Generalized solutions of a periodic Goursat problem (Q2747067)

Following, P. R. Garabedian (1964) and E. Goursat (1956) and the author's two papers (1995) and (1997), the Goursat problem in \(\mathbb{R}^2\) is NEWLINE\[NEWLINE{\partial^2u\over\partial x \partial y} (x,y)= f(x,y,u(x, y));\;u(x,0)= v(x);\;u(0,y)= w(y),NEWLINE\]NEWLINE where \(f\) is a complex valued function on \(\mathbb{R}^2\times \mathbb{C}\), smooth in the four underlying real variables, \(v\) and \(w\) intend to belong to some \(2\pi\)-periodic functions space on \(\mathbb{R}\), where functions are considered in a generalized sense. The condition \(v(0)= w(0)\) holds because of the last equalities of the system. If \(v\) and \(w\) are distributions, it is not always true that this condition make sense. It is also observed that problem may arise from the possible nonlinearities carried by \(f\).NEWLINENEWLINENEWLINEThe credibility of the investigations carried out by the author lies in the fact that, the global investigation of such a Goursat problem requires a more general framework than the one of distribution theory. Thus, it demands the study of a problem which may be solved by considering the theory of new generalized functions [cf. J. F. Colombeau (1985) and M. Oberguggenberger (1992)]. Keeping these concepts in view, the author has chosen to work in an algebra of periodic generalized functions. The author's publications of (1995) and (1996) must be referred to for better comprehension of the problem of this article. The first section contains, very explicitly, necessary results required on the differential algebras. The second section is devoted to the study of generalized Goursat problem, and results on existence and uniqueness of the solution \(u\). The problem is dealt with lucid expressions and simple mathematical concepts. Some typographical errors have been observed in the paper.