Numerical Approximation for Fractional Diffusion Equation Forced by a Tempered Fractional Gaussian Noise

Abstract

This paper discusses the fractional diffusion equation forced by a tempered fractional Gaussian noise. The fractional diffusion equation governs the probability density function of the subordinated killed Brownian motion. The tempered fractional Gaussian noise plays the role of fluctuating external source with the property of localization. We first establish the regularity of the infinite dimensional stochastic integration of the tempered fractional Brownian motion and then build the regularity of the mild solution of the fractional stochastic diffusion equation. The spectral Galerkin method is used for space approximation; after that the system is transformed into an equivalent form having better regularity than the original one in time. Then we use the semi-implicit Euler scheme to discretize the time derivative. In terms of the temporal-spatial error splitting technique, we obtain the error estimates of the fully discrete scheme in the sense of mean-squared $L^2$-norm. Extensive numerical experiments confirm the theoretical estimates.

Appendices

Proof of the Uniqueness for the Solution of Eq. (3.2)

Let $u(t,\omega )$ and $\hat{u}(t,\omega )$ be the solutions with initial values $u(0,\omega )$ and $\hat{u}(0,\omega )$, respectively, where $\omega$ denotes the path of stochastic process. By using Eq. (3.2), we have

$$\begin{aligned} \mathrm {E}\left[ \left\| u(t)-\hat{u}(t)\right\| ^2\right]= & {} \mathrm {E}\left[ \left\| S(t)u(0)-S(t)\hat{u}(0) +\int ^t_0S(t-s)f(u(s))\mathrm {d}s-\int ^t_0S(t-s)f(\hat{u}(s))\mathrm {d}s\right\| ^2\right] \\\le & {} 2\mathrm {E}\left[ \left\| u(0)-\hat{u}(0)\right\| ^2\right] +C\int ^t_0\mathrm {E}\left[ \left\| u(s)-\hat{u}(s)\right\| ^2\right] \mathrm {d}s. \end{aligned}$$

The Grönwall inequality leads to

$$\begin{aligned} \mathrm {E}\left[ \left\| u(t)-\hat{u}(t)\right\| ^2\right] \le 2\mathrm {E}\left[ \left\| u(0)-\hat{u}(0)\right\| ^2\right] \exp (Ct). \end{aligned}$$

Taking $u(0)=\hat{u}(0)$ leads to

$$\begin{aligned} \mathrm {E}\left[ \left\| u(t)-\hat{u}(t)\right\| ^2\right] =0, \quad t\in [0,T]. \end{aligned}$$

From the continuity of $t\rightarrow \left\| u(t)-\hat{u}(t)\right\| ^2$, we get

$$\begin{aligned} P\left[ \left\| u(t,\omega )-\hat{u}(t,\omega )\right\| ^2=0, \quad t\in [0,T]\right] =1, \end{aligned}$$

where P denotes the probability. The uniqueness has been proved.

Proof of the Existence for the Solution of Eq. (3.2)

Let $Y^{(0)}_t=u(0)$, $Y^{(k)}_t=Y^{(k)}_t(\omega )$, and

$$\begin{aligned} Y^{(k+1)}_t=S(t)u(0)+\int ^t_0S(t-s)f(Y^{(k)}_t)\mathrm {d}s+\int ^t_0S(t-s)\mathrm {d}B_{H,\mu }(s). \end{aligned}$$

Assumption 1 implies

$$\begin{aligned} \mathrm {E}\left[ \left\| Y^{(k+1)}_t-Y^{(k)}_t\right\| ^2\right]= & {} \mathrm {E}\left[ \left\| \int ^t_0S(t-s)f(Y^{(k)}_s)\mathrm {d}s-\int ^t_0S(t-s)f(Y^{(k-1)}_s)\mathrm {d}s\right\| ^2\right] \nonumber \\\le & {} Ct \int ^t_0\mathrm {E}\left[ \left\| Y^{(k)}_s-Y^{(k-1)}_s\right\| ^2\right] \mathrm {d}s, \end{aligned}$$

(B.1)

where $k\ge 1$ and $t\le T$. Let

$$\begin{aligned} \theta = \left\{ \begin{array}{ll} 2H, &{}\ \rho >\frac{1}{2}, \\ \frac{\gamma }{2\alpha }, &{}\ 0<\rho \le \frac{1}{2}. \end{array} \right. \end{aligned}$$

Then

$$\begin{aligned} \mathrm {E}\left[ \left\| Y^{(1)}_t-Y^{(0)}_t\right\| ^2\right]\le & {} C\mathrm {E}\left[ \left\| S(t)u(0)-u(0)\right\| ^2\right] +C\mathrm {E}\left[ \left\| \int ^t_0S(t-s)f\left( u(0)\right) \mathrm {d}s\right\| ^2\right] +Ct^{\theta }\\\le & {} Ct^2\left( 1+\mathrm {E}\left[ \left\| u(0)\right\| ^2\right] \right) +Ct^{\theta }\\\le & {} Ct^{\min \{2,\theta \}}, \end{aligned}$$

where C depends on H, $\alpha$, and $\gamma$. So by induction starting from Eq. (B.1), we obtain

$$\begin{aligned} \mathrm {E}\left[ \left\| Y^{(k+1)}_t-Y^{(k)}_t\right\| ^2\right] \le \frac{C^{k+1}t^{2k+\min \{2,\theta \}}}{k!},\quad k\ge 0, \ t\in [0,T]. \end{aligned}$$

