solution of system of coupled partial differential equations [closed]

Question

I have following system of coupled Partial differential equations. How can I solve the system by Maple? \begin{align} m_1\frac{\partial^2 u_1}{\partial t^2}+A_1\frac{\partial ^4u_1(x,t)}{\partial x^4}+k(u_1-u_2)=F_1(t) \delta(x-x_1), \end{align} \begin{align} m_2\frac{\partial^2 u_2}{\partial t^2}-A_2\frac{\partial ^2u_2(x,t)}{\partial x^2}+k(u_2-u_1)=F_2(t)\delta(x-x_2). \end{align} Code:

PDE1:=m1*diff(u1(x,t),t$2)+A1*diff(u1(x,t),x$4)+k*(u1(x,t)-u2(x,t))=F1(t)*delta(x-x1);
PDE2:=m2*diff(u2(x,t),t$2)-A2*diff(u2(x,t),x$2)+k*(u2(x,t)-u1(x,t))=F2(t)*delta(x-x2);

In here, $A_i, m_i, x_i$ and $k$ are constants where $i=1,2$ and $\delta$ is Dirac Delta function.

Boundary conditions:

$u_1(0,t)=\frac{\partial^2 u_1}{\partial x^2}(0,t)=u_1(l,t)=\frac{\partial^2 u_1}{\partial x^2}(l,t)=0$,

$u_2(0,t)=u_2(l,t)=0$

Initial conditions:

$u_i(x,0)=w_{i0}(x),$

$\frac{\partial u_i}{\partial x}(x,0)=y_{i0}(x)$ for $i=1,2.$

Since $F_1(t)$ and $F_2(t)$ are unspecified (ungiven) functions, solutions $u_1,u_2$ which we seek will be depended on $F_1(t)$ and $F_2(t)$.

Answer 1

If you replace delta() with Maple synax for the Dirac delta: Dirac() Maple's pdsolve given an error. But, not doing the replacement, it seems to make a little progress maybe:

pdsolve({PDE1, PDE2}, {u1(x, t), u2(x, t)});

Once I add the boundary conditions it stops making any progress though (and the same result if I replace your delta with Diract). It just returns NULL.

pdsolve({PDE1, PDE2, u1(0, t) = 0, u1(1, t) = 0, u2(0, t) = 0, u2(1, t) = 0, 
         (D[2](u1))(0, t) = 0, (D[2](u1))(1, t) = 0}, {u1(x, t), u2(x, t)})

I suspect that you have too many symbolic constants and unknown functions for pdsolve to handle. There may be other tricks you can play with options the pdsolve function, but its documentation is pretty impenetrable.