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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 |
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.
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. |
show all references
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. |







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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