154 HNPDE, chapter 1.1.2

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154.1 Problem 1

problem number 1241

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12]; 
 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]];
 

\[ \text {Failed} \]

Maple ✗

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); 
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'));
 

\[ \text { sol=() } \]

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154.2 Problem 2

problem number 1242

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12]; 
 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]];
 

\[ \text {Failed} \]

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); 
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'));
 

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

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154.3 Problem 3

problem number 1243

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12]; 
 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]];
 

\[ \text {Failed} \]

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); 
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'));
 

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

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154.4 Problem 4

problem number 1244

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 ClearAll[a1, a0, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12]; 
 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]];
 

\[ \text {Failed} \]

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); 
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'));
 

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

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154.5 Problem 5

problem number 1245

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[w, x, y, n, a, b, m, c, k, alpha, beta, gamma, A, C0, s]; 
 ClearAll[lambda, B, mu, d, g, B, v, f, h, q, p, delta, t]; 
 ClearAll[g1, g0, h2, h1, h0, f1, f2]; 
 ClearAll[a1, a0, b3, b2, b1, b0, c2, c1, c0, k0, k1, k2, s1, s0, k22, k11, k12, s11, s22, s12]; 
 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]];
 

\[ \text {Failed} \]

Maple ✓

 
unassign('w,x,y,a,b,n,m,c,k,alpha,beta,g,A,f,C,lambda,B,mu,d,s,t'); 
unassign('v,q,p,l,g1,g2,g0,h0,h1,h2,f2,f3,c0,c1,c2,a1,a0,b0,b1,b2'); 
unassign('k0,k1,k2,s0,s1,k22,k12,k11,s22,s12,s11'); 
