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We identify the stochastic processes associated with one-sided fractional partial differential equations on a bounded domain with various boundary conditions. This is essential for modelling using spatial fractional derivatives. We show well-posedness of the associated Cauchy problems in $C_0(Omega)$ and $L_1(Omega)$. In order to do so we develop a new method of embedding finite state Markov processes into Feller processes and then show convergence of the respective Feller processes. This also gives a numerical approximation of the solution. The proof of well-posedness closes a gap in many numerical algorithm articles approximating solutions to fractional differential equations that use the Lax-Richtmyer Equivalence Theorem to prove convergence without checking well-posedness.
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This paper derives physically meaningful boundary conditions for fractional diffusion equations, using a mass balance approach. Numerical solutions are presented, and theoretical properties are reviewed, including well-posedness and steady state solutions. Absorbing and reflecting boundary conditions are considered, and illustrated through several examples. Reflecting boundary conditions involve fractional derivatives. The Caputo fractional derivative is shown to be unsuitable for modeling fractional diffusion, since the resulting boundary value problem is not positivity preserving.
In this paper, we develop fast procedures for solving linear systems arising from discretization of ordinary and partial differential equations with Caputo fractional derivative w.r.t time variable. First, we consider a finite difference scheme to solve a two-sided fractional ordinary equation. Furthermore, we present a fast solution technique to accelerate Toeplitz matrix-vector multiplications arising from finite difference discretization. This fast solution technique is based on a fast Fourier transform and depends on the special structure of coefficient matrices, and it helps to reduce the computational work from $O(N^{3})$ required by traditional methods to $O(Nlog^{2}N)$ and the memory requirement from $O(N^{2})$ to $O(N)$ without using any lossy compression, where $N$ is the number of unknowns. Two finite difference schemes to solve time fractional hyperbolic equations with different fractional order $gamma$ are considered. We present a fast solution technique depending on the special structure of coefficient matrices by rearranging the order of unknowns. It helps to reduce the computational work from $O(N^2M)$ required by traditional methods to $O(N$log$^{2}N)$ and the memory requirement from $O(NM)$ to $O(N)$ without using any lossy compression, where $N=tau^{-1}$ and $tau$ is the size of time step, $M=h^{-1}$ and $h$ is the size of space step. Importantly, a fast method is employed to solve the classical time fractional diffusion equation with a lower coast at $O(MN$log$^2N)$, where the direct method requires an overall computational complexity of $O(N^2M)$. Moreover, the applicability and accuracy of the scheme are demonstrated by numerical experiments to support our theoretical analysis.
We study an algorithm which has been proposed by Chinesta et al. to solve high-dimensional partial differential equations. The idea is to represent the solution as a sum of tensor products and to compute iteratively the terms of this sum. This algorithm is related to the so-called greedy algorithm introduced by Temlyakov. In this paper, we investigate the application of the greedy algorithm in finance and more precisely to the option pricing problem. We approximate the solution to the Black-Scholes equation and we propose a variance reduction method. In numerical experiments, we obtain results for up to 10 underlyings. Besides, the proposed variance reduction method permits an important reduction of the variance in comparison with a classical Monte Carlo method.
In this chapter, we mainly review theoretical results on inverse source problems for diffusion equations with the Caputo time-fractional derivatives of order $alphain(0,1)$. Our survey covers the following types of inverse problems: 1. determination of time-dependent functions in interior source terms 2. determination of space-dependent functions in interior source terms 3. determination of time-dependent functions appearing in boundary conditions
When considering fractional diffusion equation as model equation in analyzing anomalous diffusion processes, some important parameters in the model related to orders of the fractional derivatives, are often unknown and difficult to be directly measured, which requires one to discuss inverse problems of identifying these physical quantities from some indirectly observed information of solutions. Inverse problems in determining these unknown parameters of the model are not only theoretically interesting, but also necessary for finding solutions to initial-boundary value problems and studying properties of solutions. This chapter surveys works on such inverse problems for fractional diffusion equations.
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