Integral and weak forms

Conservation equations can be also 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 form, extending the class of admissible solutions to include discontinuous solutions. By integrating in any space-time domain the current density form in 1-D space:

${\displaystyle y_t+j_x(y)=0}$

and by using Green's theorem, the integral form is:

${\displaystyle {\int <!--\; \int \; -->}_{-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->}}^{\mathrm{\infty <!--\; \infty \; -->}}ydx+{\int <!--\; \int \; -->}_0^{\mathrm{\infty <!--\; \infty \; -->}}j(y)dt=0}$

In a similar fashion, for the scalar multidimensional space, the integral form is:

${\displaystyle \oint <!--\; \oint \; -->y{d}^{N}r+j(y)dt=0}$

where the line integration is performed along the boundary of the domain, in an anticlock-wise manner.

Moreover, by defining a test function φ(r,t) continuously differentiable both in time and space with compact support, the weak form can be obtained pivoting on the initial condition. In 1-D space it is:

${\displaystyle {\int <!--\; \int \; -->}_0^{\mathrm{\infty <!--\; \infty \; -->}}{\int <!--\; \int \; -->}_{-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->}}^{\mathrm{\infty <!--\; \infty \; -->}}{\mathrm{\varphi <!--\; \varphi \; -->}}_{t}y+{\mathrm{\varphi <!--\; \varphi \; -->}}_{x}j(y)dxdt=-<!--\; -\; -->{\int <!--\; \int \; -->}_{-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->}}^{\mathrm{\infty <!--\; \infty \; -->}}\mathrm{\varphi <!--\; \varphi \; -->}(x,0)y(x,0)dx}$

Note that 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.