[Date Index]
[Thread Index]
[Author Index]
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
Prev by Date:
**Re: cusp fold surface**
Next by Date:
**Re: DSolve fails with Telegraph equation**
Previous by thread:
**Re: DSolve fails with Telegraph equation**
Next by thread:
**Re: DSolve fails with Telegraph equation**
| |