Distributed Order Derivatives and Relaxation Patterns

Abstract

We consider equations of the form $(D_{(\rho )}u)(t)=-\lambda u(t)$, $t>0$, where $\lambda >0$, $D_{(\rho )}$ is a distributed order derivative, that is the Caputo-Dzhrbashyan fractional derivative of order $\alpha$, integrated in $\alpha \in (0,1)$ with respect to a positive measure $\rho$. Such equations are used for modeling anomalous, non-exponential relaxation processes. In this work we study asymptotic behavior of solutions of the above equation, depending on properties of the measure $\rho$.

Cited by in corpus (11)

• Inverse Problems of Determining Sources of the Fractional Partial Differential Equations
• Growth Equation of the General Fractional Calculus
• Inverse Problems of Determining Parameters of the Fractional Partial Differential Equations
• Generalized diffusion-wave equation with memory kernel
• Relaxation patterns and semi-Markov dynamics
• Fractional Cauchy problems on bounded domains: survey of recent results
• Decay of solutions to parabolic-type problem with distributed order Caputo derivative
• Lévy mixing related to distributed order calculus, subordinators and slow diffusions
• General Fractional Calculus, Evolution Equations, and Renewal Processes
• Fractional kinetic hierarchies and intermittency
• Strong dissipativity of generalized time-fractional derivatives and quasi-linear (stochastic) partial differential equations