# Problem: codying a variational formulation

**URL:** https://community.freefem.org/t/problem-codying-a-variational-formulation/3096
**Category:** General Discussion
**Created:** [April 11, 2024, 10:59am UTC](https://community.freefem.org/t/problem-codying-a-variational-formulation/3096 "2024-04-11T10:59:08Z")
**Posts on this page:** 1
**Page:** 1

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### Author: ![mariavm](https://avatars.discourse-cdn.com/v4/letter/m/71e660/32.png) [@mariavm](https://community.freefem.org/u/mariavm)
#### Post date: [April 11, 2024, 10:59am UTC](https://community.freefem.org/t/problem-codying-a-variational-formulation/3096/1 "2024-04-11T10:59:08Z")

</div>

Hello, I am a newbie on FreeFEM and I want to solve a problem. I have this variational formula (written in LaTeX code) :  
\begin{align\*}  
\int\_\Omega \frac{du}{dt}v \text{ } dx + \int\_\Omega k(x)\nabla u \cdot\nabla v \text{ } dx = \int\_\Omega \rho u v(1-u^2) \text{ } dx.  
\end{align\*}.

I have written in FreeFEM this code:  
real rho = 1.0;  
real k = 1.0;  
real dt = 0.1;  
func u0 = 0.5; // Condición inicial

mesh Th = square(20, 20); // Domínio Omega, cuadrícula 20x20  
fespace Vh(Th, P1); // Espacio de funciones P1

Vh u=u0, v, uold; // Funciones de prueba, de base y solución inicial

// Condiciones de contorno

real T = 10; // Tiempo final

// Formulación variacional  
problem Ejercicio(u, v,solver=LU)  
= int2d(Th)( u/dt_v) // Derivada temporal  
+ int2d(Th)(k_( dx(u) \* dx(v) + dy(u) \* dy(v)))// Término de difusión  
- int2d(rho_u_v\*(1 -u\*u)) // Término no lineal  
+ on(1, u=0); // Condición de Dirichlet en el borde

for(real t = 0; t \< T; t += dt){  
Ejercicio;// uold -\>u //here the elliptic problem is solved  
uold = u; // update; equivalent to u^{n-1} = u^n  
plot(u);  
}

but it doesn´t work because in the lineal term ( - int2d(rho_u_v\*(1 -u_u))) FreeFEM doesn’t know how treat the product u_u (it interprets it as a matrix product). I’m not sure if I need to use another solver or some other way. Thank you very much in advance.