Mathematical Physics Vol 1

7. Partial differential equations

Introduction Partial differential equations (PDE) occur in various problems of physics, geometry, and technology 1 in cases where the function describing a given process or phenomenon depends on two or more independent variables. Note that only simpler problems can be described by ordinary differential equations. Problems in the field of fluid mechanics, solid state mechanics, heat propagation, electromagnetism, etc., are described by partial differential equations. The theory of partial differential equations is one of the best examples of the interconnect edness between mathematics and other scientific fields. Solving these equations is not only of formal mathematical interest but also a condition for understanding the process in the natural and other sciences. Thus, in the eyes of mathematicians, the solution of equations is a number or a function. Seen through the eyes of a physicist or biologist, a solution describes a process, and it thus has a physical meaning. There are no general methods for solving second-order partial differential equations, unlike ordinary differential equations and first-order partial differential equations, with one unknown function. 2 The reason lies primarily in the fact that second-order partial differential equations appeared in mathematical physics and mechanics almost at the very beginning of the creation of mathematical analysis (more than two centuries ago). In addition, the solutions of these equations had to satisfy both the initial and boundary conditions, so they (solutions) were sought from problem to problem. The problems posed in this way attracted the attention of mathematicians, but they sought solutions for specific equations, as general methods were not sufficiently developed. However, it is interesting that even the development of science in recent times has not brought general methods for solving partial equations of the second order. It should 1 Recently, they have also appeared in other fields of science, such as medicine, for mathematical modeling of the work of certain organs. 2 Solving these equations comes down to integrating ordinary differential equations.