Presentation3 partial differentials equation
Download as PPT, PDF5 likes1,160 views
This document provides an introduction to partial differential equations (PDEs). It defines PDEs as equations that contain derivatives of unknown functions of several variables and one or more partial derivatives. The solutions to PDEs are differentiable functions that satisfy boundary or initial conditions. PDEs are often used to express laws of physics. Examples of common PDEs discussed include the Laplace equation, Poisson equation, wave equation, heat equation, and diffusion equation.
Presentation3 partial differentials equation
- 1. PARTIAL DIFFERENTIAL EQUATIONS Dosen: Retnari Dian Mudiastuti, ST., M.Si
- 2. BASIC CONCEPTS & IDEAS Partial Differential Equations: equations that contain derivatives of some unknown functions of several variables and one or more of its partial derivatives the solution from a PDE is a differentiable function that satisfy some extra conditions of PDE, usually called either boundary conditions or initial conditions. Often to express some laws of physics which are basic to most science and engineering subjects Example: differential equation the solution
- 3. Example of PDE: Laplace Equation Imagine a solid object made of some uniform heat-conducting material (say a solid metal ball), and imagine a steady temperature distribution on its surface is maintained somehow (say with some arrangement of wires and thermostats). Then after a while the temperature at each point inside the ball will come to equilibrium (steady state), and the resulting temperature function inside the ball will then satisfy Laplace’s equation. Solution: any steady-state temperature distribution in three space
- 4. Example of PDE: POISSON EQUATION If in the temperature model we include also heat sources and sinks in the region, but still unchanging over time, the temperature function satisfies the Poisson Equation, Where f is some given function related to the sources and sinks
- 5. Example of PDE: WAVE EQUATION Here x is the space variable, t is the time, and c is the velocity with which the wave travels (depends on the medium and the type of wave, such as light, sound,etc) A solution that satisfy it:
- 6. Example of PDE: HEAT EQUATION A solution that satisfy it: represents a time-varying temperature distribution in say a uniform conducting metal rod, with insulated sides.
- 7. Example of PDE: DIFFUSION EQUATION A solution that satisfy it: represents the concentration of a dissolved substance diffusing along a uniform tube filled with liquid, or of a gas diffusing down a uniform pipe
- 8. Exercises on Partial Derivatives..
- 9. Answers..
- 10. Exercises on Chain Rule
- 11. Answers..
- 12. More exercises on PDE…
- 13. Answer..