# Solving a system of PDEs

How can we solve the set of two coupled non-homogeneous pdes? \begin{align} K_1\frac{\partial^4 u_1}{\partial x^4}+m_1\frac{\partial ^2u_1(x,t)}{\partial t^2}+k(u_1-u_2)=F_1(t) \delta(x-x_1), \end{align} \begin{align} K_2\frac{\partial^4 u_2}{\partial x^4}+m_2\frac{\partial ^2u_2(x,t)}{\partial t^2}+k(u_2-u_1)=F_2(t)\delta(x-x_2). \end{align} In here, $K_i, m_i, x_i$ and $k$ are constants where $i=1,2$.

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 $F_2(t)$.

• Take the Fourier transform in the $x$-direction, solve the resulting system of ordinary differential equations, then take the inverse Fourier transform (if possible). – Hans Engler Jul 8 '17 at 16:02