A note on the parabolic identification problem with involution and Dirichlet condition

Abstract

A space source of identification problem for parabolic equation with involution and Dirichlet condition is studied. The well-posedness theorem on the differential equation of the source identification parabolic problem is established. The stable difference scheme for the approximate solution of this problem is presented. Furthermore, stability estimates for the difference scheme of the source identification parabolic problem are presented. Numerical results are given.

On solvability of the inverse problem for a fourth-order parabolic equation with a complex-valued coefficient

“…Proof. We have to show that the resulting formal solution in the form of series (17), (18) satisfies equation (4) and conditions ( 5), (7). Let us first show that series (17), (18), as well as the formal derivative with respect to the variable t and formal derivatives up to the fourth order with respect to In [14] the validity of the estimates max X…”

“…The boundary conditions are satisfied as each term of the series (17) satisfies them. The existence of a classical solution to problem (4), ( 5), (7) has been completely proved. To prove the uniqueness of the…”

“…Let us formulate the main result of the present work. The solvability of the inverse problem (4), (7) with the Dirichlet boundary conditions ( 5) is formulated as the following theorem.…”

“…3) functions u (x, t) and f (x) satisfy equation ( 4), and the functionu (x, t) satisfies conditions ( 5), (7) in the usual sense.…”

“…The solvability of mixed problems for a perturbed wave equation with involution and with a variable complex-valued coefficient was studied in [4,5]. The solvability of inverse problems for the perturbed heat equation with involution and with a variable complex-valued coefficient was considered in [6][7][8].…”