This lesson covers the mathematical classification of partial differential equations and their physical behavior, particularly in the context of fluid dynamics and aerodynamics. It delves into the three broad categories of partial differential equations: elliptic, parabolic, and hyperbolic, and explains why this classification is crucial for developing numerical schemes in computational aerodynamics or fluid dynamics. The lesson also discusses the physical basis of these equations, their behavior, and the type of solutions they can accommodate. For instance, it explains how elliptic equations are used for equilibrium problems, while parabolic and hyperbolic equations are used for marching or propagation problems.

- Partial differential equations are classified into three categories: elliptic, parabolic, and hyperbolic. This classification is essential for developing appropriate numerical schemes in computational aerodynamics or fluid dynamics.
- Elliptic equations are used for equilibrium problems, such as steady-state problems, and are characterized by smooth solutions, even if the boundary conditions are discontinuous.
- Parabolic and hyperbolic equations are used for marching or propagation problems, such as transient heat transfer or wave propagation phenomena. These equations can accommodate discontinuities in the solution domain.
- The physical behavior of these equations is crucial in devising numerical schemes. For instance, in elliptic equations, information propagates in all directions at infinite velocities, while in hyperbolic problems, the speed of disturbance propagation is finite.

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