Solve PDE with DiracDelta function

Question

This question is related to Analytic solution of dynamic Euler–Bernoulli beam equation with compatibility condition. I think it is more appropriate to open another question on this topic.

In the question, the compatibility conditions can be introduce into equations using DiracDelta. For example, for the spring compatibility condition, the governing equation can be rewritten as

$$EI\cfrac{\partial^4 w}{\partial x^4} + \mu\cfrac{\partial^2 w}{\partial t^2} + k \delta(x-L/2) w(x,t) = 0$$

Then, we try to use Laplace Transform mentioned by @xzczd,

eqn = EI D[y[x, t], {x, 4}] + μ D[y[x, t], {t, 2}] + 
    k DiracDelta[x - L/2] y[x, t] == 0;
ic = {y[x, 0] == Sin[x/L Pi], Derivative[0, 1][y][x, 0] == 0};
bc = {y[0, t] == 0, y[L, t] == 0, Derivative[2, 0][y][0, t] == 0, 
   Derivative[2, 0][y][L, t] == 0};
teqn = With[{l = LaplaceTransform}, 
   l[{eqn, bc}, t, s] /. HoldPattern@l[u_, t, s] :> u] /. Rule @@@ ic

This step goes well, the equation is transformed to s-domain, but with DiracDelta function.

Apply DSolve to the equation returns 0

DSolve[teqn, y[x, t], x][[1, 1, -1]]

So, I thought maybe use FourierTransform to transform $x$ to $\omega$ domain may solve this problem. However, Mathematica do not return a result from FourierTransform

FourierTransform[eqn, x, s]

There are several questions related to this question but they are mostly using NDSolve. Here are two examples:

Solving a PDE containing DiracDelta

How to correctly use DSolve when the force is an impulse (dirac delta) and initial conditions are not zero

Answer 1

This is not a total answer, but it may get you a little farther with the FourierTransform. The pde:

eqn = EI D[y[x, t], {x, 4}] + μ D[y[x, t], {t, 2}] + 
   k DiracDelta[x - L/2] y[x, t] == 0

Although MMa doesn't like taking the ft of the entire pde at once, we can do it term by term.

fteq = EI FourierTransform[D[y[x, t], {x, 4}], x, s] + μ FourierTransform[D[y[x, t], {t, 2}], x, s] + 
   k FourierTransform[y[x, t] DiracDelta[x - L/2], x, s] == 0
(*EI*s^4*FourierTransform[y[x, t], x, s] + 
   μ*FourierTransform[Derivative[0, 2][y][x, t], x, s] + 
   (k*E^((I*L*s)/2)*y[L/2, t])/Sqrt[2*Pi] == 0*)

Converting this by hand to an ode MMa can work with:

fteq = EI s^4 ys[t] + μ ys''[t] + y[L/2, t] (k E^((I L s)/2))/Sqrt[2 π] == 0

sol = DSolve[fteq, ys[t], t] // Flatten

We get a solution, but not knowing y[L/2,t] this is about as far as we can go. MMa can invert the FourierTransform for a constant value for y[L/2,t] and presumably for other functions of t, but with the DiracDelta function at x = L/2, it is hard to know the value of y there.

Answer 2

After

eqn = EI D[y[x, t], {x, 4}] + μ D[y[x, t], {t, 2}] + k DiracDelta[x - L/2] y[x, t] == 0;
ic = {y[x, 0] == Sin[x/L Pi], Derivative[0, 1][y][x, 0] == 0};
bc = {y[0, t] == 0, y[L, t] == 0, Derivative[2, 0][y][0, t] == 0, Derivative[2, 0][y][L, t] == 0};
teqn = With[{l = LaplaceTransform}, l[{eqn, bc}, t, s] /. HoldPattern@l[u_, t, s] :> u] /. Rule @@@ ic

we need to do

teqns = teqn /. {t-> s}

and then

DSolve[teqns, y, {x, s}]

To have a result, suppose that instead we have (without the DiracDelta)

teqns = \[Mu] (-s Sin[(\[Pi] x)/L] + s^2 y[x, s]) + EI D[y[x,s],{x,4}] == 0

so following with

yxs = y[x,s] /. DSolve[teqns, y, {x, s}]

we obtain an output that should be concluded by

yxt = InverseLaplaceTransform[yxs, s, t]

There are many issues to handle. It seems that DSolve doesn't like the DiracDelta in the PDE and also the boundary conditions are not so easy to handle.