8.1.1 1.1

   8.1.1.1 [2003] Problem 1
   8.1.1.2 [2004] Problem 2

8.1.1.1 [2003] Problem 1

problem number 2003

Added March 23, 2019.

Problem Chapter 1.1.1.1, from Handbook of nonlinear partial differential equations by Andrei D. Polyanin, Valentin F. Zaitsev.

Solve for \(w(x,t)\) \[ w_t = a w_{xx} + b w^2 \]

Mathematica


Failed

Maple


sol=()

________________________________________________________________________________________

8.1.1.2 [2004] Problem 2

problem number 2004

Added March 23, 2019.

Problem Chapter 1.1.1.2, from Handbook of nonlinear partial differential equations by Andrei D. Polyanin, Valentin F. Zaitsev.

Solve for \(w(x,t)\)

\[ w_t = w_{xx} + a w(1-w) \]

Mathematica


\begin {align*} & \left \{w(x,t)\to \frac {1}{4} \left (1+\tanh \left (\frac {1}{12} \left (5 a t-\sqrt {6} \sqrt {a} x-12 c_3\right )\right )\right ){}^2\right \}\\& \left \{w(x,t)\to -\frac {1}{4} \left (-3+\tanh \left (\frac {1}{12} \left (5 a t-i \sqrt {6} \sqrt {a} x-12 c_3\right )\right )\right ) \left (1+\tanh \left (\frac {1}{12} \left (5 a t-i \sqrt {6} \sqrt {a} x-12 c_3\right )\right )\right )\right \}\\& \left \{w(x,t)\to -\frac {1}{4} \left (-3+\tanh \left (\frac {1}{12} \left (5 a t+i \sqrt {6} \sqrt {a} x-12 c_3\right )\right )\right ) \left (1+\tanh \left (\frac {1}{12} \left (5 a t+i \sqrt {6} \sqrt {a} x-12 c_3\right )\right )\right )\right \}\\& \left \{w(x,t)\to \frac {1}{4} \left (1+\tanh \left (\frac {1}{12} \left (5 a t+\sqrt {6} \sqrt {a} x-12 c_3\right )\right )\right ){}^2\right \}\\& \left \{w(x,t)\to \frac {1}{4} \left (1+\tanh \left (\frac {5 a t}{12}-\frac {\sqrt {a} x}{2 \sqrt {6}}+c_3\right )\right ){}^2\right \}\\& \left \{w(x,t)\to -\frac {1}{4} \left (-3+\tanh \left (\frac {5 a t}{12}-\frac {i \sqrt {a} x}{2 \sqrt {6}}+c_3\right )\right ) \left (1+\tanh \left (\frac {5 a t}{12}-\frac {i \sqrt {a} x}{2 \sqrt {6}}+c_3\right )\right )\right \}\\& \left \{w(x,t)\to -\frac {1}{4} \left (-3+\tanh \left (\frac {5 a t}{12}+\frac {i \sqrt {a} x}{2 \sqrt {6}}+c_3\right )\right ) \left (1+\tanh \left (\frac {5 a t}{12}+\frac {i \sqrt {a} x}{2 \sqrt {6}}+c_3\right )\right )\right \}\\& \left \{w(x,t)\to \frac {1}{4} \left (1+\tanh \left (\frac {5 a t}{12}+\frac {\sqrt {a} x}{2 \sqrt {6}}+c_3\right )\right ){}^2\right \}\\ \end {align*}

Maple


\[w \left (x , t\right ) = -\frac {\left (\tanh ^{2}\left (-\frac {5 a t}{12}+c_{1}+\frac {\sqrt {-6 a}\, x}{12}\right )\right )}{4}-\frac {\tanh \left (-\frac {5 a t}{12}+c_{1}+\frac {\sqrt {-6 a}\, x}{12}\right )}{2}+\frac {3}{4}\]

________________________________________________________________________________________