7.1.2 1.2

7.1.2.1 [2004] Problem 1

problem number 2004

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 [2005] Problem 2

problem number 2005

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'));
 

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

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7.1.2.3 [2006] Problem 3

problem number 2006

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'));
 

\[w \left ( x,t \right ) =1/2\,{\frac {c}{b} \left ( \tanh \left ( -1/4\,{\frac {{c}^{2}t}{b}}+1/4\,{\frac {\sqrt {2}cx}{\sqrt {ab}}}+{\it \_C1} \right ) -1 \right ) }\]

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7.1.2.4 [2007] Problem 4

problem number 2007

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'));
 

\[w \left ( x,t \right ) =1/2\,\tanh \left ( 1/4\,\sqrt {2}x+1/4\, \left ( -2\,a+1 \right ) t+{\it \_C1} \right ) +1/2\]

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7.1.2.5 [2008] Problem 5

problem number 2008

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 ) =1/2\,{\frac {1}{ \left ( 27\,{\it b0}\,{{\it b3}}^{2}-9\,{\it b1}\,{\it b2}\,{\it b3}+2\,{{\it b2}}^{3} \right ) \left ( 24\,a{\it b3}\, \left ( \RootOf \left ( 512\,{a}^{3}{{\it b3}}^{2}{{\it \_Z}}^{6}+ \left ( -384\,{a}^{2}{{\it b3}}^{2}{\it b1}+128\,{a}^{2}{\it b3}\,{{\it b2}}^{2} \right ) {{\it \_Z}}^{4}+ \left ( 72\,a{{\it b3}}^{2}{{\it b1}}^{2}-48\,a{\it b3}\,{{\it b2}}^{2}{\it b1}+8\,a{{\it b2}}^{4} \right ) {{\it \_Z}}^{2}-27\,{{\it b0}}^{2}{{\it b3}}^{3}+18\,{\it b0}\,{\it b1}\,{\it b2}\,{{\it b3}}^{2}-4\,{\it b0}\,{{\it b2}}^{3}{\it b3}-4\,{{\it b1}}^{3}{{\it b3}}^{2}+{{\it b1}}^{2}{{\it b2}}^{2}{\it b3} \right ) \right ) ^{2}-3\,{\it b1}\,{\it b3}+{{\it b2}}^{2} \right ) } \left ( -2304\, \left ( a{\it b3}\, \left ( \RootOf \left ( 512\,{a}^{3}{{\it b3}}^{2}{{\it \_Z}}^{6}+ \left ( -384\,{a}^{2}{{\it b3}}^{2}{\it b1}+128\,{a}^{2}{\it b3}\,{{\it b2}}^{2} \right ) {{\it \_Z}}^{4}+ \left ( 72\,a{{\it b3}}^{2}{{\it b1}}^{2}-48\,a{\it b3}\,{{\it b2}}^{2}{\it b1}+8\,a{{\it b2}}^{4} \right ) {{\it \_Z}}^{2}-27\,{{\it b0}}^{2}{{\it b3}}^{3}+18\,{\it b0}\,{\it b1}\,{\it b2}\,{{\it b3}}^{2}-4\,{\it b0}\,{{\it b2}}^{3}{\it b3}-4\,{{\it b1}}^{3}{{\it b3}}^{2}+{{\it b1}}^{2}{{\it b2}}^{2}{\it b3} \right ) \right ) ^{2}-1/8\,{\it b1}\,{\it