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Taken from Maple pdsolve help pages, which is taken from Landau, L.D. and Lifshitz, E.M. Translated by Sykes, J.B. and Bell, J.S. Mechanics. Oxford: Pergamon Press, 1969
Solve for \(S \left ( t,\xi ,\eta ,\phi \right ) \) \begin {align*} -{\frac {\partial }{\partial t}}S \left ( t,\xi ,\eta ,\phi \right ) &=1/2 \,{\frac {\left ( {\frac {\partial }{\partial \xi }}S \left ( t,\xi ,\eta ,\phi \right ) \right ) ^{2} \left ( {\xi }^{2}-1 \right ) }{{\sigma }^{2}m \left ( -{\eta }^{2}+{\xi }^{2} \right ) }}+1/2\,{\frac { \left ( {\frac { \partial }{\partial \eta }}S \left ( t,\xi ,\eta ,\phi \right ) \right ) ^{ 2} \left ( -{\eta }^{2}+1 \right ) }{{\sigma }^{2}m \left ( -{\eta }^{2}+{ \xi }^{2} \right ) }}+1/2\,{\frac { \left ( {\frac {\partial }{\partial \phi }}S \left ( t,\xi ,\eta ,\phi \right ) \right ) ^{2}}{{\sigma }^{2}m \left ( {\xi }^{2}-1 \right ) \left ( -{\eta }^{2}+1 \right ) }}+{\frac {a \left ( \xi \right ) +b \left ( \eta \right ) }{-{\eta }^{2}+{\xi }^{2}}} \end {align*}
Mathematica ✗
ClearAll[t, \[Zeta], \[Eta], \[Phi], a, b, s]; pde = -D[s[t, \[Zeta], \[Eta], \[Phi]], t] == ((\[Zeta]^2 - 1)*D[s[t, \[Zeta], \[Eta], \[Phi]], \[Zeta]]^2)/(2*\[Sigma]^2*m*(-\[Eta]^2 + \[Zeta]^2)) + ((-\[Eta]^2 - 1)*D[s[t, \[Zeta], \[Eta], \[Phi]], \[Eta]]^2)/(2*\[Sigma]^2*m*(-\[Eta]^2 + \[Zeta]^2)) + (1*D[s[t, \[Zeta], \[Eta], \[Phi]], \[Phi]]^2)/(2*\[Sigma]^2*m*(\[Zeta]^2 - 1)*(-\[Eta]^2 - 1)) + (a[\[Zeta]] + b[\[Zeta]])/(-\[Eta]^2 + \[Zeta]^2); sol = AbsoluteTiming[TimeConstrained[DSolve[pde, s[t, \[Zeta], \[Eta], \[Phi]], {t, \[Zeta], \[Eta], \[Phi]}], 60*10]];
\[ \text {Failed} \]
Maple ✓
S:='S';t:='t'; xi:='xi';eta:='eta';phi:='phi'; pde := -diff(S(t,xi,eta,phi),t) =1/2*diff(S(t,xi,eta,phi),xi)^2*(xi^2-1)/sigma^2/m/(xi^2-eta^2)+ 1/2*diff(S(t,xi,eta,phi),eta)^2*(1-eta^2)/m/sigma^2/(xi^2-eta^2)+ 1/2*diff(S(t,xi,eta,phi),phi)^2/m/sigma^2/(xi^2-1)/(1-eta^2)+ (a(xi)+b(eta))/(xi^2-eta^2); cpu_time := timelimit(60*10,CodeTools[Usage](assign('sol',pdsolve(pde,'build')),output='realtime'));
\[ S \left ( t,\xi ,\eta ,\phi \right ) ={\it \_c}_{{4}}\phi +{\it \_c}_{{1}}t+{\it \_C1}+{\it \_C2}+{\it \_C3}+{\it \_C4}-\int \!{\frac {\sqrt {-2\,{\eta }^{4}m{\sigma }^{2}{\it \_c}_{{1}}+2\,b \left ( \eta \right ) {\eta }^{2}m{\sigma }^{2}+2\,{\eta }^{2}{\it \_c}_{{1}}{\sigma }^{2}m-2\,{\eta }^{2}{\it \_c}_{{3}}{\sigma }^{2}m-2\,b \left ( \eta \right ) {\sigma }^{2}m+2\,{\it \_c}_{{3}}{\sigma }^{2}m-{{\it \_c}_{{4}}}^{2}}}{{\eta }^{2}-1}}\,{\rm d}\eta -\int \!{\frac {\sqrt {-2\,m{\sigma }^{2}{\xi }^{4}{\it \_c}_{{1}}-2\,a \left ( \xi \right ) m{\sigma }^{2}{\xi }^{2}+2\,{\xi }^{2}{\it \_c}_{{1}}{\sigma }^{2}m-2\,m{\sigma }^{2}{\xi }^{2}{\it \_c}_{{3}}+2\,a \left ( \xi \right ) {\sigma }^{2}m+2\,{\it \_c}_{{3}}{\sigma }^{2}m-{{\it \_c}_{{4}}}^{2}}}{{\xi }^{2}-1}}\,{\rm d}\xi \]