Re: DSolve fails with Telegraph equation

*To*: mathgroup at smc.vnet.net*Subject*: [mg69812] Re: DSolve fails with Telegraph equation*From*: dimmechan at yahoo.com*Date*: Sat, 23 Sep 2006 23:45:05 -0400 (EDT)*References*: <ef2tha$mmi$1@smc.vnet.net>

Hello. Even I am not familiar with the equation, I suppose it can be easily solved either with variables separation method or application of exponential Fourier Transnsform. Strangely, in the amazing matheworld link there is just a reference of the equation and nothing more (see http://mathworld.wolfram.com/TelegraphEquation.html) Anyway here is my attempt. I guess is not the final world in the subject. For clarity I have converted everything to InputForm. eq = D[u[x, t], {x, 2}] == A*u[x, t] + B*D[u[x, t], t] + C*D[u[x, t], {t, 2}] Derivative[2, 0][u][x, t] == A*u[x, t] + B*Derivative[0, 1][u][x, t] + C*Derivative[0, 2][u][x, t] Indeed DSolve fails to solve the equation. DSolve[eq, u[x, t], {x, t}] DSolve[Derivative[2, 0][u][x, t] == A*u[x, t] + B*Derivative[0, 1][u][x, t] + C*Derivative[0, 2][u][x, t], u[x, t], {x, t}] Let apply the separation of variables method. Then u[x_, t_] := X[x]*T[t] eq2 = D[u[x, t], {x, 2}] == A*u[x, t] + B*D[u[x, t], t] + C*D[u[x, t], {t, 2}] T[t]*Derivative[2][X][x] == A*T[t]*X[x] + B*X[x]*Derivative[1][T][t] + C*X[x]*Derivative[2][T][t] This is the result equation after you divide with X[x]*T[t] eqnew = Equal @@ (Expand[eq2[[#1]]/(T[t]*X[x])] & ) /@ {1, 2} Derivative[2][X][x]/X[x] == A + (B*Derivative[1][T][t])/T[t] + (C*Derivative[2][T][t])/T[t] As you know now you set each part of eqnew equal with a constant k. But I mention again I am not familiar with the equation. Usually, you must do an exploration about the sign of the constant k. Anyway solX = DSolve[eqnew[[1]] == k, X, x] {{X -> Function[{x}, E^(Sqrt[k]*x)*C[1] + C[2]/E^(Sqrt[k]*x)]}} solT = DSolve[eqnew[[2]] == k, T, t] {{T -> Function[{t}, E^(((-B - Sqrt[B^2 - 4*A*C + 4*C*k])*t)/(2*C))*C[1] + E^(((-B + Sqrt[B^2 - 4*A*C + 4*C*k])*t)/(2*C))*C[2]]}} And here is the desired solution u[x_, t_] = (X[x] /. solX)[[1]]*(T[t] /. solT)[[1]] (E^(((-B - Sqrt[B^2 - 4*A*C + 4*C*k])*t)/(2*C))*C[1] + E^(((-B + Sqrt[B^2 - 4*A*C + 4*C*k])*t)/(2*C))*C[2])* (E^(Sqrt[k]*x)*C[1] + C[2]/E^(Sqrt[k]*x)) And now the verification. Voila FullSimplify[D[u[x,t],{x,2}]\[Equal]A*u[x,t]+B*D[u[x,t],t]+C*D[u[x,t],{t,2}]] True Regards Dimitris