Dear Muhammad, usually the term regular is referred to a function sufficiently smooth for aims of PDE resolution. A classical solution for a nth order PDE defined on domain D, is a function in Cn(D), and in this sense regular and classical can be identified. A weak solution for a PDE is a solution in the sense of distributions, and weak solution can be a classical solution usually only in the case of regular domain D and smooth coefficients in PDE.

Gianluca