Presentation on Matlab pde toolbox

Outline
Introduction
Why we need it
PDE functions
Example problem
Solution of the problem
Code of the solution
Application
2

Introduction 14-01-07-031
What is PDE?
In mathematics, a partial differential equation (PDE) is a differential equation that contains
unknown multivariable functions and their partial derivatives. (A special case is ordinary
differential equations (ODEs), which deal with functions of a single variable and their
derivatives.)
PDEs are used to formulate problems involving functions of several variables, and are either
solved by hand, or used to create a relevant computer model.
3

Introduction
PDE Toolbox
Partial Differential Equation Toolbox™ provides functions for solving partial differential
equations (PDEs) in 2-D, 3-D, and time using finite element analysis.
It enables to specify and mesh 2-D and 3-D geometries and formulate boundary conditions and
equations.
It can solve static, time domain, frequency domain, and eigenvalue problems over the domain of
the geometry
4

Introduction: Opening PDE toolbox
Using the Apps Tab
On the MATLAB Toolstrip, click the Apps tab.
On the Apps tab, click the down arrow at the end of the Apps section.
Under Math, Statistics and Optimization, click the PDE button.
5

Introduction: Opening PDE toolbox
Using commands
To open a blank PDE app window, type pdetool in the MATLAB Command Window.
To open the PDE app with a circle already drawn in it, type pdecirc in the MATLAB Command
Window.
To open the PDE app with an ellipse already drawn in it, type pdeellip in the MATLAB Command
Window.
To open the PDE app with a rectangle already drawn in it, type pderect in the MATLAB
Command Window.
To open the PDE app with a polygon already drawn in it, type pdepoly in the MATLAB Command
Window.
6

Why do we need it 14-01-07-035
Time dependent pde
Changing with Space
7

PDE Setup
Create or import the geometry for your problem, a 2-D or 3-D region.
Set boundary conditions on the outer edges or faces of the geometry.
Specify the PDE coefficients.
Specify initial conditions.
Create a finite element mesh of the geometry.
Call the solver.
8

PDE Functions
Sol=pdepe(m, pdefun, icfun, bcfun, xmesh, tspan)
9

Problem 14-01-07-050
10
At center
𝑑𝑇
𝑑𝑟
=0
temperature convection
−
𝑑𝑇
𝑑𝑟
=
ℎ
𝑘
𝑇 − 𝑇α
So for radius r the PDE is
𝑑𝑇
𝑑𝑡
= α
1
𝑟2
𝑑
𝑑𝑟
(
𝑟2 𝑑𝑇
𝑑𝑟
)

Problem 14-01-07-046
initial condition
T₀= 5ᵒC
Boundary condition
@r=0
𝑑𝑇
𝑑𝑟
= 0
@r=0.05 −
𝑑𝑇
𝑑𝑟
=
ℎ
𝑘
𝑇 − 𝑇α
11

Solution
PDE form of MATLAB
C
𝑑𝑢
𝑑𝑡
= 𝑥−𝑚 𝑑𝑢
𝑑𝑥
(𝑥 𝑚 𝑓 𝑥, 𝑡, 𝑢,
𝑑𝑢
𝑑𝑥
+ 𝑠
Our equation is
1
α
𝑑𝑇
𝑑𝑡
= 𝑟−2 𝑑
𝑑𝑟
𝑟2 𝑑𝑇
𝑑𝑟
+ 0 𝑜𝑟
1
α
𝑑𝑢
𝑑𝑥
= 𝑟−2 𝑑
𝑑𝑥
𝑥2 𝑑𝑢
𝑑𝑥
+ 0
12

Code 14-01-07-045
function FLASH
m=2;
xmesh=linspace(0, 0.05, 20);
tspan=linspace(0,28800,32);
sol=pdepe(m, @pdefun, @icfun, @bcfun, xmesh,
tspan);
U=sol(:,:,1);
13
Function [c,f,s]=pdefun[x,t,u,DuDx]
C=1/ α;
F=DuDx;
S=0
Function u0=icfun(x)
u0=5;

code
Function [pl, ql, pr, qr]=pdebc(xl, ul, xr, ur, t)
pl=0;
ql=1;
Because matlab uses
P+qf=0
On left (center)
𝑑𝑇
𝑑𝑥
= 0 so, 0+1
𝑑𝑇
𝑑𝑥
=0
14

Plots 14-01-07-038
Surf (x, t, u)
X label(‘distance’)
Y label(‘time’)
Figure
Plot (x,u,(end,:))
X label (‘distance’)
Y label (‘temperature’)
15
Picture is downloaded from internet. But the actual curve for this equation will look like this.

Applications
 Electrostatics and Magneto statics
 Structural Mechanics
 DC Conduction and Elliptic Problems
 Heat Transfer and Diffusion
16

17
This presentation is made with the help of this video
https://www.youtube.com/watch?v=ri_1nxwupb8&t=596s

Presentation on Matlab pde toolbox

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