5.1 Classification and General Properties of PDEs

Partial differential equations (PDEs) are essential tools in mathematical physics. They model complex systems by describing how quantities change with multiple variables. Understanding PDEs helps us grasp the behavior of physical phenomena like heat flow, wave propagation, and fluid dynamics.

This section covers PDE classification, well-posed problems, and solution methods. We'll explore how to categorize PDEs, recognize their properties, and solve them using techniques like separation of variables. These skills are crucial for tackling real-world physics problems.

Classification and Properties of Partial Differential Equations (PDEs)

Classification of partial differential equations

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• Order of a PDE determined by the highest order partial derivative in the equation
  • First-order PDE has highest partial derivative of order one ($\frac{\partial u}{\partial t} + c\frac{\partial u}{\partial x} = 0$)
  • Second-order PDE has highest partial derivative of order two ($\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$)
• Linearity of a PDE depends on coefficients and nonhomogeneous term
  • Linear PDE has coefficients and nonhomogeneous term as functions of independent variables only ($a(x, y)\frac{\partial^2 u}{\partial x^2} + b(x, y)\frac{\partial^2 u}{\partial y^2} + c(x, y)u = f(x, y)$)
  • Nonlinear PDE has coefficients or nonhomogeneous term depending on the dependent variable or its derivatives ($\frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = 0$)
• Types of second-order linear PDEs classified based on the relationship between coefficients $A$, $B$, and $C$ in the equation $Au_{xx} + 2Bu_{xy} + Cu_{yy} + \text{lower order terms} = 0$
  • Elliptic PDE has $B^2 - AC < 0$ (Laplace's equation, $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0$)
  • Parabolic PDE has $B^2 - AC = 0$ (Heat equation, $\frac{\partial u}{\partial t} - \alpha^2\frac{\partial^2 u}{\partial x^2} = 0$)
  • Hyperbolic PDE has $B^2 - AC > 0$ (Wave equation, $\frac{\partial^2 u}{\partial t^2} - c^2\frac{\partial^2 u}{\partial x^2} = 0$)

Concept of well-posed problems

• Well-posed problems satisfy three conditions
  1. Existence of a solution
  2. Uniqueness of the solution
  3. Stability of the solution depends continuously on initial/boundary conditions and any nonhomogeneous terms
• Importance of well-posedness
  • Ensures the mathematical model is physically meaningful
  • Guarantees numerical methods can be used to approximate the solution

Characteristics and domains of dependence

• Characteristics are curves or surfaces along which information propagates in the solution of a PDE
  • Determined by the principal part (highest order terms) of the PDE
• Domain of dependence is the region in the independent variable space where the solution at a given point depends on the initial/boundary conditions
  • Determined by the characteristics of the PDE
• Elliptic PDEs have no real characteristics and the solution at a point depends on the entire boundary
• Parabolic PDEs have characteristics parallel to one of the coordinate axes and the solution depends on a portion of the boundary and initial condition
• Hyperbolic PDEs have real and distinct characteristics and the solution depends on a limited portion of the initial/boundary conditions

Separation of variables for PDEs

• Separation of variables assumes the solution can be written as a product of functions, each depending on a single independent variable ($u(x, t) = X(x)T(t)$)
  • Leads to ordinary differential equations (ODEs) for each function
  • Boundary conditions determine the constants in the solution
• Steps for applying separation of variables
  1. Assume a separable solution and substitute it into the PDE
  2. Separate the equation into terms, each depending on a single independent variable
  3. Equate each term to a constant (separation constant)
  4. Solve the resulting ODEs for each function
  5. Apply boundary conditions to determine constants and construct the general solution
  6. Use initial conditions (if applicable) to determine the specific solution
• Examples of PDEs solved by separation of variables
  • Heat equation $\frac{\partial u}{\partial t} = \alpha^2\frac{\partial^2 u}{\partial x^2}$ with boundary conditions $u(0, t) = u(L, t) = 0$
  • Wave equation $\frac{\partial^2 u}{\partial t^2} = c^2\frac{\partial^2 u}{\partial x^2}$ with boundary conditions $u(0, t) = u(L, t) = 0$