Equivalence of definitions of solutions for some class of fractional diffusion equations

TL;DR

This paper proves the equivalence of different definitions of weak solutions for various fractional diffusion equations, unifying variational and Laplace transform approaches to enhance understanding and solution properties.

Contribution

It establishes the equivalence of variational and Laplace transform-based definitions of weak solutions for fractional diffusion equations, unifying theoretical frameworks.

Findings

Proves the equivalence of solution definitions for fractional diffusion equations.

Shows that solutions combine advantages of variational and transform approaches.

Provides a unified framework for analyzing fractional diffusion problems.

Abstract

We study the unique existence of weak solutions for initial boundary value problems associated with different class of fractional diffusion equations including variable order, distributed order and multiterm fractional diffusion equations. So far, different definitions of weak solutions have been considered for these class of problems. This includes definition of solutions in a variational sense and definition of solutions from properties of their Laplace transform in time. The goal of the present article is to unify these two approaches by showing the equivalence of these two definitions. Such property allows also to show that the weak solutions under consideration combine the advantage of these two class of solutions which include representation of solutions by a Duhamel type of formula, suitable properties of Laplace transform of solutions, resolution of the equation in the sense of distributions and explicit link with the initial condition.