1. COMPUTATIONAL MODELING FOR ENGINEERING MECN 6040 Professor: Dr. Omar E. Meza Castillo omeza@bayamon.inter.edu http://facultad.bayamon.inter.edu/omeza Department of Mechanical Engineering

2. INTRODUCTION TO THE THEORY OF PDEs

3. LEARNING OBJECTIVES 1.Be able to distinguish between the 3 classes of 2nd order, linear PDE's. Know the physical problems each class represents and the physical/mathematical characteristics of each. 2.Be able to describe the differences between finite- difference and finite-element methods for solving PDEs. 3.Be able to solve Elliptical (Laplace/Poisson) PDEs using finite differences. 4.Be able to solve Parabolic (Heat/Diffusion) PDEs using finite differences.

4. DEFINITIONS AND TERMINOLOGY DIFFERENTIAL EQUATION An equation containing the derivative of one or more dependent variables, with respect to one or more independent variables is said to be a differential equation (DE).

5. DEFINITIONS AND TERMINOLOGY DEFINITION OF A DERIVATIVE If y=f(x), the derivative of y or f(x) with respect to x is defined as The derivative is also denoted by y’, dy/dx or f’(x)

6. THE EXPONENTIAL FUNCTION  dependent variable: y  dependent variable: y  independent variable: x  independent variable: x DEFINITIONS AND TERMINOLOGY

7. Differential Equations are CLASSIFIED by type, order and linearity. TYPE There are two main types of differential equation: “ordinary” and “partial”. DEFINITIONS AND TERMINOLOGY

8. Ordinary differential equation (ODE) Differential equations that involve only ONE independent variable are called ordinary differential equations. Examples:,  only ordinary (or total ) derivatives DEFINITIONS AND TERMINOLOGY

9. Partial differential equation (PDE) Differential equations that involve two or more independent variables are called partial differential equations. Examples:  only partial derivatives DEFINITIONS AND TERMINOLOGY

10. ORDER The order of a differential equation is the order of the highest derivative found in the DE. second order first order second order first order DEFINITIONS AND TERMINOLOGY

11.  first order  second order DEFINITIONS AND TERMINOLOGY

12. LINEAR OR NONLINEAR An n -th order differential equation is said to be linear if the function is linear in the variables is linear in the variables

13. DEFINITIONS AND TERMINOLOGY   there are no multiplications among dependent variables and their derivatives. All coefficients are functions of independent variables.

14. or or  linear first-order ordinary differential equation  linear second-order ordinary differential equation  linear third-order ordinary differential equation

15. PDE'S DESCRIBE THE BEHAVIOR OF MANY ENGINEERING PHENOMENA: ▪ Wave propagation ▪ Fluid flow (air or liquid) ▪ Air around wings, helicopter blade, atmosphere ▪ Water in pipes or porous media ▪ Material transport and diffusion in air or water ▪ Weather: large system of coupled PDE's for momentum, pressure, moisture, heat, … ▪ Vibration ▪ Mechanics of solids: ▪ stress-strain in material, machine part, structure ▪ Heat flow and distribution ▪ Electric fields and potentials ▪ Diffusion of chemicals in air or water ▪ Electromagnetism and quantum mechanics

17. CLASIFIQUE LAS SIGUIENTES ECUACIONES: Solución (a)

18. Solución (b) Solución (c)