On the probabilistic approach to the solution of generalized fractional differential equations of Caputo and Riemann-Liouville type

Abstract

This paper provides a probabilistic approach to solve linear equations involving Caputo and Riemann-Liouville type derivatives. Using the probabilistic interpretation of these operators as the generators of interrupted Feller processes, we obtain well-posedness results and explicit solutions (in terms of the transition densities of the underlying stochastic processes). The problems studied here include fractional linear differential equations, well analyzed in the literature, as well as their far reaching extensions.

29 pages

References in corpus (6)

• Fractional Calculus: Some Basic Problems in Continuum and Statistical Mechanics
• Fractional Cauchy problems on bounded domains
• The Laplace Transform Method for Linear Differential Equations of the Fractional Order
• A Wiener--Hopf Monte Carlo simulation technique for Lévy processes
• Generalized Continuous-Time Random Walks (CTRW), Subordination by Hitting Times and Fractional Dynamics
• On fully mixed and multidimensional extensions of the Caputo and Riemann-Liouville derivatives, related Markov processes and fractional differential equations

Cited by in corpus (4)

• Generalised Fractional Evolution Equations of Caputo Type
• Censored Stable Subordinators and Fractional Derivatives
• Stochastic representation of solution to nonlocal-in-time diffusion
• On the solution of two-sided fractional ordinary differential equations of Caputo type