Solve the Telegraph Equation in 1D 

Solve the telegraph equation over a 1D region.

In[1]:=

X a = 0.7; b = 0.6; c = 1.1;

Specify the initial condition.

In[2]:=

X f[x_] = D[ 0.125 Erf[ ( x - 0.5 ) /0.125 ], x ] ; Plot[f[x], {x, -1, 1}, PlotRange -> All ]

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Specify the initial speed.

In[3]:=

X vInit[x_] = - c * D[f[x], x] - ( a + b ) f[x]/2 ; Plot[vInit[x], {x, -1, 1}, PlotRange -> All]

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Solve the equation.

In[4]:=

X uif = NDSolveValue[{D[u[t, x], {t, 2}] + (a + b) D[u[t, x], {t, 1}] - c^2 D[u[t, x], {x, 2}] + a b u[t, x] == 0, u[0, x] == f[x], Derivative[1, 0][u][0, x] == vInit[x], DirichletCondition[u[t, x] == f[0], x == 0], DirichletCondition[u[t, x] == f[1], x == 1]}, u, {t, 0, 2}, {x, 0, 1}];

Animate the solution.

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In[5]:=

X framesTEQ = Table[Plot[uif[t, x], {x, 0, 1}, PlotRange -> {-1, 1.3}], {t, 0, 2, 0.1}];

In[6]:=

X Manipulate[framesTEQ[[i]], {{i, 3, "time"}, 1, Length[framesTEQ], 1}, SaveDefinitions -> True]

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