10 Diffusion Reaction in 1D

 10.1 using growth form for reaction term
 10.2 using logistic form for reaction term
 10.3 using Aleee form for reaction term

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10.1 using growth form for reaction term

problem number 80

Added December 29, 2018.

Solve for \(u(x,t)\) in \[ u_t = k u_{xx} + r u \]

with \(k=\frac{1}{10},r=1\) and \(0<x<1\) and \(t>0\).

With boundary conditions \begin{align*} u(0,t) &= 0\\ u(1,t) &=0 \end{align*}

And initial conditions \(u(x,0)=1\).

Mathematica

\[ \text{DSolve}\left [\left \{u^{(0,1)}(x,t)=\frac{1}{10} u^{(2,0)}(x,t)+u(x,t),\{u(0,t)=0,u(1,t)=0\},u(x,0)=1\right \},u(x,t),\{x,t\}\right ] \]

Maple

\[ u \left ( x,t \right ) =\sum _{n=1}^{\infty }-2\,{\frac{ \left ( \left ( -1 \right ) ^{n}-1 \right ) \sin \left ( n\pi \,x \right ){{\rm e}^{-1/10\,t \left ({\pi }^{2}{n}^{2}-10 \right ) }}}{n\pi }} \]

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10.2 using logistic form for reaction term

problem number 81

Added December 29, 2018.

Solve for \(u(x,t)\) in \[ u_t = k u_{xx} + r u\left (1- \frac{u}{\alpha } \right ) \]

with \(k=\frac{1}{100},r=\frac{1}{10},\alpha =10\) and \(0<x<1\) and \(t>0\).

With boundary conditions \begin{align*} u(0,t) &= 0\\ u(1,t) &=0 \end{align*}

And initial conditions \(u(x,0)=1\).

Mathematica

\[ \text{DSolve}\left [\left \{u^{(0,1)}(x,t)=\frac{1}{100} u^{(2,0)}(x,t)+\frac{1}{10} \left (1-\frac{1}{10} u(x,t)\right ) u(x,t),\{u(0,t)=0,u(1,t)=0\},u(x,0)=1\right \},u(x,t),\{x,t\}\right ] \]

Maple

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

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10.3 using Aleee form for reaction term

problem number 82

Added December 29, 2018.

Solve for \(u(x,t)\) in \[ u_t = k u_{xx} + \alpha u + \beta u ^2 - \gamma u^3 \]

with \(k=\frac{1}{1000},\alpha =\frac{1}{100},\beta =\frac{1}{100}, \gamma =\frac{5}{1000}\) and \(0<x<1\) and \(t>0\).

With boundary conditions \begin{align*} u(0,t) &= 0\\ u(1,t) &=0 \end{align*}

And initial conditions \(u(x,0)=1\).

Mathematica

\[ \text{DSolve}\left [\left \{u^{(0,1)}(x,t)=\frac{u^{(2,0)}(x,t)}{1000}-\frac{1}{200} u(x,t)^3+\frac{1}{100} u(x,t)^2+\frac{1}{10} u(x,t),\{u(0,t)=0,u(1,t)=0\},u(x,0)=1\right \},u(x,t),\{x,t\}\right ] \]

Maple

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