26 Burger’s PDE

 26.1 viscous fluid flow with no initial conditions
 26.2 viscous fluid flow with initial conditions
 26.3 viscous fluid flow with initial conditions as UnitBox

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26.1 viscous fluid flow with no initial conditions

problem number 160

From Mathematica symbolic PDE document.

Solve for \(u(x,t)\) \[ \frac{\partial u}{\partial t} + u(x,t) \frac{\partial u}{\partial x} = \mu \frac{\partial ^2 u}{\partial x^2} \]

Mathematica

\[ \left \{\left \{u(x,t)\to -\frac{2 c_1^2 \mu \tanh \left (c_2 t+c_1 x+c_3\right )+c_2}{c_1}\right \}\right \} \]

Maple

\[ u \left ( x,t \right ) =-2\,\mu \,{\it \_C2}\,\tanh \left ({\it \_C2}\,x+{\it \_C3}\,t+{\it \_C1} \right ) -{\frac{{\it \_C3}}{{\it \_C2}}} \]

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26.2 viscous fluid flow with initial conditions

problem number 161

From Mathematica symbolic PDE document.

Solve for \(u(x,t)\) \[ \frac{\partial u}{\partial t} + u(x,t) \frac{\partial u}{\partial x} = \mu \frac{\partial ^2 u}{\partial x^2} \]

With initial conditions

\(u\left ( x,0\right ) =\left \{ \begin{array} [c]{ccc}1 & & x< 0 \\ 0 & & x \geq 0 \end{array} \right . \)

Mathematica

\[ \left \{\left \{u(x,t)\to \frac{1}{\frac{e^{-\frac{t-2 x}{4 \mu }} \left (\text{Erf}\left (\frac{x}{2 \sqrt{\mu } \sqrt{t}}\right )+1\right )}{\text{Erf}\left (\frac{t-x}{2 \sqrt{\mu } \sqrt{t}}\right )+1}+1}\right \}\right \} \]

Maple

\[ \text{ sol=() } \]

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26.3 viscous fluid flow with initial conditions as UnitBox

problem number 162

From Mathematica DSolve help pages.

Solve for \(u(x,t)\) \[ \frac{\partial u}{\partial t} + u(x,t) \frac{\partial u}{\partial x} = \mu \frac{\partial ^2 u}{\partial x^2} \]

With initial conditions

\(u\left ( x,0\right ) =\left \{ \begin{array} [c]{ccc}1 & & |x| \leq \frac{1}{2} \\ 0 & & \text{otherwise} \end{array} \right . \)

Mathematica

\[ \left \{\left \{u(x,t)\to \frac{e^{\frac{t+1}{4 \mu }} \left (\text{Erf}\left (\frac{2 t-2 x-1}{4 \sqrt{\mu t}}\right )-\text{Erf}\left (\frac{2 t-2 x+1}{4 \sqrt{\mu t}}\right )\right )}{e^{\frac{x}{2 \mu }} \text{Erf}\left (\frac{1-2 x}{4 \sqrt{\mu t}}\right )+e^{\frac{t+1}{4 \mu }} \text{Erf}\left (\frac{2 t-2 x-1}{4 \sqrt{\mu t}}\right )-e^{\frac{t+1}{4 \mu }} \text{Erf}\left (\frac{2 t-2 x+1}{4 \sqrt{\mu t}}\right )+e^{\frac{x+1}{2 \mu }} \text{Erf}\left (\frac{2 x+1}{4 \sqrt{\mu t}}\right )-e^{\frac{x}{2 \mu }}-e^{\frac{x+1}{2 \mu }}}\right \}\right \} \]

Maple

\[ \text{ sol=() } \]