153 HNPDE, chapter 1.1.1

 153.1 Problem 1
 153.2 Problem 2

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153.1 Problem 1

problem number 1239

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

\[ \text{Failed} \]

Maple

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

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153.2 Problem 2

problem number 1240

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

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

Maple

\[ w \left ( x,t \right ) =-1/4\, \left ( \tanh \left ( -{\frac{5\,at}{12}}+1/12\,\sqrt{-6\,a}x+{\it \_C1} \right ) \right ) ^{2}-1/2\,\tanh \left ( -{\frac{5\,at}{12}}+1/12\,\sqrt{-6\,a}x+{\it \_C1} \right ) +3/4 \]