A variant of Tikhonov regularization for parabolic PDE with space derivative multiplied by a small parameter ∊

摘要

In this paper, we examine the applicability of a variant of Tikhonov regularization for parabolic PDE with its highest order space derivative multiplied by a small parameter ∊∊. The solution of the operator equation ∂u∂t-∊∂2u∂x2+a(x,t)=f(x,t) is not uniformly convergent to the solution of the operator equation ∂u∂t+a(x,t)=f(x,t), when ∊→0. This violates one of the conditions of Hadamard well-posedness and hence the perturbed parabolic operator equation is ill-posed. Since the operator is unbounded, we first discuss the general theory for unbounded operators and propose an a posteriori parameter choice rule for choosing a regularization parameter. We then apply these techniques in the context of perturbed parabolic problems. Finally, we implement our regularized scheme and compare with other basic existing schemes to assert the adaptability of the scheme as an alternate approach for solving the problem.