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'));
\[w \left ( {\it x1},{\it x2},{\it x3},t \right ) = \left ( {{\it x1}}^{2}+2\,t-2 \right ) {{\rm e}^{a}}+{\it x2}\,{\it x3}\]
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