Inverse Boundary Value Problem for a Third-Order Partial Differential Equation with Integral Conditions

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Inverse Boundary Value Problem for a Third-Order Partial Differential Equation with Integral Conditions Ziyatkhan Seyfaddin Aliyev1,2 Elmira Haci Yusifova2

· Yashar Topush Mehraliyev3 ·

Received: 26 February 2020 / Revised: 25 August 2020 / Accepted: 5 September 2020 © Iranian Mathematical Society 2020

Abstract In this paper, we study a nonlinear inverse problem for a third-order partial differential equation with integral conditions. This inverse problem is formulated as an auxiliary inverse problem which, in turn, is reduced to the operator equation in a specified Banach space using the method of spectral analysis. Finally, the existence and uniqueness of this operator equation is proved by applying the contraction mapping principle. Keywords Inverse boundary value problem · Partial differential equation of third-order · Integral boundary condition · Fixed point theorem Mathematics Subject Classification 35G31 · 35P10 · 35R30 · 47H10

1 Introduction Let Q T = {(x, t) : 0 < x < 1, 0 < t < T }. In the domain Q T , we consider the thirdorder partial differential equation:

Communicated by Majid Gazor.

B

Ziyatkhan Seyfaddin Aliyev [email protected] Yashar Topush Mehraliyev [email protected] Elmira Haci Yusifova [email protected]

1

Department of Mathematical Analysis, Baku State University, 1148 Baku, Azerbaijan

2

Department of Differential Equations, Institute of Mathematics and Mechanics National Academy of Sciences of Azerbaijan, 1141 Baku, Azerbaijan

3

Department of Differential Equations, Baku State University, 1148 Baku, Azerbaijan

123

Bulletin of the Iranian Mathematical Society

u ttt (x, t) + u x x (x, t) = a(t)u(x, t) + f (x, t).

(1.1)

For Eq. (1.1), we consider the following inverse problem: find a pair {u, a} of realvalued functions, such that u = u(x, t), (x, t) ∈ Q T , a = a(t), t ∈ [0, T ], satisfying in Q T Eq. (1.1), initial and boundary conditions: T u(x, 0) = ϕ0 (x) +

p0 (t) u(x, t) dt, 0

T

u t (x, T ) = ϕ1 (x) +

p1 (t) u(x, t) dt,

(1.2)

0

T u tt (x, 0) = ϕ2 (x) +

p2 (t) u(x, t) dt, 0 ≤ x ≤ 1, 0

u x (0, t) = u(1, t) = 0, 0 ≤ t ≤ T ,

(1.3)

and the condition of redefinition for the function u(x, t): 1 u(0, t) +

q(x) u(x, t) dx = h(t), 0 ≤ t ≤ T ,

(1.4)

0

where f (x, t), (x, t) ∈ Q T , ϕi (x), x ∈ [0, 1], pi (t), i = 0, 1, 2, h(t), t ∈ [0, T ] and q(x), x ∈ [0, 1] are given real-valued functions. Inverse problems for partial differential equations in various settings have been studied in many works (see, e.g., [3,7,15,17,18,22,24,28,29,32] and the bibliography therein). Problems of this type have many applications in various fields of science as geophysics, filtration theory, mineral exploration, biology, medicine, computed tomography, etc.; for a detailed information, see [11,13,15,17,22,26]. An important class of nonlocal problems are boundary value problems for partial differential equations with nonlocal conditions. In recent years, boundary value problems with nonlocal conditions have been actively studied (see [5,6,8,9,16,20,21,27,31] and the references th