Memory kernel reconstruction problems in the integro‐differential equation of rigid heat conductor

Abstract

The inverse problems of determining the energy–temperature relation α(t) and the heat conduction relation k(t) functions in the one‐dimensional integro‐differential heat equation are investigated. The direct problem is the initial‐boundary problem for this equation. The integral terms have the time convolution form of unknown kernels and direct problem solution. As additional information for solving inverse problems, the solution of the direct problem for x = x0 is given. At the beginning, an auxiliary problem, which is equivalent to the original problem, is introduced. Then the auxiliary problem is reduced to 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 continuous class of functions with weighted norms, we prove the main result of the article, which is a global existence and uniqueness theorem of inverse problem solutions.

Inverse problem of determining the time- dependent beam stiffness coefficient in the beam vibration equation using the finite difference method

“…Integrating by parts five times in the integrals for ϕ n and three times in the integrals for ψ n and f n (t) and taking into account the assumptions of Lemma 2.1, we obtain the relation (17). Inequality (18) is the Bessel inequality for the coefficients of the Fourier expansions of the functions ϕ If the functions ϕ(x), ψ(x) and f (x, t) satisfy the conditions of Lemma 2, then, by virtue of ( 17) and ( 18), the series ( 16) converges, and consequently the series ( 6), ( 14) and ( 15) converge absolutely and uniformly in the rectangle G, thus the sum of the series (6) satisfies the relations ( 1)-(4).…”

Kernel Determination Problem for One Parabolic Equation With Memory

“…Multidimensional inverse problems, when a kernel, in addition to the time variable, also depends on all or a part of spatial variables, are few studied. In this direction, we observe [4,5,7,9,16]. In [7], the problem of determining a kernel depending on a time variable t and an (n − 1)−dimensional spatial variable x ′ = (x 1 , .…”

“…Our investigations were devoted to the results of [4,5,7,9] under the condition of the integrodifferential heat equation of parabolic type with a variable coefficient and a particular convolution integral.…”

Inverse coefficient problem for a time - fractional wave equation with initial - boundary and integral type overdetermination conditions

“…Inverse problems for classical integro-differential wave propagation equations have been extensively studied. Nonlinear inverse coefficient problems with various types of sufficient determination conditions are often found in the literature (e.g., [15][16][17][18][19][20][21][22] and references therein). In [23][24][25][26] both the existence and uniqueness of a solution to the inverse problem are proved.…”