I want to write Matlab code using finite element method in order to solve the above problem but I didn't succeed because am not familiar with that Matlab programming however I have tried to give such code below which it dosn't work ,any help ?
-u''(x)+u(x)=(-4x^2-6) exp(x^2),u(-1)=u(1)=0, x\in ]-1,1[
[
My attempt code :
function [U] = EquaDiff2(n)
%----------------------------------
% d²u/dx² + 6 du/dx + 9 u = x(1-x)
% u(-1) = 0 u(1)= 0
syms x x1 x2 real % déclaration de variables
symboliques
function [Ke, Fe] = MatElt2Nd(x1,x2) % déclaration de la fonction , Fonction pour calculer la
mat. et vect. élementairs
%----------------------------------
x = ]-1:2/n:1['; % modification d’1 borne
d’intégration
K = zeros(n+1 ) ;
F = zeros(n+1,1) ;
for i = 1:n
j = i+1;
t = [i j];
x1 = x(i);
x2 = x(j);
[Ke,Fe] = MatElt2Nd(x1,x2);
K(t,t) = K(t,t) + Ke;
F(t) = F(t) + Fe;
end;
K(1,:) = [];
K(:,1) = [];
F(1) = [];
U = K\F;
U = [0.0;U];
t = 0:0.01:1;
return
%-------------------------------------------
% Calcul de la matrice Ke et du vecteur Fe
%-------------------------------------------
function [Ke,Fe] = MatElt2Nd(x1,x2)
Ke1 = 1/(x2-x1)*[ 1 -1 % les modifications ne
touchent
-1 1 ] ; % essentiellement que les
matrices
Ke2 =(x2-x1)* [ 2 1 % élémentaires
1 2 ] ;
N = [(x-x2)/(x1-x2) (x-x1)/(x2-x1)] % fonctions de forme
Fe =simple( int(N' * (-4*x^2-6)*exp(x^2) , x, x1, x2) ) % vecteur Fe ;
Ke = Ke1 + Ke2 ;
return
Hi, you can first rewrite your equation as a system of two first-order equations:
u'(x)=y(x)
y'(x)=u(x)+(4x^2+6)exp(x^2)
u(-1)=0 u(1)=0
Then you can define the shooting function whose parameter is z=y(-1) :
S: z --> S(z)=u(1)
To compute numerically S you simply have to call ode45.
Finally, using fsolve you can search for a zero of S, i.e. z such that S(z)=0.
Only some short MATLAB modules are required.
Best
You can follow ODE45 with examples practice that may enable you to solve any ODE problems., see e.g., https://au.mathworks.com/help/matlab/ref/ode45.html
this link may be help you
https://www.mathworks.com/matlabcentral/fileexchange/31482-using-finite-elements-method-on-a-pde-with-no-solution
Dear Richard Epenoy, Thanks for your answer , seems calling ODE45 is for forte and classical solution, But I want to solve the titled problem using finite element method for example :using assembly matrix which is the heart of the problem coming up to an apprioximate solution, in any way thanks prof for your helpful
Richard Epenoy
Hi, ode45 is use for the forward integration of your system for known initial conditions. By using fsolve you can determine the missiong condition, i.e. y(-1)
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