Re: Can Mathematica do Separation of Variables?

*To*: mathgroup at smc.vnet.net*Subject*: [mg73016] Re: Can Mathematica do Separation of Variables?*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Mon, 29 Jan 2007 04:19:35 -0500 (EST)*Organization*: The University of Western Australia*References*: <epf9mr$c44$1@smc.vnet.net>

In article <epf9mr$c44$1 at smc.vnet.net>, "Raj" <rajanikanth at gmail.com> wrote: > Can somebody please tell me if Mathematica can do separation of > variables (using DSolve) > > for ex: if I give this equation to DSolve, its not able to separate the > variables > > D[u[x,t],{t,2}]==D[u[x,t],{x,2}]+D[u[x,t],{t,1},{x,2}] There is no built-in tool for separation of variables. However, for your differential equation, deq = D[u[x,t],{t,2}] == D[u[x,t],{x,2}]+D[u[x,t],{t,1},{x,2}]; re-writing this as deq = Subtract @@ deq followed by the substitution u -> Function[{x, t}, X[x] T[t]], deq /. u -> Function[{x, t}, X[x] T[t]]; and then by dividing through by X[x] (T'[t] + T[t])) generates the separated equations: Apart[% / (X[x] (T'[t] + T[t]))] T''[t]/(T[t] + T'[t]) - X''[x]/X[x] There is another approach for such equations. One can find travelling wave solutions of the form u[x,t] = g[x - v t] as follows: deq /. u -> Function[{x, t}, g[x - v t]] // Simplify (v^2 - 1) g''[x - v t] + v g'''[x - v t] == 0 Putting x - v t -> z, the solution is immediate: DSolve[% /. x - v t -> z, g, z] {{g -> Function[{z}, -((v^2*C[1])/(E^(((-1 + v^2)*z)/v)*(1 - v^2)* (-1 + v^2))) + C[2] + z*C[3]]}} Cheers, Paul _______________________________________________________________________ Paul Abbott Phone: 61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) AUSTRALIA http://physics.uwa.edu.au/~paul