The Solvability of the Inverse Problem for the Evolution Equation with a Superstable Semigroup
Journal: Discrete and Continuous Models and Applied Computational Science (Vol.26, No. 2)Publication Date: 2018-04-21
Authors : I Tikhonov; Son Vu Nguyen;
Page : 103-118
Keywords : evolution equation; inverse problem; superstable semigroup; operator equation; Neumann series; existence and uniqueness theorem of the solution;
Abstract
For the evolution equation in a Banach space, the linear inverse source problem is studied. It is required to recover an unknown nonhomogeneous term by means of an additional nonlocal condition written in the form of a Riemann-Stieltjes integral. The main assumption is related to the superstability (quasinilpotency) of the evolution semigroup. More precisely, it is assumed that the evolutionary semigroup associated with the abstract differential equation has an infinite negative exponential type. Without other restrictions, a theorem on the solvability of the inverse problem is obtained. It is shown that the solution can be represented by a convergent Neumann series. Exact conditions under which an infinite series becomes a finite sum are found. Here, the algorithm for calculating the solution becomes finite. Model examples are considered, including an important example of the inverse problem with final overdetermination. The above results can be applicated in special parts of mathematical physics related to the theory of elasticity and the linear transport theory. As is customary, our research takes place in the general case with a choice of the complex scalar field, but the main facts are also true in the real case. The created theory allows transfer to nonlocal problems for evolution equations, when instead of the traditional initial condition special time averaging is used to find the solution.
Other Latest Articles
- Device for Periodic Modulation of Laser Radiation
- Analysis of Queueing Systems with an Infinite Number of Servers and a Small Parameter
- Construction of the Mathematical Model of Pricing for Telecommunication Services with Allowance for Congestion in Networks
- An Inviscid Analogue of the Poiseuille Problem
- On the Calculation of Electromagnetic Fields in Closed Waveguides with Inhomogeneous Filling
Last modified: 2020-08-31 19:25:59