The Feynman-Kac equation describes the distribution of the functionals of the trajectories of the particles, where the functional is defined as with being a trajectory of a particle and a prescribed function depending on specific applications Kac1949
. There are two kinds of Feynman-Kac equations: one is for the forward Feynman-Kac equation, governing the joint probability density of the functional and position; and another one is for the backward equation, just focusing on the distribution of the functionals. If the particles are with power-law waiting time and/or jump length distribution(s), the governing equations for the distribution of the functionals are so-called fractional Feynman-Kac equationsAgmon1984 ; Carmi2010 ; Wang2018 , since the fractional substantial derivative is involved in the equations. More generalizations of the Feynman-Kac equations include the models governing the distribution of the functionals of the particles undergoing the reaction and diffusion processes and of the particles with multiple internal states Hou2018 ; Xu2018 ; Xu2018-2 .
Here we solve the following backward fractional Feynman-Kac equation, presented in Carmi2010 , describing the functional distribution of the particles with power-law waiting time, i.e.,
is the joint probability density function of finding the particle onat time with the initial position of the particle at ; is the Fourier pair of ; ; stands for Laplace operator; is a bounded domain and is assumed to be bounded in in this paper; is a fixed final time; denotes the Riemann-Liouville fractional substantial derivative, whose definition Li2015 is
where denotes the Riemann-Liouville fractional derivative with the definition Podlubny1999
So far there have been many works for fractional partial differential equations, including the finite difference method, finite element method, spectral method, and so onAcosta2018 ; Bazhlekova2015 ; chen2009 ; Cheng2015 ; Deng2008 ; Deng2013 ; Ervin2006 ; li2012 ; sun2006 , but there are relatively less researches on solving fractional Feynman-Kac equation numerically Chen2018 ; Deng2015 ; Deng:17 ; Deng2017 ; Nie2019 . The main reasons are that fractional substantial derivative is a time-space coupled non-local operator and the equation covers the complex parameters which bring about many challenges on regularity and numerical analyses. To our best knowledge, numerical approximation on fractional substantial derivative is given in Chen2015 ; Ref. Deng2015 numerically solves the forward and backward fractional Feynman-Kac equations with the assumptions that the solution is regular, is a positive constant, and ( means the real part of ); Ref. Deng2017 presents the error estimate for the backward fractional Feynman-Kac equation with and ; Ref. Deng:17 provides an efficient time-stepping method to solve the forward fractional Feynman-Kac equation and makes error analysis in the measure norm. In this paper, we use the finite element method in space and convolution quadrature introduced in Lubich1988 ; Lubich1988-2 in time to solve the backward fractional Feynman-Kac equation (1.1). The main contributions are as follows.
We first provide Sobolev regularity for the solution of Eq. (1.1), i.e., Theorem 2.1 gives that the solution when is bounded in and . Compared with the previous works Chen2018 ; Deng2015 ; Deng2017 , we construct numerical scheme without any assumption on the regularity of solution in temporal and spatial directions.
Then we modify the approximation of the Riemann-Liouville fractional derivative got by convolution quadrature to approximate the Riemann-Liouville fractional substantial derivative, which skillfully overcome the trouble brought by the non-commutativity of the Riemann-Liouville fractional derivative and in error estimate for fully discrete scheme, i.e., in Eq. (3.1).
Next, a suitable modify based on the Laplace transform representation of solution is presented to guarantee the accuracy of second-order backward difference scheme (3.9) (see Sec. 3).
Besides, motivated by the error estimate in space in Bazhlekova2015 ; Jin2016 , a general idea is to get the estimate of the difference between and (for their detailed definitions, see Sec. 3 and Sec. 4). Generally, the sufficient regularity on is required to ensure the accuracy of the approximation. Here, we use for the fully discrete scheme (4.1) in Sec. 4 instead of , which weakens the requirement of regularity on to keep the accuracy of the numerical scheme.
