Proving a Partial Differential Equation (PDE) is Linear

In mathematics, a Partial Differential Equation (PDE) is a fundamental concept that describes how a quantity changes with respect to different variables in a system. Determining if a given PDE is linear is crucial for various applications, including physics, engineering, and economics. Understanding this concept is essential for applying appropriate solution methods, such as Fourier series, separation of variables, or Green's functions.

Definition and Notation

A PDE can be written as an operator acting on a function, denoted as L[f], where f is a solution to the PDE. The operator L is defined such that it transforms the function f into a new function, often representing other physical quantities like temperature, pressure, or velocity.

The notation for such a differential equation is:

L[f] g

where g is a given function or constant.

Testing for Linearity

To test whether a PDE is linear, we need to verify that the operator L satisfies the properties of linearity. A PDE is considered linear if the operator L satisfies the following:

Additivity: For any two functions f_1 and f_2, the operator L follows the rule:

L[f_1 f_2] L[f_1] L[f_2]

Homogeneity: For any function f_1, any scalar c, and f_2 defined as c f_1, the operator L satisfies:

L[c f_1] c L[f_1]

In our specific case, consider two solutions f_1 and f_2, then the following must hold:

L[f_1 f_2] L[f_1] L[f_2]

Additionally, for any scalar c, we have:

L[cf_1] c L[f_1]

Illustrative Example

Let's consider a simple linear PDE to illustrate the linearity property. Suppose we have the following PDE:

L[f] frac{partial^2 f}{partial x^2} frac{partial^2 f}{partial y^2} g

To prove that this PDE is linear, we apply the definition of linearity.

Step 1: Additivity

Consider two solutions f_1 and f_2. We need to show:

L[f_1 f_2] L[f_1] L[f_2]

Calculating L[f_1 f_2]:

L[f_1 f_2] frac{partial^2 (f_1 f_2)}{partial x^2} frac{partial^2 (f_1 f_2)}{partial y^2}

Using the linearity of differentiation:

L[f_1 f_2] frac{partial^2 f_1}{partial x^2} frac{partial^2 f_1}{partial y^2} frac{partial^2 f_2}{partial x^2} frac{partial^2 f_2}{partial y^2}

This can be written as:

L[f_1 f_2] L[f_1] L[f_2]

Step 2: Homogeneity

Consider any scalar c and a function f_1. We need to show:

L[cf_1] c L[f_1]

Calculating L[cf_1]:

L[cf_1] frac{partial^2 (cf_1)}{partial x^2} frac{partial^2 (cf_1)}{partial y^2}

Using the linearity of differentiation:

L[cf_1] c frac{partial^2 f_1}{partial x^2} c frac{partial^2 f_1}{partial y^2}

This can be written as:

L[cf_1] c left( frac{partial^2 f_1}{partial x^2} frac{partial^2 f_1}{partial y^2} right)

Thus,

L[cf_1] c L[f_1]

Conclusion

By verifying the additivity and homogeneity properties, we can conclude that the given PDE is indeed linear. Verifying these properties ensures that the PDE behaves in a predictable manner, making it amenable to various analytical and numerical techniques for solution.

Understanding the linearity of a PDE is crucial for mathematical modeling and analysis, especially in fields such as physics, engineering, and economics.