On a discrete scheme for time fractional fully nonlinear evolution equations

Abstract

We introduce a discrete scheme for second order fully nonlinear parabolic PDEs with Caputo's time fractional derivatives. We prove the convergence of the scheme in the framework of the theory of viscosity solutions. The discrete scheme can be viewed as a resolvent-type approximation.Studying differential equations with fractional derivatives is motivated by mathematical models that describe diffusion phenomena in complex media like fractals, which 2010 Mathematics Subject Classification. 35R11, 35A35, 35D40.

Approximation of an optimal control problem for the time-fractional Fokker-Planck equation

“…We conclude, by the backward version of discrete Gronwall inequality (14), the uniform convergence of u σ to u. The convergence of the gradient follows by (31).…”

“…∞ + C and therefore, applying a backward version of the discrete Gronwall's inequality (14) with ω n = D h U n ∞ , λ 1 = C, λ 2 = λ 3 = 0 and g n = C, we obtain (15) for some positive constant L.…”

“…Hamilton-Jacobi-Bellman equations with Caputo derivative have been considered from a probabilistic point of view by Kolokoltsov et al [19]. The theory of viscosity solution for this kind of equations has been constructed by Topp and Yangari [30], Giga, Namba et al [14,25]. Recently, Camilli and Goffi studied the existence and uniqueness of classical solutions to the time-fractional Hamilton-Jacobi-Bellman equation [9].…”

“…We use a classical scheme, called L1 approximation, which is naturally derived from the approximation of the fractional integral as a Riemann sum and long known to be consistent (for a concise illustration, we refer to section 3 of [29]). Moreover, the L1 approximation scheme stands out by being able to preserve at the discrete level certain desirable features of the original PDEs, such as maximum principle [14] and energy stability [13,27,29]. Therefore, approximations can be constructed to converge to suitable notions of weak solutions, which may be viscosity solutions to Hamilton-Jacobi-Bellman equations [14], or distributional solutions in Sobolev spaces to diffusion (phase field) equations [13,27,29].…”

“…Moreover, because (u, m) is a smooth function, (Ũ σ ,M σ ) is a solution of (26) with the perturbation (a n σ , b n σ ) given by the consistency error of the scheme. Hence, max 0≤n≤N { a n σ ∞ + b n σ ∞ } → 0 for σ → 0 (30) (for uniform the convergence of the approximation to the time-fractional derivative, see for example [14,Lemma 3.1]). The first step of the proof is the convergence result…”