Équations de la Physique Mathématiques
Partial differential equations, abbreviated as “PDEs” in what follows, constitute an important branch of applied mathematics. They express, in the form of equalities, the relationships that the partial derivatives of an unknown function u of several variables must satisfy in order to describe a physical phenomenon and meet a prescribed condition.
We usually classify partial differential equations into three fundamental categories: elliptic, parabolic, and hyperbolic equations.
Physics, biology, and engineering sciences require the ability to solve a wide variety of partial differential equations. The modeling of a real-world problem relies on the laws of physics (mechanics, thermodynamics, electromagnetism, acoustics, etc.), which are generally formulated in terms of balance laws that translate mathematically into ordinary differential equations or partial differential equations.
Partial differential equations also arise in many other fields: in chemistry for modeling reactions, in economics to study market behavior, and in finance to analyze derivative products (options and bonds).
Our work is divided into several chapters. In the first chapter, we present a review of vector analysis and the classification of partial differential equations. Next, we explain the method of characteristics and apply it to PDEs, in particular first- and second-order PDEs in ℝ².
The third chapter introduces the method of separation of variables, applied to the heat equation, the wave equation, and Laplace’s equation. The fourth chapter presents the finite difference method applied to the three types of partial differential equations—elliptic, parabolic, and hyperbolic—in one and two dimensions.
Finally, the fifth and last chapter is devoted to the finite difference method applied again to the three types of partial differential equations—elliptic, parabolic, and hyperbolic (as above).
- المعلم: Sabit Souhila