On the solution of fractional differential equations: a probabilistic approach

  • Thu 25 Oct 18

    14:00 - 16:00

  • Colchester Campus

    STEM Centre 3.1

  • Event speaker

    Dr Elena Hernandez-Hernandez (Warwick University)

  • Event type

    Lectures, talks and seminars

  • Event organiser

    Mathematical Sciences, Department of

  • Contact details

    Dr Harrison

Dr Elena Hernandez-Hernandez (Warwick University)

In the last decades, the theory of fractional differential equations (FDE's) has been actively studied due to its successful application at providing more accurate models to describe a variety of physical phenomena. To solve FDE's various numerical and analytical approaches have been investigated (e.g., the Laplace transform, the Mellin transform, the Fourier transform techniques, the operational calculus method). As for a probabilistic framework, some connections between probability and fractional differential equations are also known in the literature; for instance, the probabilistic interpretation of the Green (or fundamental) solution to the time-space fractional diffusion equation.

In this talk we aim to give a general overview of a probabilistic approach to study the
well-posedness for boundary value problems for fractional ordinary differential equations such as

Dβa+*u(x) = λ(x)u(x) − g(x), x є (a, b]

and, more generally, fractional partial differential equations of the type

tDβ a+*u(t, x) = Au(t,x) − g(t, x), t є (a, b], x є Rd

u(a,x) = φa(x), x є Rd

where Dβa+* stands for the fractional Caputo derivative of order β є (0, 1) and A is the infinitesimal generator of a Markov process. The approach presented here is based on the probabilistic interpretation of fractional Caputo derivatives as the generators of β−stable processes interrupted on the first attempt to cross certain boundary point.

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