Ch_06 - PDE - Summary

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

This preview shows pages 1–6. Sign up to view the full content.

Chapter 6 Partial Differential Equations

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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.
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:

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 86

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online