(B.2)

Let $m>n\ge 0$. Using Eq. (B.2), we have

$$\begin{aligned} \left\| Y^{(m)}_t-Y^{(n)}_t\right\| _{L^2(D,U)}= & {} \left\| \sum ^{m-1}_{k=n}Y^{(k+1)}_t-Y^{(k)}_t\right\| _{L^2(D,U)}\\\le & {} \sum ^{m-1}_{k=n}\left\| Y^{(k+1)}_t-Y^{(k)}_t\right\| _{L^2(D,U)}\\\le & {} \sum ^{m-1}_{k=n}\left( \frac{C^{k+1}T^{2k+\min \{2,\theta \}}}{k!}\right) ^{\frac{1}{2}}. \end{aligned}$$

As $m,n\rightarrow \infty$, then $\left\| Y^{(m)}_t-Y^{(n)}_t\right\| _{L^2(D,U)}=0$. The above deduction shows that $\left\{ Y^{(k)}_t\right\} ^\infty _{k=0}$ is a Cauchy sequence in space $L^2(D,U)$. Define

$$\begin{aligned} u(t):=\lim _{n\rightarrow \infty }Y^{(n)}_t. \end{aligned}$$

Then u(t) satisfies Eq. (3.2).

Description for the Simulation of Tempered Fractional Brownian Motion

In this paper, the Ckolesky method [7] is used to simulate tfBm. Suppose $0\le t_1\le \dots \le t_{m}\le \dots \le t_{M}=T(m=1,2,\dots , M-1)$ and the sizes of the mesh $\varDelta t=t_{m+1}-t_{m}$. Let’s consider the following vector

$$\begin{aligned} Z=\left( \beta _{H,\mu }(t_1),\beta _{H,\mu }(t_2)-\beta _{H,\mu }(t_1),\beta _{H,\mu }(t_3)- \beta _{H,\mu }(t_2),\dots ,\beta _{H,\mu }(t_{M-1})-\beta _{H,\mu }(t_M)\right) . \end{aligned}$$

The probability distribution of the vector Z is normal with mean 0 and the covariance matrix $\varSigma$. Let $\varSigma _{i,j}$ be the element of row i, column j of matrix $\varSigma$. By using Eq. (2.1), we have

$$\begin{aligned} \varSigma _{i,j}= & {} \mathrm {E}\left[ \left( \beta _{H,\mu }(t_i)-\beta _{H,\mu }(t_{i-1})\right) \left( \beta _{H,\mu }(t_j)-\beta _{H,\mu }(t_{j-1})\right) \right] \\= & {} \mathrm {E}\left[ \beta _{H,\mu }(t_i)\beta _{H,\mu }(t_j)+\beta _{H,\mu }(t_{i-1})\beta _{H,\mu }(t_{j-1}) -\beta _{H,\mu }(t_i)\beta _{H,\mu }(t_{j-1})-\beta _{H,\mu }(t_j)\beta _{H,\mu }(t_{i-1})\right] \\= & {} \frac{1}{2}\left[ C^2_{(i-j+1)\varDelta t}|(i-j+1)\varDelta t|^{2H}+C^2_{(i-j-1)\varDelta t}|(i-j-1)\varDelta t|^{2H}-2C^2_{(i-j)\varDelta t}|(i-j)\varDelta t|^{2H}\right] . \end{aligned}$$

When $\varSigma$ is a symmetric positive matrix, the covariance matrix $\varSigma$ can be written as $L(M)L(M)'$, where the matrix L(M) is lower triangular matrix and the matrix $L(M)'$ is the transpose of L(M). Let $V=(V_1,V_2,\dots ,V_M)$. The elements of the vector V are a sequence of independent and identically distributed standard normal random variables. Because $Z=L(M)V$, then Z can be simulated. Let $l_{i,j}$ be the element of row i, column j of matrix L(M). That is,

$$\begin{aligned} \varSigma _{i,j}=\sum ^j_{k=1}l_{i,k}l_{j,k}, \quad j\le i. \end{aligned}$$

As $i=j=1$, we have $l^2_{1,1}=\varSigma _{1,1}$. The $l_{i,j}$ satisfies

$$\begin{aligned} l_{i+1,1}= & {} \frac{\varSigma _{i+1,1}}{l_{1,1}},\\ l^2_{i+1,i+1}= & {} \varSigma _{i+1,i+1}-\sum ^{i}_{k=1}l^2_{i+1,k},\\ l_{i+1,j}= & {} \frac{1}{l_{j,j}}\left( \varSigma _{i+1,j}-\sum ^{j-1}_{k=1}l_{i+1,k}l_{j,k}\right) ,\quad 1<j\le i. \end{aligned}$$