Probabilistic solutions to nonlinear fractional differential equations of generalized 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.2010 Mathematics Subject Classification. 34A08, 60H30, 35S15, 34A05.

Generalised fractional evolution equations of Caputo type

“…is the joint density of (τ β 0 (t), X t,β 0+ * (s)) (see [17,Proposition 4.2]), using the notation of assumption (H2) and Remark 3.3. To obtain the last equality we used standard change of variables and identities for the stable densities ω β (⋅; ⋅, ⋅).…”

On the Solution of Two-Sided Fractional Ordinary Differential Equations of Caputo Type

“…As a continuation of our previous works, which show a new link between stochastic analysis and fractional equations (see [16]- [17], [27]), this paper appeals to a probabilistic approach to study equations involving both left-sided and right-sided generalized operators of Caputo type. We address the boundary value problem for the two-sided generalized linear equation with Caputo type derivatives −D where λ ≥ 0, u a , u b ∈ R and g is a prescribed function on [a, b].…”

Stochastic representation of solution to nonlocal-in-time diffusion

“…This operator is also known as the generator form of fractional derivatives [30,38], or a Lévy-type generator [10]. (iii) Other particular cases include the fractional derivatives of variable order, which are obtained by taking ρ as the function ρ(t, r) = −r −1−α(t) /Γ(−α(t)) with a suitable function α(t) : R → (0, 1) [27], and tempered Lévy kernels ρ(t, r) = −e −λr r −1−α /Γ(−α), α ∈ (0, 1), λ > 0, [11,51].…”