2.10.3 \(u_{tt} - u_{xx} - \frac {2}{x} u_x = 0\) with \(u(x,0)=0,u_t(x,0)=g(x)\)

problem number 104

Added Oct 6, 2019

Problem 2.4.18 from Peter Olver, Intoduction to Partial differential equations, 4th edition.

Solve for \(u(x,t)\) \[ u_{tt} - u_{xx} - \frac {2}{x} u_x = 0 \]

With \(u(x,0)=0,u_t(x,0)=g(x)\). Note, in the book, it says to assume \(g(x)\) is even function. In the code below, this assumption is not used. When I find the correct way to implement this assumption in CAS, will have to re-run these.




\[u \left (x , t\right ) = \Mapleoverset {\infty }{\Mapleunderset {n =0}{\sum }}\frac {t^{2 n +1} \left (\textbf {proc} (U) \\ \textbf {option} \,operator,\,arrow; \\ \mapleIndent {1} \mathit {diff} (\mathit {diff} (U,\,x),\,x) + 2 \ast x\hat {~}{-1} \ast \mathit {diff} (U,\,x)\\ \textbf {end\ proc};\right )^{(n )}\left (g (x )\right )}{\left (2 n +1\right )!}\]