Solving PDES in MATLAB

PEER 2013 Sept 6, 2013

One-Dimensional Heat Equation

Discretizing the Domain

Discretizing the Equations

Discretizing the Equations

Programming Heat Equation in MATLAB

% This is the finite difference implementation % for the heat equation on the domain (0,1) % u_t = u_xx, u(0,t)=0, u(1,t)=0, u(x,0)= x(1-x) clear all a = 0; % Left point b = 1; % Right point N = 10; % Number of intervals in x M = 10000; % Number of intervals in t T = 50; % Final Time del_x = (b - a)/N; % Step size in x del_t = T/M; % Step size in t x=a:del_x:b; % x grid t=0:del_t:T; % t grid lambda = del_t/del_x^2; % Defining Lambda

if lambda > 0.5 disp('Your lambda value is too big') else A = diag((1-2*lambda)*ones(N-1,1),0); A = A + diag(lambda*ones(N-2,1),1); A = A + diag(lambda*ones(N-2,1),-1); u_initial = x.*(1-x); Sol_mat(:,1) = u_initial(2:N)'; for i=1:10 Sol_mat(:,i+1) = A*Sol_mat(:,i); end u(1,:)=zeros(1,11); u(2:N,:)=Sol_mat; u(N+1,:)=zeros(1,11); for i=1:10 plot(x,u) hold on end end

PDE Toolbox MATLAB

Rectangular Plate with a circular Hole

( ) 0=∇− Tdiv

PDETOOL BOX

>> pdetool

Change generic scalar in the drop down menu to heat transfer.

Change generic scalar in the drop down menu to heat transfer.

Click the PDE button and select PDE Specification.

By default Type of PDE is Elliptic. Change values of h and Q to 0. Click OK.

Rectangle with a hole

• Click on one of the rectangle button on the main menu and use your mouse to define a rectangle.

Creating a Rectangle

Changing Rectangle Parameters

Click on the Rectangle and change the dimensions

Click the circle (ellipse) button and position a circle outside the rectangle

Create a Circle by changing radius

Drag the circle into the square R1 + E1

Set Formula Box R1 – E1

Click the Ω∂

Select Specify Boundary Conditions from the Boundary menu

Select Specify Boundary Conditions from the Boundary menu

Select the big

Select the big =

Refine by clicking

Push = again