One-dimensional inverse problems of determining the kernel of the integro- differential heat equation in a bounded domain

Abstract: The integro-differential equation of heat conduction with the time-convolution integral on the right side is considered. The direct problem is the initial-boundary problem for this integro-differential equation. Two inverse problems are studied for this direct problem consisting in determining a kernel of the integral member on two given additional conditions with respect to the solution of the direct problems, respectively. The problems are replaced with the equivalent system of the integral equations with respect to unknown functions and on the basis of contractive mapping the unique solvability inverse problem.

1 Introduction: setting up the problems Problems of determining coefficients, right sides, or other physical parameters in differential equations and partial differential equations (PDEs) equations, given additional "experimental" information about their solutions, arise quite often in various applications. These problems are inverse to the "direct" ones where a differential equation and initial and boundary data are given [1]. Inverse problems for parabolic and hyperbolic PDEs arise naturally in geophysics, oil prospecting, in the design of optical devices, and in many other areas where the interior of an object is to be imaged by measuring field in available domains. Problems of identification of memory kernels in such equations have been intensively studied starting at the end of the last century (see [2][3][4][5][6]). Nowadays, the study of inverse problems for parabolic integrodifferential equations is the subject of many studies, of which we mention previous works [7][8][9][10][11][12] as being closest to the topic of this work. We consider the initial-boundary problem of determining a function ( ) ( ) ( ) ∈ ∈ u x t x l t T , , 0, , 0, from the following equations: where a is a positive constant, and l and T are arbitrary positive numbers. When ( ) ( ) ( ) k t h x t φ x , , , , ( ) μ t 1 , and ( ) μ t 2 are given functions, this problem is called as a direct problem. In the inverse problem, it is assumed that the kernel ( ) > k t t , 0 of the integral term in (1) is unknown and it is required to determine it using additional information about the solution of the direct problem: , 0 , , 0 , . 0 0 (5) In this case, ( ) are assumed to be given functions. In the sequel, we will call the problem of determining functions ( ) ( ) ( ) ( ) ∈ ∈ u x t k t x l t T , , , 0, , 0, from equations (1)-(4) as Inverse problem 1 and the problem of determining functions ( ) ( ) For simplicity, we denote by ϑ the function u t , i.e., = u ϑ t . We differentiate the equalities equation (1) with respect to t, and using condition (2), we obtain The initial condition for ϑ will be obtained by setting = t 0 in equality (1) and using equality (2): To obtain the boundary conditions for ( ) ⋅ ϑ t , the equation differentiates equality (3) with respect to t: By differentiating the additional conditions (4) and (5) with respect to t, we obtain these conditions with respect to the function ϑ for inverse problem 1: and for inverse problem 2: We replace the initial-boundary problems (6) G x ξ t τ  l  e  πan  l  ξ  πan  l  x  , ,  2  sin  sin   n   πan  l  t τ   1   2 is the Green function of the initial-boundary problem for one-dimensional heat equation. Now we write two properties of Green function (see [13, pp. 200-221]), which will be needed in the future.
Remark 1. The integral of the Green function does not exceed 1: , which is twice continuously differentiable with respect to x and once continuously differentiable with respect to t in the domain D lT functions, We also use the usual class ( ) C D lT of continuous in D lT functions. To prove Lemma 1, we rewrite equation (11) in the form and denoting the sum of the first three summands on the right-hand side of (12) by ( ) where . lT If the conditions of Lemma 11 are fulfilled, we have that ( ) ( ) ∈ x t C D Φ , . One-dimensional inverse problems  3 Denote ( ) , , n n n1 Thus, for arbitrary = n k, we have It follows from the aforementioned estimates that the series , is a solution of equation (11). Now show that this solution is the only one. Suppose that there are two solutions ( ) is a solution to the equation Applying the Gronwall lemma here, we obtain that ( ) , 1 2 in D . lT Therefore, equation (11) has a unique solution in D lT . The lemma is proved.

