Wolfram Mathematica

Solve the Wave Equation Using Its Fundamental Solution

Define a wave operator in one spatial dimension.

In:= waveOperator = \!\( \*SubscriptBox[\(\[PartialD]\), \({t, 2}\)]\(u[x, t]\)\) - \!\( \*SubscriptBox[\(\[PartialD]\), \({x, 2}\)]\(u[x, t]\)\);

Obtain its fundamental solution using GreenFunction.

In:= gf[x_, t_, y_, s_] = GreenFunction[waveOperator, u[x, t], {x, -\[Infinity], \[Infinity]}, t, {y, s}]
Out= Plot the fundamental solution.

In:= Plot3D[gf[x, t, 0, 0] // Evaluate, {x, -4, 4}, {t, 0, 4}, ExclusionsStyle -> Orange, Mesh -> None, AxesLabel -> Automatic]
Out= Define a forcing function.

In:= f[y_, s_] := Cos[y] E^(-s)

Solve the wave equation with this forcing term by evaluating the convolution integral .

In:= sol = Integrate[ gf[x, t, y, s] f[y, s], {y, -\[Infinity], \[Infinity]}, {s, 0, \[Infinity]}, Assumptions -> t > 0 && Im[x] == 0] // FullSimplify
Out= Obtain the result using DSolveValue with homogeneous initial conditions.

In:= initialc = {u[x, 0] == 0, Derivative[0, 1][u][x, 0] == 0};
In:= DSolveValue[{waveOperator == f[x, t], initialc}, u[x, t], {x, t}]
Out= Visualize the standing wave generated by the solution.

In:= Plot[Table[sol, {t, 0, 1, 0.2}] // Evaluate, {x, -10, 10}, Filling -> Axis]
Out= 