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MathGroup Archive 2006

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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


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