Identification Problem for Telegraph-Parabolic Equations
- PDF / 557,382 Bytes
- 12 Pages / 612 x 792 pts (letter) Page_size
- 24 Downloads / 225 Views
L DIFFERENTIAL EQUATIONS
Identification Problem for Telegraph-Parabolic Equations A. Ashyralyeva,b,c,*, M. Ashyraliyevd,**, and M. A. Ashyralyyevae,*** a Near
East University, Nicosia, 99138 TRNC Turkey Friendship University of Russia (RUDN University), Moscow, 117198 Russia c Institute of Mathematics and Mathematical Modeling, Almaty, 050010 Kazakhstan d Bahcesehir University, Istanbul, 34353 Turkey e Turkmen State University, Ashgabat, 744000 Turkmenistan *e-mail: [email protected] **e-mail: [email protected] ***e-mail: [email protected]
b Peoples’
Received February 15, 2020; revised February 15, 2020; accepted April 9, 2020
Abstract—An identification problem for an equation of mixed telegraph-parabolic type with an unknown parameter depending on spatial variables is considered. The unique solvability of this problem is proved, and stability inequalities for its solution are established. As applications, stability estimates are obtained for the solutions of four identification problems for telegraph-parabolic equations with an unknown source depending on spatial variables. Keywords: source identification problem, telegraph-parabolic equation, stability, Hilbert space DOI: 10.1134/S0965542520080035
1. INTRODUCTION Theoretical methods and applications of various local and nonlocal boundary value problems for mixed-type partial differential equations have been widely studied by many researchers (see [1–3] and references therein). In particular, this concerns various nonlocal boundary value problems for hyperbolicparabolic equations and numerical methods for approximate solutions of these problems (see [4–8] and references therein). Differential equations with unknown parameters play an important role in various fields of science and technology. For this reason, such equations have been widely studied (see [9–32] and references therein). However, the theory of identification problems for mixed-type equations has not yet received due attention. The main objective of this work is to study the identification problem for an equation of mixed telegraph-parabolic type with an unknown parameter depending on spatial variables. It is well known that various boundary value problems for telegraph-parabolic equations with a parameter reduce to the following boundary value problem for a telegraph-parabolic equation with an unknown parameter p in a Hilbert space H with a self-adjoint positive definite operator A :
u''(t ) + αu'(t ) + Au(t ) = f (t ) + p, 0 < t < 1, u'(t ) + Au(t ) = g(t ) + p, −1 < t < 0, u(−1) = ϕ, u(λ) = ψ, −1 < λ ≤ 1, u(0+) = u(0−), u'(0+) = u'(0−),
(1.1)
where A ≥ δI , δ > 0, and α ≥ 0 . Here, u(t ) and p denote
u(t ) = u(t; f (t ), g(t ), ϕ, ψ),
p = p( f (t ), g(t ), ϕ, ψ).
A pair (u(t ), p) is a solution of inverse problem (1.1) if the following conditions are satisfied: 1) u(t ) ∈ D for t ∈ [−1, 1], p ∈ H , and the function Au(t ) is continuous on the interval [−1, 1]. Here, D = D( A) denotes the domain of definition of the operator A ; 1294
IDENTIFICATION PROBLEM
1295
Data Loading...
