# American Institute of Mathematical Sciences

October  2020, 13(10): 2735-2750. doi: 10.3934/dcdss.2020224

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

 School of Mathematical and Statistical Sciences, The University of Texas Rio Grande Valley, 1201 W. University Drive, Edinburg, Texas, 78539, USA

* Corresponding author: Tamer Oraby

Received  February 2019 Revised  July 2019 Published  October 2020 Early access  December 2019

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.

Citation: Stephanie Flores, Elijah Hight, Everardo Olivares-Vargas, Tamer Oraby, Jose Palacio, Erwin Suazo, Jasang Yoon. Exact and numerical solution of stochastic Burgers equations with variable coefficients. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2735-2750. doi: 10.3934/dcdss.2020224
##### References:
 [1] A. Alabert and I. Gyongy, On numerical approximation of stochastic Burgers' equation, From Stochastic Calculus to Mathematical Finance, Springer, Berlin, (2006), 1–15. doi: 10.1007/978-3-540-30788-4_1. [2] L. Bertini, N. Cancrini and G. Jona-Lasinio, The stochastic Burgers equation, Communications in Mathematical Physics, 165 (1994), 211-232.  doi: 10.1007/BF02099769. [3] L. Bertini and G. Giacomin, Stochastic Burgers and KPZ equations from particle systems, Communications in Mathematical Physics, 183 (1997), 571-607.  doi: 10.1007/s002200050044. [4] D. Blomker and A. Jentzen, Galerkin approximations for the stochastic Burgers equation, SIAM Journal of Numerical Analysis, 51 (2013), 694-715.  doi: 10.1137/110845756. [5] J. M. Burgers, A mathematical model illustrating the theory of turbulence, Advances in Applied Mechanics, Academic Press, Inc., New York, N. Y., (1948), 171–199. [6] O. Calin, An Informal Introduction to Stochastic Calculus with Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. doi: 10.1142/9620. [7] G. Casella and R. L. Berger, Statistical Inference, Biometrics, 49 (1993), 320-321.  doi: 10.2307/2532634. [8] P. Catuogno and C. Olivera, Strong solution of the stochastic Burgers equation, Applicable Analysis, 93 (2014), 646-652.  doi: 10.1080/00036811.2013.797074. [9] G. Da Prato, A. Debussche and R. Temam, Stochastic Burgers' equation, Nonlinear Differential Equations and Applications, 1 (1994), 389-402.  doi: 10.1007/BF01194987. [10] P. Düben, D. Homeier, K. Jansen, D. Mesterhazy, G. Münster and C. Urbach, Monte Carlo simulations of the randomly forced Burgers equation, EPL Journal, 84 (2008), 1-4. [11] S. Eule and R. Friedrich, A note on the forced Burgers equation, Physics Letters A: General, Atomic and Solid State Physics, 351 (2006), 238-241.  doi: 10.1016/j.physleta.2005.11.019. [12] I. Gyöngy and D. Nualart, On the stochastic Burgers' equation in the real line, The Annals of Probability, 27 (1999), 782-802.  doi: 10.1214/aop/1022677386. [13] M. Hairer and J. Voss, Approximations to the stochastic Burgers equation, Journal of Nonlinear Science, 21 (2011), 897-920.  doi: 10.1007/s00332-011-9104-3. [14] H. Holden, T. Lindstrøm, B. øksendal, J. Ubøe and T.-S. Zhang, The Burgers equation with a noisy force and the stochastic heat equation, Communications in Partial Differential Equations, 19 (1994), 119-141.  doi: 10.1080/03605309408821011. [15] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics (New York), 23. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5. [16] P. Lewis and D. Nualart, Stochastic Burgers' equation on the real line: Regularity and moment estimates, Stochastics, 90 (2018), 1053-1086.  doi: 10.1080/17442508.2018.1478834. [17] E. Pereira, E. Suazo and J. Trespalacios, Riccati-Ermakov systems and explicit solutions for variable coefficient reaction-diffusion equations, Applied Mathematics and Computation, 329 (2018), 278-296.  doi: 10.1016/j.amc.2018.01.047. [18] A. Truman and H. Z. Zhao, On stochastic diffusion equations and stochastic Burgers' equations, Journal of Mathematical Physics, 37 (1996), 283-307.  doi: 10.1063/1.531391. [19] E. Weinan, Stochastic hydrodynamics, Current Developments in Mathematics, 2000, Int. Press, Somerville, MA, (2001), 109–147.