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 ) =-2\,{\frac {\RootOf \left ( 12\, \left ( \RootOf \left ( 512\,{a}^{3}{{\it b3}}^{2}{{\it \_Z}}^{6}+ \left ( -384\,{a}^{2}{\it b1}\,{{\it b3}}^{2}+128\,{a}^{2}{{\it b2}}^{2}{\it b3} \right ) {{\it \_Z}}^{4}+ \left ( 72\,a{{\it b1}}^{2}{{\it b3}}^{2}-48\,a{\it b1}\,{{\it b2}}^{2}{\it b3}+8\,a{{\it b2}}^{4} \right ) {{\it \_Z}}^{2}-27\,{{\it b0}}^{2}{{\it b3}}^{3}+18\,{\it b0}\,{\it b1}\,{{\it b3}}^{2}{\it b2}-4\,{\it b0}\,{\it b3}\,{{\it b2}}^{3}-4\,{{\it b1}}^{3}{{\it b3}}^{2}+{{\it b1}}^{2}{\it b3}\,{{\it b2}}^{2} \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 b1}\,{{\it b3}}^{2}+128\,{a}^{2}{{\it b2}}^{2}{\it b3} \right ) {{\it \_Z}}^{4}+ \left ( 72\,a{{\it b1}}^{2}{{\it b3}}^{2}-48\,a{\it b1}\,{{\it b2}}^{2}{\it b3}+8\,a{{\it b2}}^{4} \right ) {{\it \_Z}}^{2}-27\,{{\it b0}}^{2}{{\it b3}}^{3}+18\,{\it b0}\,{\it b1}\,{{\it b3}}^{2}{\it b2}-4\,{\it b0}\,{\it b3}\,{{\it b2}}^{3}-4\,{{\it b1}}^{3}{{\it b3}}^{2}+{{\it b1}}^{2}{\it b3}\,{{\it b2}}^{2} \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 b1}\,{{\it b3}}^{2}+128\,{a}^{2}{{\it b2}}^{2}{\it b3} \right ) {{\it \_Z}}^{4}+ \left ( 72\,a{{\it b1}}^{2}{{\it b3}}^{2}-48\,a{\it b1}\,{{\it b2}}^{2}{\it b3}+8\,a{{\it b2}}^{4} \right ) {{\it \_Z}}^{2}-27\,{{\it b0}}^{2}{{\it b3}}^{3}+18\,{\it b0}\,{\it b1}\,{{\it b3}}^{2}{\it b2}-4\,{\it b0}\,{\it b3}\,{{\it b2}}^{3}-4\,{{\it b1}}^{3}{{\it b3}}^{2}+{{\it b1}}^{2}{\it b3}\,{{\it b2}}^{2} \right ) \right ) ^{2}a{{\it b2}}^{2}+{{\it \_Z}}^{2}{\it b3} \right ) \left ( 24\,a{\it b3}\, \left ( \RootOf \left ( 512\,{a}^{3}{{\it b3}}^{2}{{\it \_Z}}^{6}+ \left ( -384\,{a}^{2}{\it b1}\,{{\it b3}}^{2}+128\,{a}^{2}{{\it b2}}^{2}{\it b3} \right ) {{\it \_Z}}^{4}+ \left ( 72\,a{{\it b1}}^{2}{{\it b3}}^{2}-48\,a{\it b1}\,{{\it b2}}^{2}{\it b3}+8\,a{{\it b2}}^{4} \right ) {{\it \_Z}}^{2}-27\,{{\it b0}}^{2}{{\it b3}}^{3}+18\,{\it b0}\,{\it b1}\,{{\it b3}}^{2}{\it b2}-4\,{\it b0}\,{\it b3}\,{{\it b2}}^{3}-4\,{{\it b1}}^{3}{{\it b3}}^{2}+{{\it b1}}^{2}{\it b3}\,{{\it b2}}^{2} \right ) \right ) ^{2}-3\,{\it b1}\,{\it b3}+{{\it b2}}^{2} \right ) \tanh \left ( \RootOf \left ( 12\, \left ( \RootOf \left ( 512\,{a}^{3}{{\it b3}}^{2}{{\it \_Z}}^{6}+ \left ( -384\,{a}^{2}{\it b1}\,{{\it b3}}^{2}+128\,{a}^{2}{{\it b2}}^{2}{\it b3} \right ) {{\it \_Z}}^{4}+ \left ( 72\,a{{\it b1}}^{2}{{\it b3}}^{2}-48\,a{\it b1}\,{{\it b2}}^{2}{\it b3}+8\,a{{\it b2}}^{4} \right ) {{\it \_Z}}^{2}-27\,{{\it b0}}^{2}{{\it b3}}^{3}+18\,{\it b0}\,{\it b1}\,{{\it b3}}^{2}{\it b2}-4\,{\it b0}\,{\it b3}\,{{\it b2}}^{3}-4\,{{\it b1}}^{3}{{\it b3}}^{2}+{{\it b1}}^{2}{\it b3}\,{{\it b2}}^{2} \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 