Introduction to Partial Differential Equations

- What are PDE's
- PDE's are Differential equations where the unknown function depends on more than one variable
- examples
- u
_{t}= u_{xx}(heat equation in one dimension) - u
_{t}= u_{xx}+ u_{yy}(heat equation in two dimensions) - u
_{rr}+ 1/r u_{r }+ 1/r_{2}u_{θθ}(Laplace's Equation in Polar Coordinates) - Why are PDE's Useful?
- relate space and time
- used to describe laws of physics
- Solving Partial Differential Equations
- Separation of Variables
- Integral Transforms
- Change of Coordinates
- Transformation of the Dependent Variable
- Numerical Methods
- Perturbation Methods
- Impulse Response
- Integral Equations
- Calculus of Varaiations
- 6 basic classifications of PDEs
- order of equation
- u
_{t}= u_{xx }(second order) - Number of variables
- u
_{t}= u_{xx }(two variables: x and t) - Linearity
- Linear equations are when all the dependent variables u and its derivates are not multiplied against one another or squared
- A u
_{xx }+B u_{xy }+C u_{yy }+D u_{x }+E u_{y }+F u = G - Where A,B,C,D,E,F,G are constants or given functions of x and y
- Homogeneity
- equations are homogeneous if G is 0 for all x and y, if it is nonzero it is non-homogeneous
- Kinds of Coefficients
- constant coefficients versus variable coefficients
- Three basic types of linear equations
- parabolic
- describes heat flow and diffusion
- satsify B
^{2 }- 4AC = 0 - hyperbolic
- describe vibrating systems
- B
^{2 }- 4AC > 0 - elliptic
- steady state phenomena
- B
^{2 }- 4AC < 0

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