I'm trying to solve a coupled set of PDEs using Finite element method. But I require the weak form of the equations to do it. Can anyone shed some light onto how to derive the weak form of these equations.
$$\frac{\partial f}{\partial t} + \frac{\partial{fg}}{\partial x} = 0 \\ \frac{\partial{fg}}{\partial t} + \frac{\partial{f^2 g}}{\partial x} = \frac{3}{R} \frac{\partial }{\partial x}(f \frac{\partial g}{\partial x}) + C f $$
It is an open boundary problem related to fluid flow, but initial conditions are defined. feel free to assume convenient initial values for solving.
f and g are respectively the area of cross-section and velocity of a fluid. They are both functions of x and t
Assuming $t>0$ and $x \in [a,b] \subset \mathbb{R},$ we have:
$$ \int^b_a \left( \frac{\partial f}{\partial t} + \frac{\partial \, f g}{\partial x}\right)\omega_1 \mathrm{d}x = 0,$$ where $\omega_1$ is a test function. On the other hand, for the second equation we have to apply 1st Green's identity, which yields:
$$\begin{align} \int^a_b \left( \frac{\partial \, f g}{\partial t} + \frac{\partial \, f^2 g}{\partial x} \right) \omega_2 \,\mathrm{d}x & = \\ \frac{3}{R} \left( \left. f \frac{\partial g}{\partial x} \omega_2\right|_{x=b} - \left. f \frac{\partial g}{\partial x} \omega_2 \right|_{x=a} - \int^a_b f \frac{\partial g}{\partial x} \frac{\partial \omega_2}{\partial x} \, \mathrm{d}x\right) + \int_a^b C f \omega_2 \, \mathrm{d}x &, \end{align}$$ where $\omega_2$ is another test function. The particular choice of what kind of function they would be is up to you. For example, you can set $\omega_i$ as the set of piecewise linear functions (hat functions) on a partition of $[a,b]$.
Now, information of what happens in the boundary is required for the formulation to be complete.
Two considerations:
Note that your system of PDEs is not linear, i.e., contains terms as $f^2$ or $f g$. It is then required to use a method for solving non-linear equations, such as Newton-Raphson. You can always sum the two equations above to come up with something like: $$ \mathsf{F}(f,g; \omega_1, \omega_2) = 0, $$ which can be solved with the iteration: $$ - \mathbf{J}_{\mathsf{F}}(\mathbf{f};\mathbf{\omega}) \Delta \mathbf{f} = \mathsf{F}(\mathbf{f};\omega), \quad \mathbf{f} = (f,g), \quad \omega = (\omega_1,\omega_2), $$ $\mathbf{J}$ is the matrix jacobian of $\mathsf{F}$ with respect to $f$ and $g$ and $\Delta \mathbf{f}$ is the difference between iterations.
The FEM approach is all about spatial discretization of the equations. Note that you must apply a time-marching method, such as implicit/backward Euler, to march in time.
Hope this helps!