7.1.2 1.2

7.1.2.1 [2003] Problem 1

problem number 2003

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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7.1.2.2 [2004] Problem 2

problem number 2004

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'));
 

\[w \left (x , t\right ) = \frac {\sqrt {b a}\, \left (-1+\tanh \left (-\frac {3 a t}{4}+\frac {\sqrt {2}\, \sqrt {a}\, x}{4}+c_1 \right )\right )}{2 b}\]

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7.1.2.3 [2005] Problem 3

problem number 2005

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'));
 

\[w \left (x , t\right ) = \frac {c \left (-1+\tanh \left (-\frac {c^{2} t}{4 b}+\frac {\sqrt {2}\, c x}{4 \sqrt {b a}}+c_1 \right )\right )}{2 b}\]

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7.1.2.4 [2006] Problem 4

problem number 2006

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'));
 

\[w \left (x , t\right ) = \frac {\tanh \left (\frac {\sqrt {2}\, x}{4}+\frac {\left (-2 a +1\right ) t}{4}+c_1 \right )}{2}+\frac {1}{2}\]

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7.1.2.5 [2007] Problem 5

problem number 2007

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'));
 

\[\text {Expression too large to display}\]

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