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# Exact and numerical solution of stochastic Burgers equations with variable coefficients

• * Corresponding author: Tamer Oraby
• We will introduce exact and numerical solutions to some stochastic Burgers equations with variable coefficients. The solutions are found using a coupled system of deterministic Burgers equations and stochastic differential equations.

Mathematics Subject Classification: Primary: 60H15, 35R60; Secondary: 65M06, 65M50.

 Citation: • • Figure 1.  Two realizations of the stochastic process in equation (28)

Figure 2.  Two realizations of the stochastic process in equation (29)

Figure 3.  (a) and (b): Two realizations of the stochastic mesh resulting from solving equation (7) with $C(t) = t+1$ and $E(t) = t+2$ for $t\in[0,2]$ with $\Delta t = 0.0408$ when $z\in[-1,1]$ with $\Delta z = .1$. (c) and (d): Two realizations of the stochastic mesh resulting from solving equation (9) with $B(t) = \exp(t)$, $R(t) = 1$ and $E(t) = 1$ for $t\in[0,2]$ with $\Delta t = 0.0408$ when $z\in[-1,1]$ with $\Delta z = .1$. Notice the uniformity over space since the noise is space uniform

Figure 4.  Stencil of the numerical scheme with the realization of the incremental trajectory $dX_t$ when it is positive (a) and negative (b). The stencil is shown for $\Delta t = k$ and $\Delta x = h$ which are fixed

Figure 5.  Two realizations of the stochastic meshes (a) and (b), and their respective simulated numerical solutions over those two meshes (c) and (d)

Figure 6.  Two realizations of the two processes $Z_t$ and $\dot{Z}_t$ that solve equation (31) (a) and (b), the stochastic meshes (c) and (d), and their respective simulated numerical solutions over those two meshes (e) and (f)

Figure 7.  The relative frequency of the times the absolute error of the SFEM is smaller than the absolute error of the stochastic mesh (SM) method for the solution of (25) at each pair $(t_i,z_j)$ for $i = 0,\ldots,m$ and $j = 0,\ldots,n$ for (a) (m, n) = (20, 20) giving $P_{\text{max}} = .059$, (b) (m, n) = (20, 30) giving $P_{\text{max}} = .053$, (c) (m, n) = (30, 20) giving $P_{\text{max}} = .077$, (d) (m, n) = (30, 30) giving $P_{\text{max}} = .0597$, (e) (m, n) = (40, 20) giving $P_{\text{max}} = .139$, (f) (m, n) = (40, 30) giving $P_{\text{max}} = .087$

Table 1.  The maximum values of the mean absolute errors over $[0,1]\times[-1,1]$ for different values of $n$ and $m$ show that the stochastic mesh method (SM) is better than the stochastic forward Euler method (SFEM) in overall

 n m MMAE for SM MMAE for SFEM 20 20 0.032 0.047 20 30 0.033 0.042 20 40 0.032 0.038 30 20 0.030 0.050 30 30 0.030 0.046 30 40 0.033 0.040 40 20 0.035 0.052 40 30 0.033 0.047 40 40 0.033 0.039
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