Editing Conservation law (section)

==Integral and weak forms==
Conservation equations can usually also be expressed in integral form: the advantage of the latter is substantially that it requires less smoothness of the solution, which paves the way to [[weak formulation|weak form]], extending the class of admissible solutions to include discontinuous solutions.<ref name="Toro"/>{{rp|p=62–63}} By integrating in any space-time domain the current density form in 1-D space:
<math display="block"> y_t + j_x (y)= 0 </math>
and by using [[Green's theorem]], the integral form is:
<math display="block"> \int_{- \infty}^\infty y \, dx + \int_0^\infty j (y) \, dt = 0 </math>

In a similar fashion, for the scalar multidimensional space, the integral form is:
<math display="block"> \oint \left[y \, d^N r + j (y) \, dt\right] = 0 </math>
where the line integration is performed along the boundary of the domain, in an anticlockwise manner.<ref name="Toro" />{{rp|p=62–63}}

Moreover, by defining a [[test function]] ''φ''('''r''',''t'') continuously differentiable both in time and space with compact support, the [[weak formulation|weak form]] can be obtained pivoting on the [[initial condition]]. In 1-D space it is:
<math display="block"> \int_0^\infty \int_{-\infty}^\infty \phi_t y + \phi_x j(y) \,dx \,dt = - \int_{-\infty}^\infty \phi(x,0) y(x,0) \, dx </math>

In the weak form all the partial derivatives of the density and current density have been passed on to the test function, which with the former hypothesis is sufficiently smooth to admit these derivatives.<ref name="Toro"/>{{rp|p=62–63}}