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
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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
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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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