Problem of Determining a Multidimensional Kernel in One Parabolic Integro–differential Equation

Abstract

The multidimensional parabolic integro-differential equation with the time-convolution in- tegral on the right side is considered. The direct problem is represented by the Cauchy problem for this equation. In this paper it is studied the inverse problem consisting in finding of a time and spatial dependent kernel of the integrated member on known in a hyperplane xn = 0 for t > 0 to the solution of direct problem. With use of the resolvent of kernel this problem is reduced to the investigation of more convenient inverse problem. The last problem is replaced with the equivalent system of the integral equations with respect to unknown functions and on the bases of contractive mapping principle it is proved the unique solvability to the direct and inverse problems

Kernel Determination Problem for One Parabolic Equation With Memory

“…Among works devoted to finding the kernel depending on one time variable (one-dimensional inverse problem), we note [4,14,16,19]. 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.…”

“…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.…”