2.15.6 \(w_t = w_{x_1 x_1} + w_{x_2 x_2} + w_{x_3 x_3}\)

problem number 138

Added December 20, 2018.

Example 25, Taken from https://www.mapleprimes.com/posts/209970-Exact-Solutions-For-PDE-And-Boundary--Initial-Conditions-2018

Linear PDE, initial conditions at \(t=1\). Solve for \(w(x_1,x_2,x_3,t)\) \[ w_t = w_{x_1 x_1} + w_{x_2 x_2} + w_{x_3 x_3} \] With initial condition \(w(x_1,x_2,x_3,1) = e^a x_1^2 +x_2 x_3\)

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x1, x2, x3, t], t] == D[w[x1, x2, x3, t], {x1, 2}] + D[w[x1, x2, x3, t], {x2, 2}] + D[w[x1, x2, x3, t], {x3, 2}]; 
ic  = w[x1, x2, x3, 1] == Exp[a]*x1^2 + x2*x3; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[{pde, ic}, w[x1, x2, x3, t], {x1, x2, x3, t}], 60*10]];
 

\[\left \{\left \{w(\text {x1},\text {x2},\text {x3},t)\to e^a \left (2 t+\text {x1}^2-2\right )+\text {x2} \text {x3}\right \}\right \}\]

Maple ✓

restart; 
pde := diff(w(x1, x2, x3, t), t) = diff(w(x1, x2, x3, t), x1$2)+diff(w(x1, x2, x3, t), x2$2)+diff(w(x1, x2, x3, t), x3$2); 
ic  := w(x1, x2, x3, 1) = exp(a)*x1^2+x2*x3; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde, ic],w(x1,x2,x3,t))),output='realtime'));
 

\[w \left (\mathit {x1} , \mathit {x2} , \mathit {x3} , t\right ) = \mathit {x2} \mathit {x3} +\left (\mathit {x1}^{2}+2 t -2\right ) {\mathrm e}^{a}\]

____________________________________________________________________________________