Ch_06 - PDE Supplement

Ch_06 - PDE Supplement - Partial Differential Equations...

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Partial Differential Equations Supplements

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Table of Contents 1. Introduction 1.1 Classifications 1.2 Diffusion Type PDE’s 1.3 Boundary Conditions 1.4 Solution Methods 1.5 Integral Transforms 1.6 Stürm-Liouville Problems 1.7 Legendre’s Equation. Legendre Polynomials 1.8 Bessel’s Equation. Bessel Functions 1.9 Orthogonal Functions 1.10 Orthogonal Series Part I: Cartesian Coordinates System 2. One Dimensional Heat Equation in Cartesian Coordinates 2.1 Dimensionless Parameters 2.2 Homogeneous IBVP in 1-D Finite Regions 2.2A Method of Separation of Variables 2.2B Solution with Normalized Eigenfunction 2.2C Evaluation of Kernel and Eigenvalues in 1-D Finite Region 2.2D Integral Transform Method 2.2F Finite Fourier Cosine and Sine Transform Method 2.2E Laplace Transform Method 2.3 Homogeneous IBVP in 1-D Semi-Infinite Region 2.3A Method of Combination of Variables 2.3B Method of Separation of Variables 2.3C Evaluation of Kernel in 1-D Semi-Infinite Region 2.3D Fourier Cosine and Sine Integral Method 2.3E Fourier Cosine and Sine Transform Method 2.3F Laplace Transform Method 2.4 Homogeneous IBVP in 1-D Infinite Region 2.4A Fourier Integral Method 2.4B Fourier Transform Method
Partial Differential Equations Table of Contents 3. Transient Problems 3.1 Nonhomogeneous BVP of Heat Conduction in Finite Regions 3.2 3.3 3.4 3.5 3.6 3.7 3.8

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Partial Differential Equations 1. Introduction. Some Definitions, Formulas, Methods, and Solutions 1.1 Classifications Definition: Characteristic Equation For any partial differential equation of two independent variables of x and y , 222 22 ,,, , uuu u u AB C F x y u x yx y xy         If 2 4 0 BA C  : Hyperbolic Partial Differential Equation (wave equation) 2 0 C  : Parabolic Partial Differential Equation (1-D heat equation) 2 0 C  : Elliptic Partial Differential Equation (Laplace equation) Types of Partial Differential Equations Hyperbolic PDE 2 tt xx uc u : One dimensional vibrating string  2 , tt xx uf x t  : Wave equation with forced vibration 2 tt xx t u h u  : One dimensional vibrating string with friction 2 tt xx t u h u k u : Transmission line equation tt u  : Wave equation in higher dimensions tt t u h u  : Wave equation with friction Parabolic PDE tx x uu : One dimensional diffusion equation   , x f x t : Diffusion with heat source (or loss) x k u : Diffusion with lateral heat-concentration loss x x h u : Diffusion-convection equation Elliptic PDE 2 0 u  : Laplace’s equation 2 uk : Poisson’s equation 0   : Helmholtz’s equation 2 0 ukEVu  : Schrodinger’s equation
Partial Differential Equations 1. Introduction Polyanin, Manzhirov, “ Handbook of Mathematics for Engineers & Scientists ”, CRC, 2007, Part 1, Ch.14.

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This note was uploaded on 04/27/2011 for the course CHEM 101 taught by Professor Suh during the Spring '11 term at University of Toronto- Toronto.

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Ch_06 - PDE Supplement - Partial Differential Equations...

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