Ch_06 - PDE - Summary

Ch_06 - PDE - Summary - Chapter 6 Partial Differential...

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Chapter 6 Partial Differential Equations
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Table of Contents 1. Classification of Partial Differential Equations Part I: Cartesian Coordinate System 2. Hyperbolic Equations in Cartesian Coordinates 2.1 Separation of Variables: Use of Fourier Series 2.2 D’Alembert’s Solution of the Wave Equations 3. Parabolic Equations in Cartesian Coordinates 3.1 1-D Finite Regions; Solution by Fourier Series 3.2 1-D Semi-Infinite Region; Solution by Combination of Variables 3.3 1-D Semi-Infinite Region; Solution by Fourier Cosine and Sine Transforms 3.4 1-D Finite Region; Solution by Fourier Integrals 3.5 1-D Finite Region; Solution by Fourier Transforms 3.6 Transient Problems (Non-Homogeneous B.C’s) 3.7 Parabolic Equations with Generation 3.8 1-D Heat Equation with Convection 4. Multi-dimensional Steady State Problems 4.1 Solution by Fourier Series 4.2 Solution by Fourier Integrals and Transforms 4.3 2-D Poisson Equations (Non-homogeneous PDE) 5. Two Dimensional Partial Differential Equations 5.1 2-D Hyperbolic Equations 5.2 2-D Homogeneous Parabolic Equations 5.3 2-D Transient Problems (Non-Homogeneous B.C’s) Part II: Cylindrical and Spherical Coordinate System 6. Homogeneous Problems in Cylindrical Coordinates 6.1 Homogeneous Wave Equations in , rt Variables 6.2 Homogeneous Heat Equations in Variables 6.3 Steady State Problems in r θ Variables 6.4 Steady State Problems in rz Variables
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Table of Contents 7. Non-Homogeneous Problems in Cylindrical Coordinates 7.1 Heat Equations in , rt Variables with Heat Generation 8. Two Dimensional Unsteady Problems in Cylindrical Coordinates 8.1 Homogeneous Equations in , , θ Variables 8.2 Homogeneous Equations in rzt Variables 9. Homogeneous Problems in Spherical Coordinates 9.1 Homogeneous Equations in Variables 9.2 Steady State Radial Conduction Appendix
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Partial Differential Equations If there are two or more independent variables and equation contains differential coefficients with respect to each of these, the equation is said to be a “ Partial Differential Equation Let (,) uu x t = be a function of 2 variables of x and t . The followings are partial differential equations: 2 2 2 22 2 2 2 i) 0 ii) iii) iv) 0 xt tx t x x x x ∂∂ += =+ = + ++ = ± Linear Operator L If L satisfies the following conditions, then, {} x t u = + L is a linear operator. i.e., { } { } { } 12 1 2 (,) ux t ux t t t cuxt c uxt + = LL L Therefore, all above equations are linear. Non-linear terms are such like: x , x t , 2 () x u In above equations: i) is linear and homogeneous ii) is linear and non-homogeneous iii) is also linear and homogeneous. In equation ii) or iv), 2 or + are non-homogeneous term.
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Partial Differential Equations 1. Classification of PDEs 1. Classification of Partial Differential Equations ± Definition: For any partial differential equation of two independent variables of x and y , 222 22 (, ) uuu u u A BC D E F u G x y xy x y ∂∂∂∂ ++ + + + = ∂∂ 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) EXAMPLE:
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Ch_06 - PDE - Summary - Chapter 6 Partial Differential...

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