Question

In each of Problems 21 through 24 determine the order of the given partial differential equation; also state whether the equation is linear or nonlinear. Partial derivatives are denoted by subscripts. $$ u_{x x}+u_{y y}+u_{z z}=0 $$

Short answer

Answer: The given partial differential equation is of order 2 and is linear.

Step-by-step solution

Determine the order of the PDE

The given partial differential equation is: $$ u_{xx} + u_{yy} + u_{zz} = 0 $$ Here, \(u_{xx}\) denotes the second partial derivative of \(u\) with respect to \(x\), \(u_{yy}\) denotes the second partial derivative of \(u\) with respect to \(y\), and \(u_{zz}\) denotes the second partial derivative of \(u\) with respect to \(z\). The highest order of derivatives in this equation is 2, therefore, the order of the given PDE is 2.

Identify if the PDE is linear or nonlinear

To determine if the PDE is linear or nonlinear, we need to examine the coefficients of the derivatives and check if there are any products or nonlinear functions of the unknown function \(u\). In this case, the coefficients of the derivatives are all 1, and there are no products or nonlinear functions of \(u\). Therefore, the given PDE is linear. The given partial differential equation is of order 2 and is linear.

Key concepts

Linear Partial Differential Equation

In the realm of mathematics, partial differential equations (PDEs) are a type of differential equation that contain unknown multivariable functions and their partial derivatives. A linear partial differential equation is one in which the dependent variable and all its derivatives appear linearly, meaning they are not multiplied together, raised to any power other than one, or composed with any non-linear functions.

The equation provided in the exercise, \( u_{xx} + u_{yy} + u_{zz} = 0 \), is a classic example of a linear PDE. The linearity comes from the fact that the highest exponent of any of the function's derivatives is one, and the coefficients of \( u_{xx} \), \( u_{yy} \), and \( u_{zz} \) are constants (in this case, all equal to 1). Moreover, the terms do not contain any products or compositions of the derivatives and the function \( u \). Linear PDEs are essential as they frequently appear in physics and engineering, representing various phenomena like waves, heat flow, and quantum mechanics. Their properties often allow for more straightforward analytical or numerical solutions compared to nonlinear counterparts.

Order of a Differential Equation

The order of a differential equation is determined by the highest derivative that appears in the equation. In the case of partial differential equations, this means looking at the derivatives with respect to all variables involved and identifying the one with the highest order.

For the equation \( u_{xx} + u_{yy} + u_{zz} = 0 \), the subscript \( xx \) in \( u_{xx} \) indicates a second derivative with respect to \( x \) is present, which similarly applies for \( u_{yy} \) and \( u_{zz} \) with respect to \( y \) and \( z \), respectively. Since the second derivatives are the highest present, the equation is deemed to be of second order. The order is an essential characteristic as it influences the complexity of the problem and the methods employed for finding solutions.

Second Partial Derivatives

The concept of second partial derivatives arises in the context of functions with several independent variables. These derivatives measure how the function changes as each variable is varied, while all other variables are held constant.

In our equation, \( u_{xx} \) is the second partial derivative of \( u \) with respect to \( x \) twice, indicating how the function's rate of change with respect to \( x \) itself changes. Similarly, \( u_{yy} \) and \( u_{zz} \) represent the rates of change in the \( y \) and \( z \) directions, respectively. These second derivatives provide vital information about the function's curvature and behavior in different directions. Such information is crucial in multiple disciplines, for instance, in engineering for assessing the stress on materials or in economics for evaluating how an outcome might change with various influencing factors.