Introduction to Simple Partial Differential Equations

What is a Simple Partial Differential Equation?

A partial differential equation (PDE) is an equation involving the partial derivatives of an unknown function of several variables. While complex and intricate PDEs play crucial roles in physics, engineering, and other sciences, there are also simpler PDEs that serve as excellent starting points for understanding the basics. These simple PDEs often arise from basic physical principles and provide a foundation for more advanced study.

Example of a Simple PDE

One such simple PDE is discussed in the book Transport Phenomena by B. Bird, where basic PDEs are derived from first physical principles. Consider the following PDE on (mathbb{R}^n):

[frac{partial^n u}{partial x_1partial x_2cdotspartial x_n} 0]

This is a straightforward and easy PDE. However, to make it more concrete, we can consider a simpler case with only two variables.

Simple PDE in Two Variables

On (mathbb{R}^2), the PDE

[frac{partial^2 u}{partial x partial y} 0]

has a general solution given by

[u_{xy} Ax^B y]

where (A) and (B) are any two-times differentiable functions. Therefore, the following are all solutions to the PDE (1) in (mathbb{R}^2):

[u_1_{xy} alpha xy, alpha in mathbb{R}] [u_2_{xy} e^x e^{-2y}] [u_3_{xy} cosh x - 6 - 5 sin(pi y)]

The Laplacian: A Fundamental Simple PDE

Another simple and fundamental PDE is the Laplacian equation, which plays a significant role in many areas of physics and engineering. On (mathbb{R}^n), the Laplacian is given by:

[ abla^2 B 0]

The Laplacian in Cartesian coordinates is:

[frac{partial^2 B}{partial x^2} frac{partial^2 B}{partial y^2} frac{partial^2 B}{partial z^2} 0]

A simple solution to this equation is:

[B xyz]

If you have studied vector calculus, you would recognize this as:

[ abla cdot ( abla B) 0]

In physics, the Laplacian equation is a special case of the diffusion equation, which models the evolution of a scalar field over time. When the system is in equilibrium, the time derivative is zero:

[frac{partial B}{partial t} 0]

If, for example, (B) is a temperature, this PDE models how the temperature of a solid evolves over time. Under equilibrium conditions, it simplifies to:

[ abla^2 B 0]

To find the specific solution, you would need boundary conditions, such as:

(B(y 1) w) (B(x 0) u) Other relevant boundary conditions as needed.

These simple PDEs, including the Laplacian, form the bedrock of more complex physical models and are essential for building a strong foundation in the field of differential equations.