b1}\,{{\it b3}}^{2}+128\,{a}^{2}{{\it b2}}^{2}{\it b3} \right ) {{\it \_Z}}^{4}+ \left ( 72\,a{{\it b1}}^{2}{{\it b3}}^{2}-48\,a{\it b1}\,{{\it b2}}^{2}{\it b3}+8\,a{{\it b2}}^{4} \right ) {{\it \_Z}}^{2}-27\,{{\it b0}}^{2}{{\it b3}}^{3}+18\,{\it b0}\,{\it b1}\,{{\it b3}}^{2}{\it b2}-4\,{\it b0}\,{\it b3}\,{{\it b2}}^{3}-4\,{{\it b1}}^{3}{{\it b3}}^{2}+{{\it b1}}^{2}{\it b3}\,{{\it b2}}^{2} \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 b1}\,{{\it b3}}^{2}+128\,{a}^{2}{{\it b2}}^{2}{\it b3} \right ) {{\it \_Z}}^{4}+ \left ( 72\,a{{\it b1}}^{2}{{\it b3}}^{2}-48\,a{\it b1}\,{{\it b2}}^{2}{\it b3}+8\,a{{\it b2}}^{4} \right ) {{\it \_Z}}^{2}-27\,{{\it b0}}^{2}{{\it b3}}^{3}+18\,{\it b0}\,{\it b1}\,{{\it b3}}^{2}{\it b2}-4\,{\it b0}\,{\it b3}\,{{\it b2}}^{3}-4\,{{\it b1}}^{3}{{\it b3}}^{2}+{{\it b1}}^{2}{\it b3}\,{{\it b2}}^{2} \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 b1}\,{{\it b3}}^{2}+128\,{a}^{2}{{\it b2}}^{2}{\it b3} \right ) {{\it \_Z}}^{4}+ \left ( 72\,a{{\it b1}}^{2}{{\it b3}}^{2}-48\,a{\it b1}\,{{\it b2}}^{2}{\it b3}+8\,a{{\it b2}}^{4} \right ) {{\it \_Z}}^{2}-27\,{{\it b0}}^{2}{{\it b3}}^{3}+18\,{\it b0}\,{\it b1}\,{{\it b3}}^{2}{\it b2}-4\,{\it b0}\,{\it b3}\,{{\it b2}}^{3}-4\,{{\it b1}}^{3}{{\it b3}}^{2}+{{\it b1}}^{2}{\it b3}\,{{\it b2}}^{2} \right ) x+{\it \_C1} \right ) }{27\,{\it b0}\,{{\it b3}}^{2}-9\,{\it b1}\,{\it b3}\,{\it b2}+2\,{{\it b2}}^{3}}}+{\frac {-16\, \left ( \RootOf \left ( 512\,{a}^{3}{{\it b3}}^{2}{{\it \_Z}}^{6}+ \left ( -384\,{a}^{2}{\it b1}\,{{\it b3}}^{2}+128\,{a}^{2}{{\it b2}}^{2}{\it b3} \right ) {{\it \_Z}}^{4}+ \left ( 72\,a{{\it b1}}^{2}{{\it b3}}^{2}-48\,a{\it b1}\,{{\it b2}}^{2}{\it b3}+8\,a{{\it b2}}^{4} \right ) {{\it \_Z}}^{2}-27\,{{\it b0}}^{2}{{\it b3}}^{3}+18\,{\it b0}\,{\it b1}\,{{\it b3}}^{2}{\it b2}-4\,{\it b0}\,{\it b3}\,{{\it b2}}^{3}-4\,{{\it b1}}^{3}{{\it b3}}^{2}+{{\it b1}}^{2}{\it b3}\,{{\it b2}}^{2} \right ) \right ) ^{2}a{\it b2}+9\,{\it b0}\,{\it b3}-{\it b1}\,{\it b2}}{48\,a{\it b3}\, \left ( \RootOf \left ( 512\,{a}^{3}{{\it b3}}^{2}{{\it \_Z}}^{6}+ \left ( -384\,{a}^{2}{\it b1}\,{{\it b3}}^{2}+128\,{a}^{2}{{\it b2}}^{2}{\it b3} \right ) {{\it \_Z}}^{4}+ \left ( 72\,a{{\it b1}}^{2}{{\it b3}}^{2}-48\,a{\it b1}\,{{\it b2}}^{2}{\it b3}+8\,a{{\it b2}}^{4} \right ) {{\it \_Z}}^{2}-27\,{{\it b0}}^{2}{{\it b3}}^{3}+18\,{\it b0}\,{\it b1}\,{{\it b3}}^{2}{\it b2}-4\,{\it b0}\,{\it b3}\,{{\it b2}}^{3}-4\,{{\it b1}}^{3}{{\it b3}}^{2}+{{\it b1}}^{2}{\it b3}\,{{\it b2}}^{2} \right ) \right ) ^{2}-6\,{\it b1}\,{\it b3}+2\,{{\it b2}}^{2}}} \]