Quasi‐boundary value method for non‐well posed problem for aparabolic equation with integral boundary condition

Abstract

In this paper we study the problem of control by the initial conditions of the heat equation with an integral boundary condition. This problem is ill-posed. Perturbing the final condition we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed. We also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions. Finally, we give explicit convergence rates.

A Nonlinear Case of the 1-D Backward Heat Problem: Regularization and Error Estimate

“…The authors show that the stability estimate of the method is of order −1 . Very recently, in [6], the quasi-boundary method was used to solve a backward heat equation with an integral boundary condition.…”

A regularized method for two dimensional nonlinear heat equation backward in time

“…be the norm in L 2 (I). In the section, we shall study the existence, the uniqueness and the stability of a solution of Problem (4)- (6). In fact, one has Theorem 1 Let ϕ ∈ L 2 (I) and let f ∈…”

“…and u is the unique solution of Problem (4)- (6). Proof First, using the Galerkin method (see, e.g., [10] ), we can show that the assumption on u t holds if u(., ., 0) ∈ H 1 0 (I).…”

“…e 2s(n 2 +m 2 ) f 2 nm (u)(s)ds and C is defined in Theorem 4. Proof Let u be the solution of problem (4)-(6) corresponding to ϕ and let w be the solution of problem (4)- (6) corresponding to ϕ where ϕ, ϕ are in right hand side of (6).…”

“…> is inner product in L 2 (I). We note the reader that if f = 0, then the problem (4)- (6) has been considered in [4] under the same form. Moreover, this problem is different as compared to the problem (7)- (9) in [17] (See p.874).…”