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.
M.Allen, A nondivergence parabolic problem with a fractional time derivative, Differential Integral Equations31(3–4) (2018), 215–230.
2.
M.Allen, L.Caffarelli and A.Vasseur, A parabolic problem with a fractional time derivative, Arch. Ration. Mech. Anal.221(2) (2016), 603–630. doi:10.1007/s00205-016-0969-z.
3.
G.Barles and P.E.Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal.4(3) (1991), 271–283.
4.
F.Camilli and A.Goffi, Existence and regularity results for viscous Hamilton–Jacobi equations with Caputo time-fractional derivative, 2019, preprint.
5.
Z.-Q.Chen, Time fractional equations and probabilistic representation, Chaos Solitons Fractals102 (2017), 168–174. doi:10.1016/j.chaos.2017.04.029.
6.
Z.-Q.Chen, P.Kim, T.Kumagai and J.Wang, Heat kernel estimates for time fractional equations, preprint.
7.
M.G.Crandall, H.Ishii and P.-L.Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N. S.)27(1) (1992), 1–67. doi:10.1090/S0273-0979-1992-00266-5.
8.
Y.Giga, Surface Evolution Equations, a Level Set Approach, Monographs in Mathematics, Vol. 99, Birkhäuser Verlag, Basel, 2006.
9.
Y.Giga and T.Namba, Well-posedness of Hamilton–Jacobi equations with Caputo’s time fractional derivative, Comm. Partial Differential Equations42(7) (2017), 1088–1120. doi:10.1080/03605302.2017.1324880.
10.
R.Jensen, P.-L.Lions and P.E.Souganidis, A uniqueness result for viscosity solutions of second order fully nonlinear partial differential equations, Proc. Amer. Math. Soc.102(4) (1988), 975–978. doi:10.1090/S0002-9939-1988-0934877-2.
11.
A.Kubica and M.Yamamoto, Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients, Fract. Calc. Appl. Anal.21(2) (2018), 276–311. doi:10.1515/fca-2018-0018.
12.
Y.Luchko, Maximum principle for the generalized time-fractional diffusion equation, J. Math. Anal. Appl.351(1) (2009), 218–223. doi:10.1016/j.jmaa.2008.10.018.
13.
R.Metzler and J.Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep.339(1) (2000), 1–77. doi:10.1016/S0370-1573(00)00070-3.
14.
T.Namba, On existence and uniqueness of viscosity solutions for second order fully nonlinear PDEs with Caputo time fractional derivatives, NoDEA Nonlinear Differential Equations Appl.25(3) (2018), 23. doi:10.1007/s00030-018-0513-y.
15.
T.Namba, Analysis for viscosity solutions with special emphasis on anomalous effects, January 2017, thesis, University of Tokyo.
16.
A.Oberman, Convergent difference schemes for degenerate elliptic and parabolic equations: Hamilton–Jacobi equations and free boundary problems, SIAM J. Numer. Anal.44(2) (2006), 879–895. doi:10.1137/S0036142903435235.
17.
K.Sakamoto and M.Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl.382(1) (2011), 426–447. doi:10.1016/j.jmaa.2011.04.058.
18.
E.Topp and M.Yangari, Existence and uniqueness for parabolic problems with Caputo time derivative, J. Differential Equations262(12) (2017), 6018–6046. doi:10.1016/j.jde.2017.02.024.
19.
R.Zacher, Weak solutions of abstract evolutionary integro-differential equations in Hilbert spaces, Funkcial. Ekvac.52(1) (2009), 1–18. doi:10.1619/fesi.52.1.