Inverse problem 1
Using the additional condition for inverse problem 1, from (11) we have Differentiating this equality with respect to t, we arrive at equation: , where ( ) ⋅ δ is the Dirac's delta function, and taking into account the following relations: In what follows, we denote Next we write equality (14) as the integral equation of the second order with respect to unknown function ( ) k t We represent the system of equations (11) and (15) in the form where is the vector function and 1 2 is defined by the right sides of equations (11) and (15): The following notations were introduced in equalities (17) and (18): , , , , 1 , d . To prove Theorem 1, we define for the unknown vector function ( ) ( ) ∈ g x t C D , lT the following weight norm: The number ≥ σ 0 will be chosen later. Denote by ( ) B g ρ , 0 the ball of vector functions g with center at the point g 0 and radius > ρ 0, i.e., ( : .
The number > ρ 0 will also be chosen later.
if the numbers σ and ρ will be chosen in suitable way. Remind that operator A is contractive if the following two conditions are met (see [15, pp. 87-97]): Note that the weight norm ‖⋅‖ σ is equivalent to the usual norm ‖⋅‖: The convolution operator is commutative and invariant with respect to multiplication by − e σt : The last formula implies the estimation be an element of ( ) B g ρ , 0 , i.e., e G x ξ t τ g τ φ ξ ξ τ G x ξ t τ g α g ξ τ α α ξ τ G x ξ t τ g τ e φ ξ e ξ τ e Gx ξ t τ g αe g ξ τ αe α ξ τ  If we choose σ as i.e., the first condition of contractive mapping for A 1 is satisfied. Now we carry out the estimations for A 2 :   Using relations (19)-(24), we estimate I 2 as follows: One-dimensional inverse problems  7 Conducting the similar estimates alike as for the case I 1 we have for I 3  Now we can choose ρ σ , such that there hold the inequalities: . As a result, we conclude that if σ, ρ satisfy the conditions into itself, i.e., ( ) ∈ Ag B g ρ , .
0 Second, we check the second condition of contractive mapping. In accordance with (17) for the first component of operator A we obtain Ag G x ξ t τ g τ g τ φ ξ ξ τe G x ξ t τ g α g ξ τ α g α g ξ τ α α ξ τe Here the integrand in the last integral can be estimated as follows: Therefore, It is obvious that if we choose σ as i.e., the second condition of contractive mapping for A 1 is satisfied.
The second component of A can be estimated in the following form: We denote the summands in this equality by J J J , , , 1 2 3 respectively, and carry out the estimates for them separately. The estimate for J 1 has the form Taking into account the relation estimate J 2 and J 3 as follows: Summing the obtained estimates for Now we choose numbers σ ρ , so that the expression at ∥ ∥ − g g σ 1 2 becomes less than 1, i.e., the inequality

Inverse problem 2
In Section 1, Inverse problem 2 was reduced to the problem of determining the kernel ( ) ( ) ∈ k t t T , 0 , from equations (6)- (8) and (10). To obtain the integral equation for ( ) k t in this case we use equation (11) for solution direct problem and additional condition (10). As a result, we have We represent the system of equations (11) and (27) in the form of operator equation where is the vector function, and 1 2 are defined by the right parts of equations (11) and (27). Proof. We introduce the vector function by formula Then, in accordance with equalities (11) and (27), the components of operator A will have the form: The conditions of contractive mapping for operator A 1 were received in the previous section. Here, it is shown that A 2 has the property of a contraction mapping operator. Let ( ) ( ) ∈ g x t B D , T . Then, it is easy to see that The second component Ag can be estimated in the analogous way: is a contraction map. According to the principle of contracting operators, therefore, if σ, ρ are taken from condition ( ) > σ σ σ max , 1 2 , ( ) < ρ κ κ min , 3 4 , then the operator A in the set ( ) B g ρ , 0 has a unique fixed point. Hence, the theorem is proved. □ Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.