7.1.1 1.1

7.1.1.1 [2001] Problem 1

problem number 2001

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 ✗

ClearAll["Global`*"]; 
pde =  D[w[x, t], t] == a*D[w[x, t], {x, 2}] + b*w[x, t]^2; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, t], {x, t}], 60*10]];
 

Failed

Maple ✗

restart; 
pde :=  diff(w(x,t),t)= a*diff(w(x,t),x$2) + b*w(x,t)^2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,t))),output='realtime'));
 

sol=()

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7.1.1.2 [2002] Problem 2

problem number 2002

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 ✓

ClearAll["Global`*"]; 
pde =  D[w[x, t], t] == D[w[x, t], {x, 2}] + a*w[x, t]*(1 - w[x, t]); 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, t], {x, t}], 60*10]];
 

\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 ✓

restart; 
pde :=  diff(w(x,t),t)= diff(w(x,t),x$2) + a*w(x,t)*(1-w(x,t)); 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,t))),output='realtime'));
 

\[w \left (x , t\right ) = -\frac {\tanh \left (-\frac {5 a t}{12}+\frac {\sqrt {6}\, \sqrt {-a}\, x}{12}+c_1 \right )^{2}}{4}-\frac {\tanh \left (-\frac {5 a t}{12}+\frac {\sqrt {6}\, \sqrt {-a}\, x}{12}+c_1 \right )}{2}+\frac {3}{4}\]

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