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##### References:
 [1] A. Alabert and I. Gyongy, On numerical approximation of stochastic Burgers' equation, From Stochastic Calculus to Mathematical Finance, Springer, Berlin, (2006), 1–15. doi: 10.1007/978-3-540-30788-4_1. [2] L. Bertini, N. Cancrini and G. Jona-Lasinio, The stochastic Burgers equation, Communications in Mathematical Physics, 165 (1994), 211-232.  doi: 10.1007/BF02099769. [3] L. Bertini and G. Giacomin, Stochastic Burgers and KPZ equations from particle systems, Communications in Mathematical Physics, 183 (1997), 571-607.  doi: 10.1007/s002200050044. [4] D. Blomker and A. Jentzen, Galerkin approximations for the stochastic Burgers equation, SIAM Journal of Numerical Analysis, 51 (2013), 694-715.  doi: 10.1137/110845756. [5] J. M. Burgers, A mathematical model illustrating the theory of turbulence, Advances in Applied Mechanics, Academic Press, Inc., New York, N. Y., (1948), 171–199. [6] O. Calin, An Informal Introduction to Stochastic Calculus with Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. doi: 10.1142/9620. [7] G. Casella and R. L. Berger, Statistical Inference, Biometrics, 49 (1993), 320-321.  doi: 10.2307/2532634. [8] P. Catuogno and C. Olivera, Strong solution of the stochastic Burgers equation, Applicable Analysis, 93 (2014), 646-652.  doi: 10.1080/00036811.2013.797074. [9] G. Da Prato, A. Debussche and R. Temam, Stochastic Burgers' equation, Nonlinear Differential Equations and Applications, 1 (1994), 389-402.  doi: 10.1007/BF01194987. [10] P. Düben, D. Homeier, K. Jansen, D. Mesterhazy, G. Münster and C. Urbach, Monte Carlo simulations of the randomly forced Burgers equation, EPL Journal, 84 (2008), 1-4. [11] S. Eule and R. Friedrich, A note on the forced Burgers equation, Physics Letters A: General, Atomic and Solid State Physics, 351 (2006), 238-241.  doi: 10.1016/j.physleta.2005.11.019. [12] I. Gyöngy and D. Nualart, On the stochastic Burgers' equation in the real line, The Annals of Probability, 27 (1999), 782-802.  doi: 10.1214/aop/1022677386. [13] M. Hairer and J. Voss, Approximations to the stochastic Burgers equation, Journal of Nonlinear Science, 21 (2011), 897-920.  doi: 10.1007/s00332-011-9104-3. [14] H. Holden, T. Lindstrøm, B. øksendal, J. Ubøe and T.-S. Zhang, The Burgers equation with a noisy force and the stochastic heat equation, Communications in Partial Differential Equations, 19 (1994), 119-141.  doi: 10.1080/03605309408821011. [15] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics (New York), 23. Springer-Verlag, Berlin, 1992. doi: 10.1007/978-3-662-12616-5. [16] P. Lewis and D. Nualart, Stochastic Burgers' equation on the real line: Regularity and moment estimates, Stochastics, 90 (2018), 1053-1086.  doi: 10.1080/17442508.2018.1478834. [17] E. Pereira, E. Suazo and J. Trespalacios, Riccati-Ermakov systems and explicit solutions for variable coefficient reaction-diffusion equations, Applied Mathematics and Computation, 329 (2018), 278-296.  doi: 10.1016/j.amc.2018.01.047. [18] A. Truman and H. Z. Zhao, On stochastic diffusion equations and stochastic Burgers' equations, Journal of Mathematical Physics, 37 (1996), 283-307.  doi: 10.1063/1.531391. [19] E. Weinan, Stochastic hydrodynamics, Current Developments in Mathematics, 2000, Int. Press, Somerville, MA, (2001), 109–147.
Two realizations of the stochastic process in equation (28)
Two realizations of the stochastic process in equation (29)
(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
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
Two realizations of the stochastic meshes (a) and (b), and their respective simulated numerical solutions over those two meshes (c) and (d)
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)
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 $P_{\text{max}} = .087$
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
 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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