Title: PARTIAL DIFFERENTIAL EQUATIONS
1PARTIAL DIFFERENTIAL EQUATIONS
2Introduction
- 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.
6B2 4AC Category Example
lt 0 Elliptic Laplace equation (Steady state with two spatial dimension)
0 Parabolic Heat conduction equation (time variable with one spatial dimension)
gt 0 Hyperbolic Wave equation (time variable with one spatial dimension)
7Elliptic Equations
- Typically used to characterize steady-state
distribution of an unknown in two spatial
dimensions.
8Laplace Equation
The PDE as an expression of the conservation of
energy
9- Need to reformulate the equation in terms of
temperature. Use Fouriers Law - and
- substituting back results in
(Laplace equation)
10Parabolic Equations
Hot
Cool
Heat balance (the amount of heat stored in the
element) over a unit time, Dt
11Input Output Storage
Dividing by volume of the element (DxDyDz) and Dt
Taking the limit yields
12Substituting Fouriers Law
Gives
13Solution
A grid used for the finite difference solution of
elliptic PDEs in two independent variables.
14Numerical Differentiation using Centred-Finite
Divided Difference
- First Derivative
- Second Derivative
- Third Derivative
15Solution
16Finite 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.
17Group Assignment
Group Task
Group A Problem 1
Group B Problem 2
Group C Problem 3
Group D Problem 4