HNPDE, chapter 1.1.1

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153.1 Problem 1

Problem Chapter 1.1.1.1, from Handbook of nonlinear partial diﬀerential equations by Andrei D. Polyanin, Valentin F. Zaitsev.

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

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

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153.2 Problem 2

Problem Chapter 1.1.1.2, from Handbook of nonlinear partial diﬀerential equations by Andrei D. Polyanin, Valentin F. Zaitsev.

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]*(1 - w[x, t]); 
 sol = AbsoluteTiming[TimeConstrained[DSolve[pde, w[x, t], {x, t}], 60*10]];

\[ \left \{\left \{w(x,t)\to \frac {1}{4} \left (\tanh \left (\frac {5 a t}{12}-\frac {\sqrt {a} x}{2 \sqrt {6}}-c_3\right )+1\right ){}^2\right \},\left \{w(x,t)\to -\frac {1}{4} \left (-3+\tanh \left (\frac {5 a t}{12}-\frac {i \sqrt {a} x}{2 \sqrt {6}}-c_3\right )\right ) \left (1+\tanh \left (\frac {5 a t}{12}-\frac {i \sqrt {a} x}{2 \sqrt {6}}-c_3\right )\right )\right \},\left \{w(x,t)\to -\frac {1}{4} \left (-3+\tanh \left (\frac {5 a t}{12}+\frac {i \sqrt {a} x}{2 \sqrt {6}}-c_3\right )\right ) \left (1+\tanh \left (\frac {5 a t}{12}+\frac {i \sqrt {a} x}{2 \sqrt {6}}-c_3\right )\right )\right \},\left \{w(x,t)\to \frac {1}{4} \left (\tanh \left (\frac {5 a t}{12}+\frac {\sqrt {a} x}{2 \sqrt {6}}-c_3\right )+1\right ){}^2\right \},\left \{w(x,t)\to \frac {1}{4} \left (\tanh \left (\frac {5 a t}{12}-\frac {\sqrt {a} x}{2 \sqrt {6}}+c_3\right )+1\right ){}^2\right \},\left \{w(x,t)\to -\frac {1}{4} \left (-3+\tanh \left (\frac {5 a t}{12}-\frac {i \sqrt {a} x}{2 \sqrt {6}}+c_3\right )\right ) \left (1+\tanh \left (\frac {5 a t}{12}-\frac {i \sqrt {a} x}{2 \sqrt {6}}+c_3\right )\right )\right \},\left \{w(x,t)\to -\frac {1}{4} \left (-3+\tanh \left (\frac {5 a t}{12}+\frac {i \sqrt {a} x}{2 \sqrt {6}}+c_3\right )\right ) \left (1+\tanh \left (\frac {5 a t}{12}+\frac {i \sqrt {a} x}{2 \sqrt {6}}+c_3\right )\right )\right \},\left \{w(x,t)\to \frac {1}{4} \left (\tanh \left (\frac {5 a t}{12}+\frac {\sqrt {a} x}{2 \sqrt {6}}+c_3\right )+1\right ){}^2\right \}\right \} \]

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

\[ w \left ( x,t \right ) =-1/4\, \left ( \tanh \left ( -{\frac {5\,at}{12}}+1/12\,\sqrt {-6\,a}x+{\it \_C1} \right ) \right ) ^{2}-1/2\,\tanh \left ( -{\frac {5\,at}{12}}+1/12\,\sqrt {-6\,a}x+{\it \_C1} \right ) +3/4 \]

153 HNPDE, chapter 1.1.1

153.1 Problem 1

153.2 Problem 2