- #1
imranq
- 57
- 1
Homework Statement
Hey I'm trying to get a sense of this problem, just starting pde class:
[tex]au_x+bu_y+cu=0[/tex]
Homework Equations
The Attempt at a Solution
Dunno what to do with that last term
A partial differential equation is a mathematical equation that includes partial derivatives of a multivariable function. It describes the relationship between the function and its derivatives, and is commonly used to model physical systems in fields such as physics, engineering, and economics.
This notation represents a linear partial differential equation with two independent variables, x and y, and one dependent variable, u. The coefficients a, b, and c determine the relationship between the partial derivatives of u with respect to x and y, and the value of u itself. The equation as a whole is typically solved for u.
There are various methods for solving PDEs, including separation of variables, the method of characteristics, and numerical methods such as finite difference or finite element methods. The specific method used depends on the type of PDE and the boundary/initial conditions given.
PDEs are important in science because they provide a powerful tool for modeling and understanding complex physical systems. They allow us to describe how a system changes over time and space, and can be used to make predictions about its behavior. PDEs are used in a wide range of fields including physics, chemistry, biology, and engineering.
PDEs have many real-world applications, such as modeling heat transfer, fluid dynamics, electromagnetic waves, and diffusion processes. They are also used in finance to model stock prices and in geology to study the behavior of earthquakes. PDEs are essential in many areas of science and engineering where understanding the behavior of complex systems is critical.