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.

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..