Existence and Uniqueness of a Solution of a System of Nonlinear Integral Equations
- PDF / 365,862 Bytes
- 8 Pages / 612 x 792 pts (letter) Page_size
- 69 Downloads / 229 Views
GRAL AND INTEGRO-DIFFERENTIAL EQUATIONS
Existence and Uniqueness of a Solution of a System of Nonlinear Integral Equations A. M. Denisov1∗ 1
Lomonosov Moscow State University, Moscow, 119991 Russia e-mail: ∗ [email protected]
Received March 23, 2020; revised March 23, 2020; accepted May 14, 2020
Abstract— We consider a system of nonlinear integral equations arising when studying the inverse coefficient problem for a system of partial differential equations. The inverse problem is to determine two coefficients of the system based on additional information on one of the solution components. We prove the local existence and uniqueness of the solution of the nonlinear system of integral equations and, as a consequence, the existence and uniqueness of the solution of the inverse problem. DOI: 10.1134/S0012266120090049
1. SYSTEM OF NONLINEAR INTEGRAL EQUATIONS Consider the following problem for functions u(x, t) and a(x, t): ux + at = 0,
(x, t) ∈ QT ,
at = γ(t)(ϕ(t)u − a),
(x, t) ∈ QT ,
(1.1) (1.2)
u(0, t) = µ(t),
0 ≤ t ≤ T,
(1.3)
a(x, 0) = ψ(x),
0 ≤ x ≤ l,
(1.4)
where positive numbers T and l and continuous functions γ(t), ϕ(t), µ(t), and ψ(t) are given and Qτ = {(x, t) : 0 ≤ x ≤ l, 0 ≤ t ≤ τ }. Problem (1.1)–(1.4) can be viewed as a mathematical model for the sorption dynamics process [1, p. 174; 2, p. 6] in which the properties of the absorbent vary over time. Consider the inverse problem. Let the functions µ(t) and ψ(x) be given, and let the functions γ(t) and ϕ(t) be unknown. It is required to determine the functions γ(t), ϕ(t), u(x, t), and a(x, t) given some additional information on one of the components of the solution of problem (1.1)–(1.4): u(l, t) = g(t),
0 ≤ t ≤ T,
(1.5)
ux (l, t) = p(t),
0 ≤ t ≤ T.
(1.6)
We will assume that the given functions µ(t), ψ(x), g(t), and p(t) satisfy the following conditions: µ, g, p ∈ C[0, T ]; ψ ∈ C[0, l]; µ(t) > 0, g(t) > 0, and p(t) < 0 for 0 ≤ t ≤ T ; ψ(x) ≥ 0 for 0 ≤ x ≤ l, ψ(l) = 0, and ψ(x) is not identically zero. It follows from Eqs. (1.1) and (1.2) and conditions (1.3)–(1.6) that this inverse problem is an evolution problem with respect to the variable t. Therefore, we give a definition of its solution for an arbitrary t0 ∈ (0, T ]. Definition. A quadruple of functions (γ(t), ϕ(t), u(x, t), a(x, t)) is called a solution of the inverse problem for t ∈ [0, t0 ] if γ, ϕ ∈ C[0, t0 ], u, ux , a, at ∈ C(Qt0 ), one has the inequalities γ(t) > 0, ϕ(t) > 0 for 0 ≤ t ≤ t0 and the inequalities u(x, t) > 0, a(x, t) ≥ 0 for (x, t) ∈ Qt0 , and the functions γ(t), ϕ(t), u(x, t), and a(x, t) satisfy Eqs. (1.1) and (1.2) and conditions (1.3)–(1.6) in Qt0 . Let us reduce the inverse problem to a system of nonlinear integral equations. 1140
EXISTENCE AND UNIQUENESS OF A SOLUTION
1141
Let a quadruple of functions (γ(t), ϕ(t), u(x, t), a(x, t)) be a solution of the inverse problem for t ∈ [0, t0 ]. Integrating Eq. (1.2) with allowance for condition (1.4), we obtain Zt a(x, t) = ψ(x) exp − γ(θ) dθ 0 (1.7) Zt Zt + exp − γ(θ) dθ γ(τ )ϕ(τ )u(x, τ ) dτ, (
Data Loading...
