A nonhomogeneous backward heat problem: Regularization and error estimates

Abstract

We consider the problem of finding the initial temperature, from the final temperature, in the nonhomogeneous heat equation $$displaylines{ u_t-u_{xx}= f(x,t),quad (x,t)in (0,pi)imes (0,T),\ u(0,t)= u(pi,t)= 0, quad (x,t) in (0,pi)imes(0,T). }$$ This problem is known as the backward heat problem and is severely ill-posed. Our goal is to present a simple and convenient regularization method, and sharp error estimates for its approximate solutions. We illustrate our results with a numerical example.

Keywords

• Backward heat problem
• ill-posed problem
• nonhomogeneous heat equation
• contraction principle