A specific inverse problem for the Volterra convolution equation

Abstract

This article attends to the specific inverse problem of determining the difference kernel k of a second kind linear Volterra integral equation of convolution type from a prescribed quotient μ of resolvent r and kernel k of the equation. A problem of this type arises, for example, in the theory of viscoelasticity and leads to a nonlinear integral equation of the third kind , for which several existence theorems are proved. Moreover, the applicability of Tikhonov's regularization method is studied. Finally, the case of non-convolution Volterra equations is also briefly discussed.

Keywords:

• Volterra equation
• Convolution
• Inverse problem
• Nonlinear ill-posed problem
• Regularization
• Linear viscoelasticity

MSC 2000 Subject Classifications::

• 45Q05
• 47J06
• 65R32
• 45G10
• 74D05