Regularization and error estimate for the nonlinear backward heat problem using a method of integral equation

We consider the inverse time problem for the nonlinear heat equation in the form ut−uxx=f(x,t,u(x,t)),(x,t)∈(0,π)×(0,T),u(0,t)=u(π,t)=0t∈(0,T). The nonlinear problem is severely ill-posed. We shall use the method of integral equation to regularize the problem and to get some error estimates. We show that the approximate problems are well-posed and that their solution uϵ(x,t)uϵ(x,t) converges on [0,T][0,T] if and only if the original problem has a unique solution. We obtain several other results, including some explicit convergence rates. Some numerical tests illustrate that the proposed method is feasible and effective.