b3}+1/24\,{{\it b2}}^{2} \right ) ^{2}\RootOf \left ( 12\, \left ( \RootOf \left ( 512\,{a}^{3}{{\it b3}}^{2}{{\it \_Z}}^{6}+ \left ( -384\,{a}^{2}{{\it b3}}^{2}{\it b1}+128\,{a}^{2}{\it b3}\,{{\it b2}}^{2} \right ) {{\it \_Z}}^{4}+ \left ( 72\,a{{\it b3}}^{2}{{\it b1}}^{2}-48\,a{\it b3}\,{{\it b2}}^{2}{\it b1}+8\,a{{\it b2}}^{4} \right ) {{\it \_Z}}^{2}-27\,{{\it b0}}^{2}{{\it b3}}^{3}+18\,{\it b0}\,{\it b1}\,{\it b2}\,{{\it b3}}^{2}-4\,{\it b0}\,{{\it b2}}^{3}{\it b3}-4\,{{\it b1}}^{3}{{\it b3}}^{2}+{{\it b1}}^{2}{{\it b2}}^{2}{\it b3} \right ) \right ) ^{4}{a}^{2}{\it b3}-6\, \left ( \RootOf \left ( 512\,{a}^{3}{{\it b3}}^{2}{{\it \_Z}}^{6}+ \left ( -384\,{a}^{2}{{\it b3}}^{2}{\it b1}+128\,{a}^{2}{\it b3}\,{{\it b2}}^{2} \right ) {{\it \_Z}}^{4}+ \left ( 72\,a{{\it b3}}^{2}{{\it b1}}^{2}-48\,a{\it b3}\,{{\it b2}}^{2}{\it b1}+8\,a{{\it b2}}^{4} \right ) {{\it \_Z}}^{2}-27\,{{\it b0}}^{2}{{\it b3}}^{3}+18\,{\it b0}\,{\it b1}\,{\it b2}\,{{\it b3}}^{2}-4\,{\it b0}\,{{\it b2}}^{3}{\it b3}-4\,{{\it b1}}^{3}{{\it b3}}^{2}+{{\it b1}}^{2}{{\it b2}}^{2}{\it b3} \right ) \right ) ^{2}a{\it b1}\,{\it b3}+2\, \left ( \RootOf \left ( 512\,{a}^{3}{{\it b3}}^{2}{{\it \_Z}}^{6}+ \left ( -384\,{a}^{2}{{\it b3}}^{2}{\it b1}+128\,{a}^{2}{\it b3}\,{{\it b2}}^{2} \right ) {{\it \_Z}}^{4}+ \left ( 72\,a{{\it b3}}^{2}{{\it b1}}^{2}-48\,a{\it b3}\,{{\it b2}}^{2}{\it b1}+8\,a{{\it b2}}^{4} \right ) {{\it \_Z}}^{2}-27\,{{\it b0}}^{2}{{\it b3}}^{3}+18\,{\it b0}\,{\it b1}\,{\it b2}\,{{\it b3}}^{2}-4\,{\it b0}\,{{\it b2}}^{3}{\it b3}-4\,{{\it b1}}^{3}{{\it b3}}^{2}+{{\it b1}}^{2}{{\it b2}}^{2}{\it b3} \right ) \right ) ^{2}a{{\it b2}}^{2}+{{\it \_Z}}^{2}{\it b3} \right ) \tanh \left ( \RootOf \left ( 12\, \left ( \RootOf \left ( 512\,{a}^{3}{{\it b3}}^{2}{{\it \_Z}}^{6}+ \left ( -384\,{a}^{2}{{\it b3}}^{2}{\it b1}+128\,{a}^{2}{\it b3}\,{{\it b2}}^{2} \right ) {{\it \_Z}}^{4}+ \left ( 72\,a{{\it b3}}^{2}{{\it b1}}^{2}-48\,a{\it b3}\,{{\it b2}}^{2}{\it b1}+8\,a{{\it b2}}^{4} \right ) {{\it \_Z}}^{2}-27\,{{\it b0}}^{2}{{\it b3}}^{3}+18\,{\it b0}\,{\it b1}\,{\it b2}\,{{\it b3}}^{2}-4\,{\it b0}\,{{\it b2}}^{3}{\it b3}-4\,{{\it b1}}^{3}{{\it b3}}^{2}+{{\it b1}}^{2}{{\it b2}}^{2}{\it b3} \right ) \right ) ^{4}{a}^{2}{\it b3}-6\, \left ( \RootOf \left ( 512\,{a}^{3}{{\it b3}}^{2}{{\it \_Z}}^{6}+ \left ( -384\,{a}^{2}{{\it b3}}^{2}{\it b1}+128\,{a}^{2}{\it b3}\,{{\it b2}}^{2} \right ) {{\it \_Z}}^{4}+ \left ( 72\,a{{\it b3}}^{2}{{\it b1}}^{2}-48\,a{\it b3}\,{{\it b2}}^{2}{\it b1}+8\,a{{\it b2}}^{4} \right ) {{\it \_Z}}^{2}-27\,{{\it b0}}^{2}{{\it b3}}^{3}+18\,{\it b0}\,{\it b1}\,{\it b2}\,{{\it b3}}^{2}-4\,{\it b0}\,{{\it b2}}^{3}{\it b3}-4\,{{\it b1}}^{3}{{\it b3}}^{2}+{{\it b1}}^{2}{{\it b2}}^{2}{\it b3} \right ) \right ) ^{2}a{\it b1}\,{\it b3}+2\, \left ( \RootOf \left ( 512\,{a}^{3}{{\it b3}}^{2}{{\it \_Z}}^{6}+ \left ( -384\,{a}^{2}{{\it b3}}^{2}{\it b1}+128\,{a}^{2}{\it b3}\,{{\it b2}}^{2} \right ) {{\it \_Z}}^{4}+ \left ( 72\,a{{\it b3}}^{2}{{\it b1}}^{2}-48\,a{\it b3}\,{{\it b2}}^{2}{\it b1}+8\,a{{\it b2}}^{4} \right ) {{\it \_Z}}^{2}-27\,{{\it b0}}^{2}{{\it b3}}^{3}+18\,{\it b0}\,{\it b1}\,{\it b2}\,{{\it b3}}^{2}-4\,{\it b0}\,{{\it b2}}^{3}{\it b3}-4\,{{\it b1}}^{3}{{\it b3}}^{2}+{{\it b1}}^{2}{{\it b2}}^{2}{\it b3} \right ) \right ) ^{2}a{{\it b2}}^{2}+{{\it \_Z}}^{2}{\it b3} \right ) t+\RootOf \left ( 512\,{a}^{3}{{\it b3}}^{2}{{\it \_Z}}^{6}+ \left ( -384\,{a}^{2}{{\it b3}}^{2}{\it b1}+128\,{a}^{2}{\it b3}\,{{\it b2}}^{2} \right ) {{\it \_Z}}^{4}+ \left ( 72\,a{{\it b3}}^{2}{{\it b1}}^{2}-48\,a{\it b3}\,{{\it b2}}^{2}{\it b1}+8\,a{{\it b2}}^{4} \right ) {{\it \_Z}}^{2}-27\,{{\it b0}}^{2}{{\it b3}}^{3}+18\,{\it b0}\,{\it b1}\,{\it b2}\,{{\it b3}}^{2}-4\,{\it b0}\,{{\it b2}}^{3}{\it b3}-4\,{{\it b1}}^{3}{{\it b3}}^{2}+{{\it b1}}^{2}{{\it b2}}^{2}{\it b3} \right ) x+{\it \_C1} \right ) -432\, \left ( \left ( \RootOf \left ( 512\,{a}^{3}{{\it b3}}^{2}{{\it \_Z}}^{6}+ \left ( -384\,{a}^{2}{{\it b3}}^{2}{\it b1}+128\,{a}^{2}{\it b3}\,{{\it b2}}^{2} \right ) {{\it \_Z}}^{4}+ \left ( 72\,a{{\it b3}}^{2}{{\it b1}}^{2}-48\,a{\it b3}\,{{\it b2}}^{2}{\it b1}+8\,a{{\it b2}}^{4} \right ) {{\it \_Z}}^{2}-27\,{{\it b0}}^{2}{{\it b3}}^{3}+18\,{\it b0}\,{\it b1}\,{\it b2}\,{{\it b3}}^{2}-4\,{\it b0}\,{{\it b2}}^{3}{\it b3}-4\,{{\it b1}}^{3}{{\it b3}}^{2}+{{\it b1}}^{2}{{\it b2}}^{2}{\it b3} \right ) \right ) ^{2}a{\it b2}-{\frac {9\,{\it b0}\,{\it b3}}{16}}+1/16\,{\it b1}\,{\it b2} \right ) \left ( {\it b0}\,{{\it b3}}^{2}-1/3\,{\it b1}\,{\it b2}\,{\it b3}+{\frac {2\,{{\it b2}}^{3}}{27}} \right ) \right ) }\]

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