Burgers equation

The Burgers equation is the partial differential equation used in hydrodynamics . This equation is known in various fields of applied mathematics . The equation is named after Johann Martinus Burgers (1895-1981). It is a special case of the Navier - Stokes equations in the one-dimensional case.

Let the fluid flow velocity u and its kinematic viscosity be given $\nu$ ${\ displaystyle \ nu}$ $\ nu$ . The Burgers equation in general form is written as follows:

{\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}=\nu {\frac {\partial ^{2}u}{\partial x^{2}}}

{\ displaystyle {\ frac {\ partial u} {\ partial t}} + u {\ frac {\ partial u} {\ partial x}} = \ nu {\ frac {\ partial ^ {2} u} {\ partial x ^ {2}}}}

{\ displaystyle {\ frac {\ partial u} {\ partial t}} + u {\ frac {\ partial u} {\ partial x}} = \ nu {\ frac {\ partial ^ {2} u} {\ partial x ^ {2}}}}

.

If the effect of viscosity can be neglected, i.e. $\nu =0$ ${\ displaystyle \ nu = 0}$ ${\ displaystyle \ nu = 0}$ , the equation takes the form:

{\frac {\partial u}{\partial t}}+u{\frac {\partial u}{\partial x}}=0

{\ displaystyle {\ frac {\ partial u} {\ partial t}} + u {\ frac {\ partial u} {\ partial x}} = 0}

{\ displaystyle {\ frac {\ partial u} {\ partial t}} + u {\ frac {\ partial u} {\ partial x}} = 0}

.

In this case, we obtain the Hopf equation — the quasilinear transport equation — the simplest equation describing discontinuous flows or flows with shock waves .

If a $\nu$ ${\ displaystyle \ nu}$ $\ nu$ materially and not equal $0$ ${\ displaystyle 0}$ ${\ displaystyle 0}$ , the equation reduces to the case $\nu =1$ ${\ displaystyle \ nu = 1}$ $\ nu = 1$ : for $\nu <0$ ${\ displaystyle \ nu <0}$ ${\ displaystyle \ nu <0}$ need to make a replacement first $u\to -u$ ${\ displaystyle u \ to -u}$ ${\ displaystyle u \ to -u}$ , $x\to -x$ ${\ displaystyle x \ to -x}$ ${\ displaystyle x \ to -x}$ , and for any sign $\nu$ ${\ displaystyle \ nu}$ $\ nu$ : $u\to {\sqrt {|\nu |}}\,u$ ${\ displaystyle u \ to {\ sqrt {| \ nu |}} \, u}$ ${\ displaystyle u \ to {\ sqrt {| \ nu |}} \, u}$ , $x\to {\sqrt {|\nu |}}\,x$ ${\ displaystyle x \ to {\ sqrt {| \ nu |}} \, x}$ ${\ displaystyle x \ to {\ sqrt {| \ nu |}} \, x}$ .

The Burgers equation can be linearized by the Hopf-Cole transform. For this (at $\nu =1$ ${\ displaystyle \ nu = 1}$ $\ nu = 1$ ) you need to make a replacement function:

u={\frac {\partial \ln w}{\partial x}}=w_{x}/w

{\ displaystyle u = {\ frac {\ partial \ ln w} {\ partial x}} = w_ {x} / w}

{\ displaystyle u = {\ frac {\ partial \ ln w} {\ partial x}} = w_ {x} / w}

.

In this case, the solutions of the Burgers equation are reduced to positive solutions of the linear heat equation :

u(x,t)=2{\frac {\partial }{\partial x}}\ln {\Bigl \{}(4\pi t)^{-1/2}\int _{-\infty }^{\infty }\exp {\Bigl [}-{\frac {(x-x')^{2}}{4t}}-{\frac {1}{2}}\int _{0}^{x'}u(x'',0)dx''{\Bigr ]}dx'{\Bigr \}}.

{\ displaystyle u (x, t) = 2 {\ frac {\ partial} {\ partial x}} \ ln {\ Bigl \ {} (4 \ pi t) ^ {- 1/2} \ int _ {- \ infty} ^ {\ infty} \ exp {\ Bigl [} - {\ frac {(x-x ') ^ {2}} {4t}} - {\ frac {1} {2}} \ int _ { 0} ^ {x '} u (x``, 0) dx' '{\ Bigr]} dx' {\ Bigr \}}.}

{\ displaystyle u (x, t) = 2 {\ frac {\ partial} {\ partial x}} \ ln {\ Bigl \ {} (4 \ pi t) ^ {- 1/2} \ int _ {- \ infty} ^ {\ infty} \ exp {\ Bigl [} - {\ frac {(x-x ') ^ {2}} {4t}} - {\ frac {1} {2}} \ int _ { 0} ^ {x '} u (x``, 0) dx' '{\ Bigr]} dx' {\ Bigr \}}.}

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Burgers' Equation at EqWorld: The World of Mathematical Equations.

Burgers equation

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