Origin Of Theory Of Partial Differential Equations

The purpose of this dissertation is to probe and to analyze the historical development of theory of PDEs systematically on the basis of reading original sources and of investigating previous researches on the subject. The major contributions made in the dissertation are as follows.The history of some basic problems, from the times of Cauchy to the times of Hadamard and Petrovskii, such as the existence and the uniqueness of the Cauchy problem's solution of general PDEs, the existence of a solution of the Dirichlet problem of the Laplace equations ,the concept of well-posed solution of PDEs and the classification of PDEs is systematically and entirely studied and is viewed, and it is the first time to do so in the study of the history of PDEs.The major contributions from the most representative authors in the different periods of the history of PDEs are evaluated as objectively as possible about their motivations of the work, origins of ideas, approaches and characteristic by all-sided using comparing approach. For example, C. B. KoBaneBCKaa's innovation and its historical meaning are disclosed by being compared with Cauchy's results and contrasted with Weierstrass' work. The difference between the classification of du Bois-Reymond and that of Petrovskii is pointed out by comparing with each other.The establishment of the concept of well-posedness is discussed and it is concluded that Hadamard set out the concept in four steps: germination of the idea, giving the notion "well-posed", the formation of the concept of continuity of the solution with respect to the data, and establishing the concept of well-posedness.The history of the Dirichelet principle is investigated in the point of view of the existence. It is concluded that Riemann was a old style mathematician, and Weierstrass' existence meant a new king of thinking, so the latter's critic to the former meant the interaction of two different idea, which was helpful to the development of the theory of PDEs.