Difference between revisions of "NumMethodsPDEs"
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=Looking at Nature= | =Looking at Nature= | ||
| + | |||
| + | <math>u_x = \frac{\partial u}{\partial x}</math> | ||
A second-order linear equation in two variables: | A second-order linear equation in two variables: | ||
| − | <math> Au_{xx} + Bu_{xy} + Cu_{yy} + Du_x + Eu_y + Fu = G </math> | + | <math>Au_{xx} + Bu_{xy} + Cu_{yy} + Du_x + Eu_y + Fu = G</math> |
* '''Parabolic''': <math>B^2 - 4AC = 0</math>. This family of equations describe heat flow and diffusion processes. | * '''Parabolic''': <math>B^2 - 4AC = 0</math>. This family of equations describe heat flow and diffusion processes. | ||
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Taylor Series: | Taylor Series: | ||
| − | <math> f(x + h) = f(x) + \frac{f'(x)}{1!}h + \frac{f^{(2)}(x)}{2!}h^2 + \cdots + \frac{f^{(n)}(x)}{n!}h^n + R_n(x)</math> | + | <math>f(x + h) = f(x) + \frac{f'(x)}{1!}h + \frac{f^{(2)}(x)}{2!}h^2 + \cdots + \frac{f^{(n)}(x)}{n!}h^n + R_n(x)</math> |
Revision as of 17:29, 17 January 2011
Numerical Methods for PDEs: Solving PDEs on a computer
Introduction
Looking at Nature
[math]\displaystyle{ u_x = \frac{\partial u}{\partial x} }[/math]
A second-order linear equation in two variables:
[math]\displaystyle{ Au_{xx} + Bu_{xy} + Cu_{yy} + Du_x + Eu_y + Fu = G }[/math]
- Parabolic: [math]\displaystyle{ B^2 - 4AC = 0 }[/math]. This family of equations describe heat flow and diffusion processes.
- Hyperbolic: [math]\displaystyle{ B^2 - 4AC \gt 0 }[/math]. Describe vibrating systems and wave motion.
- Elliptic: [math]\displaystyle{ B^2 - 4AC \lt 0 }[/math]. Steady-state phenomena.
Taylor Series:
[math]\displaystyle{ f(x + h) = f(x) + \frac{f'(x)}{1!}h + \frac{f^{(2)}(x)}{2!}h^2 + \cdots + \frac{f^{(n)}(x)}{n!}h^n + R_n(x) }[/math]