2.15.7 Linear PDE, initial conditions at $t=t_0$

problem number 139

Added December 20, 2018.

Example 26, Taken from https://www.mapleprimes.com/posts/209970-Exact-Solutions-For-PDE-And-Boundary--Initial-Conditions-2018

Solve for $w(x_1,x_2,x_3,t)$

$$\frac{\partial w}{\partial t}=\frac{\partial w^2}{\partial x_1{x}_2}+\frac{\partial w^2}{\partial x_1{x}_3}+\frac{\partial w^2}{\partial x_3^2}+\frac{\partial w^2}{\partial x_2{x}_3}$$

With initial condition $w(x_1,x_2,x_3,t_0)=e^{x_1}+x_2-3x_3$

Mathematica ✓

ClearAll["Global`*"]; 
pde =  D[w[x1, x2, x3, t], t] == D[w[x1, x2, x3, t], x1, x2] + D[w[x1, x2, x3, t], x1, x3] + D[w[x1, x2, x3, t], {x3, 2}] - D[w[x1, x2, x3, t], x2, x3]; 
ic  = w[x1, x2, x3, t0] == Exp[x1] + x2 - 3*x3; 
sol =  AbsoluteTiming[TimeConstrained[DSolve[{pde, ic}, w[x1, x2, x3, t], {x1, x2, x3, t}], 60*10]];
 

$$\left\{\{w(\text{x1},\text{x2},\text{x3},t)\to e^{\text{x1}}+\text{x2}-3\text{x3}\}\right\}$$

Maple ✓

restart; 
pde := diff(w(x1, x2, x3, t), t)= diff(w(x1,x2,x3,t),x1,x2)+diff(w(x1,x2,x3,t),x1,x3)+diff(w(x1,x2,x3,t),x3$2)-diff(w(x1,x2,x3,t),x2,x3); 
ic  := w(x1, x2, x3, t0) = exp(x1)+x2-3*x3; 
cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve([pde, ic],w(x1,x2,x3,t))),output='realtime'));
 

$$w(\mathit{x}\mathit{1},\mathit{x}\mathit{2},\mathit{x}\mathit{3},t)=\mathit{x}\mathit{2}-3\mathit{x}\mathit{3}+{\mathrm{e}}^{\mathit{x}\mathit{1}}$$

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