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Math Notes

University Mathematics

ConceptComplete

Classification and Examples - Core Definitions

Partial differential equations (PDEs) are equations involving partial derivatives of an unknown function with respect to multiple independent variables. They form the foundation of mathematical physics and appear throughout science and engineering.

DefinitionPartial Differential Equation

A partial differential equation is an equation of the form F(x1,…,xn,u,∂u∂x1,…,∂u∂xn,∂2u∂xi∂xj,…)=0F(x_1, \ldots, x_n, u, \frac{\partial u}{\partial x_1}, \ldots, \frac{\partial u}{\partial x_n}, \frac{\partial^2 u}{\partial x_i \partial x_j}, \ldots) = 0 where u=u(x1,…,xn)u = u(x_1, \ldots, x_n) is an unknown function of nn independent variables, and FF is a given function of its arguments.

The order of a PDE is the highest order of partial derivative appearing in the equation. A PDE is linear if it is linear in the unknown function and all its derivatives. The general form of a second-order linear PDE in two variables is: A∂2u∂x2+B∂2u∂x∂y+C∂2u∂y2+D∂u∂x+E∂u∂y+Fu=GA\frac{\partial^2 u}{\partial x^2} + B\frac{\partial^2 u}{\partial x \partial y} + C\frac{\partial^2 u}{\partial y^2} + D\frac{\partial u}{\partial x} + E\frac{\partial u}{\partial y} + Fu = G

DefinitionClassification of Second-Order PDEs

A second-order linear PDE with constant coefficients is classified by the discriminant Δ=B2−4AC\Delta = B^2 - 4AC as:

  • Elliptic if Δ<0\Delta < 0
  • Parabolic if Δ=0\Delta = 0
  • Hyperbolic if Δ>0\Delta > 0
ExampleClassic PDEs

The three fundamental types of second-order PDEs are:

  1. Laplace equation (elliptic): ∇2u=∂2u∂x2+∂2u∂y2=0\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0
  2. Heat equation (parabolic): ∂u∂t=k∇2u\frac{\partial u}{\partial t} = k\nabla^2 u
  3. Wave equation (hyperbolic): ∂2u∂t2=c2∇2u\frac{\partial^2 u}{\partial t^2} = c^2\nabla^2 u
Remark

The classification determines the fundamental nature of solutions: elliptic equations describe steady-state phenomena, parabolic equations model diffusion processes, and hyperbolic equations govern wave propagation. This distinction is essential for understanding solution behavior and choosing appropriate numerical methods.

Understanding the classification of PDEs is crucial because it dictates the type of boundary conditions needed for well-posedness, the qualitative behavior of solutions, and the appropriate analytical and numerical methods for solving them.