8.1.2 1.2

8.1.2.1 [2005] Problem 1

problem number 2005

Added March 23, 2019.

Problem Chapter 1.1.2.1, from Handbook of nonlinear partial differential equations by Andrei D. Polyanin, Valentin F. Zaitsev.

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

\[ w_t = a w_{xx} - b w^3 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, t], t] == a*D[w[x, t], {x, 2}] - b*w[x, t]^3; 
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)^3; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,t))),output='realtime'));
 

sol=()

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8.1.2.2 [2006] Problem 2

problem number 2006

Added March 23, 2019.

Problem Chapter 1.1.2.2, from Handbook of nonlinear partial differential equations by Andrei D. Polyanin, Valentin F. Zaitsev.

Solve for \(w(x,t)\) \[ w_t = w_{xx} + a w - b w^3 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, t], t] == D[w[x, t], {x, 2}] + a*w[x, t] - b*w[x, t]^3; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, t], {x, t}], 60*10]];
 

Failed

Maple ✓

restart; 
pde :=  diff(w(x,t),t)= diff(w(x,t),x$2) +a*w(x,t)- b*w(x,t)^3; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,t))),output='realtime'));
 

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

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8.1.2.3 [2007] Problem 3

problem number 2007

Added March 23, 2019.

Problem Chapter 1.1.2.3, from Handbook of nonlinear partial differential equations by Andrei D. Polyanin, Valentin F. Zaitsev.

Solve for \(w(x,t)\) \[ w_t = a w_{xx} - b w^3 - c w^2 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde =  D[w[x, t], t] == a*D[w[x, t], {x, 2}] - b*w[x, t]^3 - c*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)^3- c*w(x,t)^2; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,t))),output='realtime'));
 

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

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8.1.2.4 [2008] Problem 4

problem number 2008

Added March 23, 2019.

Problem Chapter 1.1.2.4, from Handbook of nonlinear partial differential equations by Andrei D. Polyanin, Valentin F. Zaitsev.

Solve for \(w(x,t)\) \[ w_t = w_{xx} -w(1-w)(a-w) \]

Mathematica ✗

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

Failed

Maple ✓

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

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

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8.1.2.5 [2009] Problem 5

problem number 2009

Added March 23, 2019.

Problem Chapter 1.1.2.5, from Handbook of nonlinear partial differential equations by Andrei D. Polyanin, Valentin F. Zaitsev.

Solve for \(w(x,t)\) \[ w_t = a w_{xx} +b_0+b_1 w+ b_2 w^2+ b_3 w^3 \]

Mathematica ✗

ClearAll["Global`*"]; 
pde = D[w[x, t], t] == a*D[w[x, t], {x, 2}] + b0 + b1*w[x, t] + b2*w[x, t]^2 + b3*w[x, t]^3; 
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) +b0+b1*w(x,t)+b2*w(x,t)^2+b3*w(x,t)^3; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,w(x,t))),output='realtime'));
 

