Introduction

The following are notes from Partial Differential Equations for Scientists and Engineers written by Farlow.

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/r₂ 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 B² - 4AC = 0
• hyperbolic
• describe vibrating systems
• B² - 4AC > 0
• elliptic
• steady state phenomena
• B² - 4AC < 0

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