Distributional solution
A distributional solution or solution in the distributional sense is a distribution that has the same property for partial differential equations as the regular distribution generated by a classical solution . It generalizes classic solutions in the sense that for every classic PDGL solution, the associated distribution is a distributional solution.
definition
Be a distribution, and
a linear partial differential operator . is called a distributional solution of the partial differential equation if and only if:
In this equation, a linear partial differential operator on distributions is understood.
If there is a distributional solution of , and there is a locally integrable function , so that (ie if there is a regular distribution that is generated by), then the function is sometimes also called (instead of ) a distributional solution.
properties
Distributional solution from a classic solution
Is a classic solution from
- ,
so the regular distribution generated by it is always a distributional solution.
Classic solution from a distribution solution
Be , ie is a Green function for . If - times is continuously differentiable and - times continuously differentiable, and if then , where is the level of the highest derivative that occurs in, then is
a classic solution from
- .
literature
- Lars Hörmander : The Analysis of Linear Partial Differential Operators. Volume 1: Distribution Theory and Fourier Analysis. Second edition. Springer-Verlag, Berlin et al. 1990, ISBN 3-540-52345-6 ( basic teaching of mathematical sciences 256).
- Werner: functional analysis, chapter on locally convex spaces