#### 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

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=()

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##### 8.1.1.2 [2004] Problem 2

problem number 2004

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

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

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