This paper investigates the inverse problem of finding a time-dependent diffusion coefficient in a parabolic equation with the periodic boundary and integral overdetermination conditions. Under some assumption on the data, the existence, uniqueness, and continuous dependence on the data of the solution are shown by using the generalized Fourier method. The accuracy and computational efficiency of the proposed method are verified with the help of the numerical examples.

Denote the domain

Consider the equation

The pair

The parameter identification in a parabolic differential equation from the data of integral overdetermination condition plays an important role in engineering and physics [

Boundary value problems for parabolic equations in one or two local classical conditions are replaced by heat moments [

Various statements of inverse problems on determination of thermal coefficient in one-dimensional heat equation were studied in [

Boundary value problems and inverse problems for parabolic equations with periodic boundary conditions are investigated in [

In the present work, one heat moment is used with periodic boundary condition for the determination of thermal coefficient. The existence and uniqueness of the classical solution of the problem (

This paper organized as follows. In Section

We have the following assumptions on the data of the problem (

where

Let the assumptions (

(1) The inverse problem (

(2) The solution of inverse problem (

By applying the standard procedure of the Fourier method, we obtain the following representation for the solution of (

The assumptions

Now let us show that there exists

Under assumptions (

Let

Let

If we consider these estimates in

From (

We use the finite difference method with a predictor-corrector-type approach, that is suggested in [

We subdivide the intervals

Equation (

Now, let us construct the predicting-correcting mechanism. First, multiplying (

Then from (

Consider the inverse problem (

It is easy to check that the analytical solution of the problem (

Let us apply the scheme which was explained in the previous section for the step sizes

In the case when

The analytical and numerical solutions of

The analytical and numerical solutions of

Consider the inverse problem (

It is easy to check that the analytical solution of the problem (

Let us apply the scheme which was explained in the previous section for the step sizes

In the case when

The analytical and numerical solutions of

The analytical and numerical solutions of

The inverse problem regarding the simultaneously identification of the time-dependent thermal diffusivity and the temperature distribution in one-dimensional heat equation with periodic boundary and integral overdetermination conditions has been considered. This inverse problem has been investigated from both theoretical and numerical points of view. In the theoretical part of the paper, the conditions for the existence, uniqueness, and continuous dependence on the data of the problem have been established. In the numerical part, the sensitivity of the Crank-Nicolson finite-difference scheme combined with an iteration method with the examples has been illustrated.