator equation become ill-posed. Since we are dealing with unbounded operators, first we discuss the general theory for unbounded operators for iterated regularization scheme and propose an a posteriori parameter choice rule for choosing a regularization parameter in the iterative scheme. We then apply these techniques in the context of perturbed parabolic problems. Finally, we implement our iterative scheme and compare with other basic existing schemes to assert the adaptability of the scheme as an alternate approach for solving the problem.