Subordination approach to multi-term time-fractional diffusion-wave equations

Abstract

This paper is concerned with the fractional evolution equation with a discrete distribution of Caputo time-derivatives such that the largest and the smallest orders, $\alpha$ and ${\alpha }_m$, satisfy the conditions $1<\alpha \le 2$ and $\alpha -{\alpha }_m\le 1$. First, based on a study of the related propagation function, the nonnegativity of the fundamental solutions to the spatially one-dimensional Cauchy and signaling problems is proven and propagation speed of a disturbance is discussed. Next, we study the equation with a general linear spatial differential operator defined in a Banach space and suppose it generates a cosine family. A subordination principle is established, which implies the existence of a unique solution and gives an integral representation of the solution operator in terms of the corresponding cosine family and a probability density function. Explicit representation of the probability density function is derived. The subordination principle is applied for obtaining regularity results. The analytical findings are supported by numerical work.

References in corpus (5)

• Distributed Order Calculus and Equations of Ultraslow Diffusion
• Distributed-order fractional Cauchy problems on bounded domains
• Propagation Speed of the Maximum of the Fundamental Solution to the Fractional Diffusion-Wave Equation
• Solutions of the main boundary value problems for the time-fractional telegraph equation by the Green function method
• Completely monotone functions and some classes of fractional evolution equations

Cited by in corpus (3)

• Generalized diffusion-wave equation with memory kernel
• Strong positivity property and a related inverse source problem for multi-term time-fractional diffusion equations
• A Boundary-Value Problem for the 3-D Fractional Wave Equation with Singularity