\[w \left (x , t\right ) = \frac {-2304 \left (a \mathit {b3} \RootOf \left (512 \textit {\_Z}^{6} a^{3} \mathit {b3}^{2}-27 \mathit {b0}^{2} \mathit {b3}^{3}+18 \mathit {b0} \mathit {b1} \mathit {b2} \,\mathit {b3}^{2}-4 \mathit {b0} \,\mathit {b2}^{3} \mathit {b3} -4 \mathit {b1}^{3} \mathit {b3}^{2}+\mathit {b1}^{2} \mathit {b2}^{2} \mathit {b3} +\left (-384 a^{2} \mathit {b1} \,\mathit {b3}^{2}+128 a^{2} \mathit {b2}^{2} \mathit {b3} \right ) \textit {\_Z}^{4}+\left (72 a \,\mathit {b1}^{2} \mathit {b3}^{2}-48 a \mathit {b1} \,\mathit {b2}^{2} \mathit {b3} +8 a \,\mathit {b2}^{4}\right ) \textit {\_Z}^{2}\right )^{2}-\frac {\mathit {b1} \mathit {b3}}{8}+\frac {\mathit {b2}^{2}}{24}\right )^{2} \RootOf \left (12 a^{2} \mathit {b3} \RootOf \left (512 \textit {\_Z}^{6} a^{3} \mathit {b3}^{2}-27 \mathit {b0}^{2} \mathit {b3}^{3}+18 \mathit {b0} \mathit {b1} \mathit {b2} \,\mathit {b3}^{2}-4 \mathit {b0} \,\mathit {b2}^{3} \mathit {b3} -4 \mathit {b1}^{3} \mathit {b3}^{2}+\mathit {b1}^{2} \mathit {b2}^{2} \mathit {b3} +\left (-384 a^{2} \mathit {b1} \,\mathit {b3}^{2}+128 a^{2} \mathit {b2}^{2} \mathit {b3} \right ) \textit {\_Z}^{4}+\left (72 a \,\mathit {b1}^{2} \mathit {b3}^{2}-48 a \mathit {b1} \,\mathit {b2}^{2} \mathit {b3} +8 a \,\mathit {b2}^{4}\right ) \textit {\_Z}^{2}\right )^{4}-6 a \mathit {b1} \mathit {b3} \RootOf \left (512 \textit {\_Z}^{6} a^{3} \mathit {b3}^{2}-27 \mathit {b0}^{2} \mathit {b3}^{3}+18 \mathit {b0} \mathit {b1} \mathit {b2} \,\mathit {b3}^{2}-4 \mathit {b0} \,\mathit {b2}^{3} \mathit {b3} -4 \mathit {b1}^{3} \mathit {b3}^{2}+\mathit {b1}^{2} \mathit {b2}^{2} \mathit {b3} +\left (-384 a^{2} \mathit {b1} \,\mathit {b3}^{2}+128 a^{2} \mathit {b2}^{2} \mathit {b3} \right ) \textit {\_Z}^{4}+\left (72 a \,\mathit {b1}^{2} \mathit {b3}^{2}-48 a \mathit {b1} \,\mathit {b2}^{2} \mathit {b3} +8 a \,\mathit {b2}^{4}\right ) \textit {\_Z}^{2}\right )^{2}+2 a \,\mathit {b2}^{2} \RootOf \left (512 \textit {\_Z}^{6} a^{3} \mathit {b3}^{2}-27 \mathit {b0}^{2} \mathit {b3}^{3}+18 \mathit {b0} \mathit {b1} \mathit {b2} \,\mathit {b3}^{2}-4 \mathit {b0} \,\mathit {b2}^{3} \mathit {b3} -4 \mathit {b1}^{3} \mathit {b3}^{2}+\mathit {b1}^{2} \mathit {b2}^{2} \mathit {b3} +\left (-384 a^{2} \mathit {b1} \,\mathit {b3}^{2}+128 a^{2} \mathit {b2}^{2} \mathit {b3} \right ) \textit {\_Z}^{4}+\left (72 a \,\mathit {b1}^{2} \mathit {b3}^{2}-48 a \mathit {b1} \,\mathit {b2}^{2} \mathit {b3} +8 a \,\mathit {b2}^{4}\right ) \textit {\_Z}^{2}\right )^{2}+\textit {\_Z}^{2} \mathit {b3} \right ) \tanh \left (t \RootOf \left (12 a^{2} \mathit {b3} \RootOf \left (512 \textit {\_Z}^{6} a^{3} \mathit {b3}^{2}-27 \mathit {b0}^{2} \mathit {b3}^{3}+18 \mathit {b0} \mathit {b1} \mathit {b2} \,\mathit {b3}^{2}-4 \mathit {b0} \,\mathit {b2}^{3} \mathit {b3} -4 \mathit {b1}^{3} \mathit {b3}^{2}+\mathit {b1}^{2} \mathit {b2}^{2} \mathit {b3} +\left (-384 a^{2} \mathit {b1} \,\mathit {b3}^{2}+128 a^{2} \mathit {b2}^{2} \mathit {b3} \right ) \textit {\_Z}^{4}+\left (72 a \,\mathit {b1}^{2} \mathit {b3}^{2}-48 a \mathit {b1} \,\mathit {b2}^{2} \mathit {b3} +8 a \,\mathit {b2}^{4}\right ) \textit {\_Z}^{2}\right )^{4}-6 a \mathit {b1} \mathit {b3} \RootOf \left (512 \textit {\_Z}^{6} a^{3} \mathit {b3}^{2}-27 \mathit {b0}^{2} \mathit {b3}^{3}+18 \mathit {b0} \mathit {b1} \mathit {b2} \,\mathit {b3}^{2}-4 \mathit {b0} \,\mathit {b2}^{3} \mathit {b3} -4 \mathit {b1}^{3} \mathit {b3}^{2}+\mathit {b1}^{2} \mathit {b2}^{2} \mathit {b3} +\left (-384 a^{2} \mathit {b1} \,\mathit {b3}^{2}+128 a^{2} \mathit {b2}^{2} \mathit {b3} \right ) \textit {\_Z}^{4}+\left (72 a \,\mathit {b1}^{2} \mathit {b3}^{2}-48 a \mathit {b1} \,\mathit {b2}^{2} \mathit {b3} +8 a \,\mathit {b2}^{4}\right ) \textit {\_Z}^{2}\right )^{2}+2 a \,\mathit {b2}^{2} \RootOf \left (512 \textit {\_Z}^{6} a^{3} \mathit {b3}^{2}-27 \mathit {b0}^{2} \mathit {b3}^{3}+18 \mathit {b0} \mathit {b1} \mathit {b2} \,\mathit {b3}^{2}-4 \mathit {b0} \,\mathit {b2}^{3} \mathit {b3} -4 \mathit {b1}^{3} \mathit {b3}^{2}+\mathit {b1}^{2} \mathit {b2}^{2} \mathit {b3} +\left (-384 a^{2} \mathit {b1} \,\mathit {b3}^{2}+128 a^{2} \mathit {b2}^{2} \mathit {b3} \right ) \textit {\_Z}^{4}+\left (72 a \,\mathit {b1}^{2} \mathit {b3}^{2}-48 a \mathit {b1} \,\mathit {b2}^{2} \mathit {b3} +8 a \,\mathit {b2}^{4}\right ) \textit {\_Z}^{2}\right )^{2}+\textit {\_Z}^{2} \mathit {b3} \right )+x \RootOf \left (512 \textit {\_Z}^{6} a^{3} \mathit {b3}^{2}-27 \mathit {b0}^{2} \mathit {b3}^{3}+18 \mathit {b0} \mathit {b1} \mathit {b2} \,\mathit {b3}^{2}-4 \mathit {b0} \,\mathit {b2}^{3} \mathit {b3} -4 \mathit {b1}^{3} \mathit {b3}^{2}+\mathit {b1}^{2} \mathit {b2}^{2} \mathit {b3} +\left (-384 a^{2} \mathit {b1} \,\mathit {b3}^{2}+128 a^{2} \mathit {b2}^{2} \mathit {b3} \right ) \textit {\_Z}^{4}+\left (72 a \,\mathit {b1}^{2} \mathit {b3}^{2}-48 a \mathit {b1} \,\mathit {b2}^{2} \mathit {b3} +8 a \,\mathit {b2}^{4}\right ) \textit {\_Z}^{2}\right )+c_{1}\right )-432 \left (\mathit {b0} \,\mathit {b3}^{2}-\frac {1}{3} \mathit {b1} \mathit {b2} \mathit {b3} +\frac {2}{27} \mathit {b2}^{3}\right ) \left (a \mathit {b2} \RootOf \left (512 \textit {\_Z}^{6} a^{3} \mathit {b3}^{2}-27 \mathit {b0}^{2} \mathit {b3}^{3}+18 \mathit {b0} \mathit {b1} \mathit {b2} \,\mathit {b3}^{2}-4 \mathit {b0} \,\mathit {b2}^{3} \mathit {b3} -4 \mathit {b1}^{3} \mathit {b3}^{2}+\mathit {b1}^{2} \mathit {b2}^{2} \mathit {b3} +\left (-384 a^{2} \mathit {b1} \,\mathit {b3}^{2}+128 a^{2} \mathit {b2}^{2} \mathit {b3} \right ) \textit {\_Z}^{4}+\left (72 a \,\mathit {b1}^{2} \mathit {b3}^{2}-48 a \mathit {b1} \,\mathit {b2}^{2} \mathit {b3} +8 a \,\mathit {b2}^{4}\right ) \textit {\_Z}^{2}\right )^{2}-\frac {9 \mathit {b0} \mathit {b3}}{16}+\frac {\mathit {b1} \mathit {b2}}{16}\right )}{2 \left (27 \mathit {b0} \,\mathit {b3}^{2}-9 \mathit {b1} \mathit {b2} \mathit {b3} +2 \mathit {b2}^{3}\right ) \left (24 a \mathit {b3} \RootOf \left (512 \textit {\_Z}^{6} a^{3} \mathit {b3}^{2}-27 \mathit {b0}^{2} \mathit {b3}^{3}+18 \mathit {b0} \mathit {b1} \mathit {b2} \,\mathit {b3}^{2}-4 \mathit {b0} \,\mathit {b2}^{3} \mathit {b3} -4 \mathit {b1}^{3} \mathit {b3}^{2}+\mathit {b1}^{2} \mathit {b2}^{2} \mathit {b3} +\left (-384 a^{2} \mathit {b1} \,\mathit {b3}^{2}+128 a^{2} \mathit {b2}^{2} \mathit {b3} \right ) \textit {\_Z}^{4}+\left (72 a \,\mathit {b1}^{2} \mathit {b3}^{2}-48 a \mathit {b1} \,\mathit {b2}^{2} \mathit {b3} +8 a \,\mathit {b2}^{4}\right ) \textit {\_Z}^{2}\right )^{2}-3 \mathit {b1} \mathit {b3} +\mathit {b2}^{2}\right )}\]

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