Finally, we provide a complete error analysis for the proposed numerical scheme and obtain the optimal convergence rates in - and -norm.
The rest of the paper is organized as follows. We first provide some preliminaries and a regularity estimate for the solution of Eq. (1.1) in Sec. 2. Section 3 presents the approximation of the Riemann-Liouville fractional substantial derivative by backward Euler and second-order backward difference convolution quadratures and gives the error estimates of the time semi-discrete schemes. In Sec. 4, we use the finite element method to discretize the Laplace operator and provide the error estimate for the fully discrete scheme with the non-smooth initial data. In Sec. 5, we verify the effectiveness of the algorithm by numerical experiments. We conclude the paper with some discussions in the last section.
First, we introduce with a zero Dirichlet boundary condition. For any , denote the space with the norm Thomee2006
wherenorm) of operator . Thus , , and . For and , we define sectors and in the complex plane as
and the contour is defined by
oriented with an increasing imaginary part, where denotes the imaginary unit and . Then we denote as the operator norm from to and define and as and respectively in the following. Throughout this paper, denotes a generic positive constant, whose value may differ at each occurrence; and let arbitrary small.
where denotes Caputo fractional substantial derivative defined by Li2015
and means the Caputo fractional derivative with its definition Podlubny1999
Then we recall the Laplace transform for the fractional substantial derivative.
Lemma 2.1 (Li2015 )
The Laplace transform of the Riemann-Liouville fractional substantial derivative with is given by
and the Laplace transform of the Caputo fractional substantial derivative with is given by
where and ‘’ stands for taking the Laplace transform. And in the following we denote as .
According to Lemma 2.1, the solution of Eq. can be written as
By the definition of , it is easy to see that
Before we provide the regularity estimate for the solution of Eq. (2.1), the following lemma about is also needed.
Lemma 2.2 (Deng:17 )
Let be bounded in . By choosing sufficiently close to and sufficiently large (depending on the value ), we have the following results:
For all and , we have , and
where and denote two positive constants. So and are both analytic function of .
The operator is well-defined, bounded, and analytic with respect to , satisfying
Assume is bounded in . If and is the solution of Eq. (2.1), then we have the estimate
3 Temporal discretization and error analysis
In this section, we first use the backward Euler and second-order backward difference convolution quadratures introduced in Lubich1988 ; Lubich1988-2 to discretize the Riemann-Liouville fractional substantial derivative and get the first- and second-order schemes in time. Then we provide the complete error analysis.
Let the time step size with , , , and . Firstly, we use the relationship between Caputo fractional derivative and Riemann-Liouville fractional derivative
to reformulate Eq. (2.1) with Riemann-Liouville fractional substantial derivative, i.e.,
3.1 Backward Euler scheme and error estimate
We use backward Euler convolution quadrature to discretize the time fractional substantial derivative and get the first-order accuracy in time. Introduce as the numerical approximation of solution . Then we can obtain the temporal semi-discrete scheme
Multiplying on both sides of the first formula of (3.2) and summing from to lead to
Simple calculation implies
which is followed by . Furthermore, we have
Using Cauchy’s integral formula yields
where , , , and the second equality follows by taking . Deforming the contour to , one has
Next, we provide a lemma about defined in (3.5).
Lemma 3.1 (Deng:17 )
Let be bounded in . By choosing sufficiently close to and sufficiently large depending on , there exists a positive constant depending on and such that the following estimates hold when :
For , we have , and
The operator is well-defined, bounded, and analytic with respect to , satisfying
where . Here, means the imaginary part of and the real part of .
For the real number , the following estimate holds
3.2 Second-order backward difference scheme and error estimate
In this subsection, we use second-order backward difference convolution quadrature to discretize the time fractional substantial derivative and obtain the second-order accuracy in time. Similarly, introduce as the numerical approximation of the solution , and let
According to (2.2), we have
Using , , and to approximate , , and respectively, we have