← Previous revision | Revision as of 12:04, 8 May 2024 |
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| ==Other forms== | ==Other forms== |
| ===Multi-dimensional Burgers' equation=== | |
| In two or more dimensions, the Burgers' equation becomes | |
 | :<math>\frac{\partial u}{\partial t} + u \cdot \nabla u = \nu \nabla^2 u.</math> |
| ===Generalized Burgers' equation=== | ===Generalized Burgers' equation=== |
| The generalized Burgers' equation extends the quasilinear convective to more generalized form, i.e., | The generalized Burgers' equation extends the quasilinear convective to more generalized form, i.e., |
| <math display="block">\frac{\partial u}{\partial t} + c(u) \frac{\partial u}{\partial x} = \nu\frac{\partial^2 u}{\partial x^2}.</math> |
| :<math>\frac{\partial u}{\partial t} + c(u) \frac{\partial u}{\partial x} = \nu\frac{\partial^2 u}{\partial x^2}.</math> |
| where <math>c(u)</math> is any arbitrary function of u. The inviscid <math>\nu=0</math> equation is still a quasilinear hyperbolic equation for <math>c(u)>0</math> and its solution can be constructed using [[method of characteristics]] as before.<ref>Courant, R., & Hilbert, D. Methods of Mathematical Physics. Vol. II.</ref> | where <math>c(u)</math> is any arbitrary function of u. The inviscid <math>\nu=0</math> equation is still a quasilinear hyperbolic equation for <math>c(u)>0</math> and its solution can be constructed using [[method of characteristics]] as before.<ref>Courant, R., & Hilbert, D. Methods of Mathematical Physics. Vol. II.</ref> |
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| Added space-time noise <math>\eta(x,t) = \dot W(x,t)</math>, where <math>W</math> is an <math>L^2(\mathbb R)</math> [[Wiener process]], forms a stochastic Burgers' equation<ref>{{cite journal |first1=W. |last1=Wang |first2=A. J. |last2=Roberts | title=Diffusion Approximation for Self-similarity of Stochastic Advection in Burgers' Equation |journal=Communications in Mathematical Physics |volume=333 |pages=1287–1316 |year=2015 |issue=3 |doi=10.1007/s00220-014-2117-7 |arxiv=1203.0463 |bibcode=2015CMaPh.333.1287W |s2cid=119650369 }}</ref> | Added space-time noise <math>\eta(x,t) = \dot W(x,t)</math>, where <math>W</math> is an <math>L^2(\mathbb R)</math> [[Wiener process]], forms a stochastic Burgers' equation<ref>{{cite journal |first1=W. |last1=Wang |first2=A. J. |last2=Roberts | title=Diffusion Approximation for Self-similarity of Stochastic Advection in Burgers' Equation |journal=Communications in Mathematical Physics |volume=333 |pages=1287–1316 |year=2015 |issue=3 |doi=10.1007/s00220-014-2117-7 |arxiv=1203.0463 |bibcode=2015CMaPh.333.1287W |s2cid=119650369 }}</ref> |
| <math display="block">\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \nu \frac{\partial^2 u}{\partial x^2}-\lambda\frac{\partial\eta}{\partial x}.</math> |
| :<math>\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \nu \frac{\partial^2 u}{\partial x^2}-\lambda\frac{\partial\eta}{\partial x}.</math> | |
| This [[stochastic PDE]] is the one-dimensional version of [[Kardar–Parisi–Zhang equation]] in a field <math>h(x,t)</math> upon substituting <math>u(x,t)=-\lambda\partial h/\partial x</math>. | This [[stochastic PDE]] is the one-dimensional version of [[Kardar–Parisi–Zhang equation]] in a field <math>h(x,t)</math> upon substituting <math>u(x,t)=-\lambda\partial h/\partial x</math>. |
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