Numerical reconstruction of the kernel function in generalized non-convolutional fractional operators

Abstract

This paper deals with the numerical reconstruction of the kernel function 𝜅(𝑡,𝑧) for a new class of generalized non-convolutional fractional operators. These operators are defined in integral form of Volterra type and the general kernel is not always reducible to classical convolutiontype expressions. The work focuses on the reconstruction of the kernel starting from the generalized Sonin condition, which connects the kernel of the integral operator with that of the corresponding differential operator. Taking the generalized Sonin condition as the starting point, we propose a constructive numerical method by approximating the functional identity obtained from the integration of the product of both kernels, operation that is achieved by proper integration on segment lines of the domain. By casting the theoretical requirements as one-dimensional integral estimations on parameterized trajectories (segment lines), the problem becomes a nonlinear inverse problem for the determination of one of the kernels, 𝜅, provided that the other one is known. Our approach is based on the selection of a proper partition of the domain, and the consideration of the auxiliary functions 𝜑𝑡,𝑧 and 𝜙𝑡,𝑧, which are combined in such a way that it is possible to compute the unknown functions 𝜓𝑡,𝑧 and 𝛹𝑡,𝑧, provided that the given restrictions on the integrals in the generalized Sonin condition are fulfilled. The feasibility of reconstructing the kernel in some chosen test examples is illustrated by some numerical procedures, as a first step toward the implementation of generalized fractional models in numerical simulations, and also as a tool for the selection of proper kernels in the definition of generalized fractional operators.