1. PARTIAL DIFFERENTIAL EQUATIONS

2. Introduction
Given a function u that depends on both x and y, the partial derivatives of u w.r.t. x and y are:

3. An equation involving partial derivatives of an unknown function of two or more independent variables is called Partial Differential Equation (PDE). Examples:
The order of a PDE is that of the highest-order partial derivative appearing in the equation.

4. A PDE is linear if it is linear in the unknown function and all its derivatives, with coefficients depending only on the independent variables
e.g.
x’’ + ax’ + bx + c = 0 – linear
x’ = t2x – linear
x’’ = 1/x – nonlinear

5. For linear, two independent variables second order equations can be expressed as:
where A, B and C are functions of x and y and D is a function of x, y, ¶u/¶x and ¶u/¶y.
Above equation can be classified into categories in the next slide based on values of A, B, and C.

6. B2 – 4AC Category Example < 0 Elliptic
Laplace equation (Steady state with two spatial dimension)
= 0
Parabolic
Heat conduction equation (time variable with one spatial dimension)
> 0
Hyperbolic
Wave equation (time variable with one spatial dimension)

7. Elliptic Equations
Typically used to characterize steady-state distribution of an unknown in two spatial dimensions.

8. Laplace Equation
The PDE as an expression of the conservation of energy

9. Need to reformulate the equation in terms of temperature
Need to reformulate the equation in terms of temperature. Use Fourier’s Law:
and
substituting back results in
(Laplace equation)

10. Parabolic Equations Heat conduction
Hot
Cool
Heat balance (the amount of heat stored in the element) over a unit time, Dt

11. Input – Output = Storage
Dividing by volume of the element (DxDyDz) and Dt
Taking the limit yields:

12. Substituting Fourier’s Law:
Gives:

13. Solution Finite Difference
A grid used for the finite difference solution of elliptic PDEs in two independent variables.

14. Numerical Differentiation using Centred-Finite Divided Difference
First Derivative
Second Derivative
Third Derivative

15. Solution
Finite Element

16. Finite Element Analysis
Two interpretations
Physical Interpretation:
The continous physical model is divided into finite pieces called elements and laws of nature are applied on the generic element. The results are then recombined to represent the continuum.
Mathematical Interpretation:
The differentional equation representing the system is converted into a variational form, which is approximated by the linear combination of a finite set of trial functions.

17. Group Assignment Group Task Group A Problem 1 Group B Problem 2
Group C
Problem 3
Group D
Problem 4