Inverse problem of determining the kernel in an integro-differential equation of parabolic type

Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki

Nuriddinov

2020

The inverse problem of determining a multidimensional kernel of an integral term depending on a time variable $t$ and $ (n-1)$-dimensional spatial variable $x'=\left(x_1,\ldots, x_ {n-1}\right)$ in the $n$-dimensional heat equation with a variable coefficient of thermal conductivity is investigated. The direct problem is the Cauchy problem for this equation. The integral term has the time convolution form of kernel and direct problem solution. As additional information for solving the inverse problem, the solution of the direct problem on the hyperplane $x_n = 0$ is given. At the beginning, the properties of the solution to the direct problem are studied. For this, the problem is reduced to solving an integral equation of the second kind of Volterra-type and the method of successive approximations is applied to it. Further the stated inverse problem is reduced to two auxiliary problems, in the second one of them an unknown kernel is included in an additional condition outside integral. Then the auxiliary problems are replaced by an equivalent closed system of Volterra-type integral equations with respect to unknown functions. Applying the method of contraction mappings to this system in the Hölder class of functions, we prove the main result of the article, which is a local existence and uniqueness theorem of the inverse problem solution.

Section: Introductionsupporting

confidence: 82%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On investigation of the inverse problem for a parabolic integro-differential equation with a variable coefficient of thermal conductivity

Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki

Nuriddinov

2020

“…To close the system of integral equations (3.8), (4.1)-(4.3), we use the obvious equalities (4.4)-(4.6). Under the assumptions of the theorem, the validity of the inverse transformations is established in the usual way (see [12]). Therefore, Lemma 2 is proved.…”

Section: In View Of (24) Eqs (21)-(23) For the New Functions V(xmentioning

confidence: 99%

Inverse problem of determining the one-dimensional kernel of the viscoelasticity equation in a bounded domain

Safarov

2015

Math Notes

Self Cite

The one-dimensional integro-differential equation arising in the theory of viscoelasticity with constant density and Lam´e coefficients is considered. The direct problem is to determine the displacement function from the initial boundary-value problem for this equation, provided that the initial conditions are zero. The spatial domain is the closed interval [0, l], and the boundary condition is given by the stress function in the form of a concentrated perturbation source at the left endpoint of this interval and as zero at the right endpoint. For the direct problem, we study the inverse problem of determining the kernel appearing in the integral term of the equation. To find it, we introduce an additional condition for the displacement function at x = 0. The inverse problem is replaced by an equivalent system of integral equations for the unknown functions. The contraction mapping principle is applied to this system in the space of continuous functions with weighted norms. A theorem on the global unique solvability is proved.

“…The subject matter of previous studies 1-10 is related most closely to that of the present paper. These papers deal with problems of determining the kernel depending only on the temporal variable (one-dimensional inverse problem) for the cases of distributed [1][2][3][4] and concentrated [5][6][7][8][9][10] wave excitation sources. These problems can be reduced to the solution of integral equations of Volterra type for the unknown functions.…”

Section: Introductionmentioning

confidence: 99%

The problem of determining the one‐dimensional matrix kernel of the system of viscoelasticity equations

Math Methods in App Sciences

Totieva

2018

The integro‐differential system of viscoelasticity equations with a source of explosive type is considered. It is assumed that the coefficients of the equations depend only on one spatial variable. The problem of determining the matrix kernel included in the integral terms of the equations is studied. The solution of the problem is reduced to a series of inverse problems for scalar hyperbolic equations. For each case, the inverse problem is replaced by an equivalent system of integral equations for unknown functions. The principle of constricted mapping in the space of continuous functions with weighted norms to the latter is applied. The theorems of global unique solvability are proved, and the stability estimates of solution to the inverse problems are obtained.