Question

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

Short answer

The differential equation is parabolic.

Step-by-step solution

Identifying the Type of PDE

To classify the given partial differential equation (PDE), we analyze the highest order derivatives. The equation involves second and first order derivatives with respect to one variable and a first order derivative with respect to another variable. Since the highest order derivative is of second order, we are dealing with a second order PDE. However, the given equation lacks the second order derivative with respect to y, \( \frac{\partial^2 u}{\partial y^2} \) and mixed second order derivatives such as \( \frac{\partial^2 u}{\partial x \partial y} \) or \( \frac{\partial^2 u}{\partial y \partial x} \). The equation takes the general form \( A\frac{\partial^2 u}{\partial x^2} + B\frac{\partial u}{\partial x} + C\frac{\partial u}{\partial y} = 0 \), where A, B, C are constants. We can now compare this form to standard forms of second order PDEs to classify it.

Key concepts

Second Order Partial Differential Equations

Second order partial differential equations (PDEs) are equations that involve second order derivatives with respect to one or more variables. These equations provide a description of various phenomena such as heat conduction, wave propagation, and quantum mechanics, just to name a few.

A general second order PDE in two independent variables x and y can be represented as: \[A\frac{{\text{d}}^2 u}{{\text{d}}x^2} + B\frac{{\text{d}}^2 u}{{\text{d}}x{\text{d}}y} + C\frac{{\text{d}}^2 u}{{\text{d}}y^2} + D\frac{{\text{d}}u}{{\text{d}}x} + E\frac{{\text{d}}u}{{\text{d}}y} + Fu + G = 0\],where A, B, C, D, E, F, and G are functions of x and y, or constants, and u is the unknown function of x and y.

When approaching such equations, it’s essential to consider the highest order derivatives and their coefficients. In our exercise, we have a second order derivative with respect to x, \[\frac{{\text{d}}^2 u}{{\text{d}}x^2}\],and first order derivatives with respect to x and y. The lack of the second order derivative with respect to y and the absence of mixed derivative terms simplifies the classification process.

Elliptic Partial Differential Equations

Elliptic partial differential equations form a subclass of second order PDEs characterized by certain properties of their coefficients. A famous example of an elliptic PDE is the Laplace equation, given by\[abla^2 u = 0\],where \(abla^2\)is the Laplacian operator. To identify an elliptic PDE, one must look at the second order derivative coefficients. These equations do not display any time-evolving behavior and are typically associated with steady-state processes.

In the exercise presented, we lack sufficient information to conclude if the equation is elliptic since we need to consider the equation's discriminant, derived from the coefficients of the second order derivatives. Specifically, an elliptic PDE must fulfill the condition that the discriminant, which is \[B^2 - 4AC\],is negative. In context, our exercise equation does not present a term like \[\frac{{\text{d}}^2 u}{{\text{d}}y^2}\],which would be necessary to complete the analysis for ellipticity.

PDE Classification

Classification of PDEs is crucial because it informs us about the nature of the solutions and the appropriate methods for solving them. Second order PDEs are primarily classified as elliptic, parabolic, or hyperbolic, each relating to different physical phenomena.

The classification of a second order PDE is determined by the sign of the discriminant within the equation’s leading part—the part involving the highest order derivatives. For an elliptic PDE, the discriminant is negative, for a parabolic PDE it is zero, and for a hyperbolic PDE, it’s positive.

Returning to our exercise, to classify the given PDE, we would typically calculate the discriminant. The lack of certain second order terms makes it difficult to classify using standard methods. Instead, we can infer properties based on existing terms and the behavior they suggest. However, with the information given, it’s challenging to categorically classify the PDE without additional context or terms.