## 2 Materials and Methods
### Variational Formulation  
Although differential equations mathematically model several problems, many need analytical solutions. In these cases, the variational formulation of the equation is used, among other techniques, for obtaining numerical solutions. This reformulation makes it possible to obtain approximate values of the equation at pre-determined points in its domain, enabling a general understanding of the phenomenon studied. The variational formulation rewrites the classical formulation of the differential equation (or system of equations), so its solutions must be sought in a space with fewer differentiability restrictions. These solutions are called weak solutions [6]. For linear differential equations, it is possible to state that the solution to an initial value problem exists in any interval that contains the starting point and in which the coefficients of the differential equation are continuous. If any of the coefficients is discontinuous at specific points, the solution may be discontinuous or even cease to exist at those points in the interval [19]. Thus, to satisfy the differential equation in its classical model (3), the solution function \(C\) must be, obligatorily, at least of class \(C^{2}\); in addition to being continuous throughout the domain, its first and second derivatives must also be continuous. In practice, however, the function that models the source term or initial conditions is only sometimes continuous concerning time or space. In other words, environmental problems generally do not have the regularity conditions that the classical formulation requires [7; 20]. Approximation methods, whose instruments do not have the regularity requirements of the robust formulation, are an alternative to these problems [7]. Thus, the natural option is to use the variational formulation (also called weak formulation of the problem), which \"weakens\" the hypotheses of the solution's regularity, \"widening\" the conditions necessary to find or construct the sought solution [14]. In variational methods, the objective is to obtain an approximate solution of a differential equation within a convenient space of functions of finite dimension. In this subspace, the approximate solution is an exact projection of the solution. It can be written as a linear combination of a finite number of known functions, which form the basis of this subspace. Thus, with the resolution of the system of algebraic equations resulting from the discretization process, the coefficients of this linear expansion are sought to satisfy the differential equation. Thus, the problem must be rewritten in its weak form [21]. However, before obtaining the variational formulation (or weak formulation) of equation (3), it is necessary to understand some important definitions and state [PERSON]'s theorem, which is presented below. Assuming \(u\in\Omega\) and \(v\in\Omega\) functions, with \(\Omega\subset\mathbb{R}^{2}\). If  
* \(u\) and its first and second derivatives are continuous;
* \(v\) and its first derivatives are continuous. [PERSON]'s theorem guarantees that the following equality holds [22]:  
\[\int_{\Omega}(-\Delta u)vdx=\int_{\Omega}\
abla u\cdot\
abla vdx-\int_{ \partial\Omega}(\eta\cdot\
abla u)vd\partial\Omega.\]  
where \(\eta\) is the unit vector out of \(\partial\Omega\). Let \(L^{2}(\Omega)\) be the space of functions \(u\in\Omega\), which are Lebesgue integrable, over \(\Omega\), with \(\Omega\subset\mathbb{R}^{2}\). We have  
\[L^{2}(\Omega)=\bigg{\{}u:\Omega\to\mathbb{R},\text{with}\int_{\Omega}|u(x,y)| ^{2}\text{d}\mu<\infty\bigg{\}}.\]  
It considers \(u\in L^{2}(\Omega)\), so \(u\) has weak derivatives of all orders. It is not possible to say that these weak derivatives also belong to \(L^{2}(\Omega)\). Thus, it is necessary to define the space \(H^{m}(\Omega)\), so that if the function \(u(x,y)\) belongs to \(H^{m}(\Omega)\), then \(u(x,y)\) and its derivatives\(\frac{\partial^{i}u}{\partial x^{i}}\) and \(\frac{\partial^{i}u}{\partial y^{i}}\), with \(1\leq i\leq m\), belong to \(L^{2}(\Omega)\)[14].