Question

Classify the following differential equations (as elliptic, etc.) $$ \frac{\partial^{2} u}{\partial x^{2}}-2 \frac{\partial^{2} u}{\partial x \partial y}+\frac{\partial^{2} u}{\partial y^{2}}=0 $$

Short answer

The given PDE is hyperbolic.

Step-by-step solution

Identify the Partial Differential Equation (PDE)

Firstly, recognize that the given equation is a second-order linear partial differential equation (PDE).

Determine the Classification of the PDE

The classification of a second-order linear PDE depends on the discriminant \( B^{2} - 4AC \) using the general form \( Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_{x} + Eu_{y} + Fu = G \) where \( A \) is the coefficient of \( u_{xx} \) (second partial derivative with respect to \( x \) alone), \( B \) is the coefficient of \( u_{xy} \) (mixed partial derivative), and \( C \) is the coefficient of \( u_{yy} \) (second partial derivative with respect to \( y \) alone).

Key concepts

Elliptic Partial Differential Equations

Elliptic partial differential equations (PDEs) are a class of equations that arise in a variety of scientific fields, such as physics and engineering. They are essential for describing phenomena like heat distribution, electrostatics, and steady-state fluid flow.

An elliptic PDE is generally characterized by the absence of time-dependency and by having solutions that are smooth, provided the boundary conditions are also smooth. A classic example of an elliptic PDE is Laplace's equation, which appears in many physical theories.

The equation in our exercise is an example of a second-order linear elliptic PDE, if the discriminant condition for an elliptic equation is satisfied. This condition is that the discriminant must be negative, which means the coefficients of the equation must satisfy the inequality \( B^2 - 4AC < 0 \). We will assess this in the following sections with regards to the provided equation.

Second-order Linear PDEs

Second-order linear PDEs are those partial differential equations that involve second derivatives of the unknown function with respect to the independent variables. These PDEs are classified based on their behavior and the properties of their solutions into three main types: elliptic, hyperbolic, and parabolic.

The fundamental form of a second-order linear PDE is presented as \( Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_{x} + Eu_{y} + Fu = G \), where the coefficients can be functions of the independent variables, constants, or a mixture of both. The \'linear\' term suggests that the unknown function and its derivatives appear linearly in the equation.

In the particular equation from the exercise, there are no first-order derivative terms or 'forcing term' involved. This simplification helps in direct comparison to the canonical form. The solution to such PDEs heavily relies on boundary and/or initial conditions provided with the problem.

Partial Differential Equation Discriminant

The discriminant is a crucial aspect in classifying second-order linear PDEs. It is determined by examining the coefficients of the highest-order derivatives in the equation. For the general second-order linear PDE \( Au_{xx} + 2Bu_{xy} + Cu_{yy} + \ldots \), classified as elliptic, hyperbolic, or parabolic, the discriminant is given by \( B^2 - 4AC \).

Based on the discriminant value:\( B^2 - 4AC \), the classification is as follows:

• If \( B^2 - 4AC < 0 \), the equation is elliptic.
• If \( B^2 - 4AC = 0 \), the equation is parabolic.
• If \( B^2 - 4AC > 0 \), the equation is hyperbolic.

For the exercise provided, the coefficients can be identified as \( A = 1 \), \( B = -1 \), and \( C = 1 \). Therefore, the discriminant is \( (-1)^2 - 4 \times 1 \times 1 = 1 - 4 = -3 \), which is less than zero. This confirms that the PDE is indeed elliptic, matching the characteristics of the elliptic type mentioned in the previous sections.