# How to solve a system of coupled Partial differential equations

I have following system of coupled Partial differential equations. How can I solve the system by Mathematica.

\begin{align} m_1\frac{\partial^2 u_1}{\partial t^2}+A_1\frac{\partial ^2u_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}

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 $F_2(t)$.

Code:

PDE1 = m1*D[u1[x, t], {t, 2}] + A1*D[u1[x, t], {x, 4}] +
k*(u1[x, t] - u2[x, t]) == F1[t] DiracDelta[x - x1];
PDE2 = m2*D[u2[x, t], {t, 2}] - A2*D[u2[x, t], {x, 2}] +
k*(u2[x, t] - u1[x, t]) == F2[t] DiracDelta[x - x2];

• Try with DSolve first if failed then try with NDSolve – zhk Jul 10 '17 at 14:11
• Your equations contain the fourth derivative of $u_1$ with respect to $x$, but only the second derivative of $u_2$ with respect to $x$. Is this intentional, or is it a typo? – Michael Seifert Jul 10 '17 at 15:50
• No, it is not a typo. – HD239 Jul 11 '17 at 6:22
• Sorry, I forgot one boundary condition. ( I added) But it has just 4 initial conditionals. – HD239 Jul 11 '17 at 15:47
• Take the Laplace transform in t of the equations, leaving two coupled equations in x, which can be solved either numerically or symbolically, depending on the details of w[x] and y[x]. Then, perform the inverse Laplace transform. This last part could be difficult. – bbgodfrey Jul 12 '17 at 5:00