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

problem number 145

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


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

Maple


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

________________________________________________________________________________________