Solving Integro-differential equation with limited integral

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ash
ash el 25 de Jun. de 2015
Comentada: Hewa selman el 22 de Dic. de 2021
Hi,
How can I solve this equation numerically using matlab
w''''=w''*int(w'^2,0,1)
I tried using the standard form of ODE function, the only problem I faced is how to represent that limited integral Thanks

Respuesta aceptada

Torsten
Torsten el 25 de Jun. de 2015
Write your integro-differential equation as
w1'=w2
w2'=w3
w3'=w4
w4'=w3*integral_{t=0}^{t=1}w2^2(t') dt'
Then discretize the interval [0:1] in n subintervals 0=t(1)<t(2)<...<t(n)=1.
Compute the derivatives as
wj'(t(i))=(wj(t(i+1))-wj(t(i)))/dt (j=1,2,3,4)
and compute the integral using the trapezoidal rule.
You'll arrive at a polynomial system (order 3) of equations for the unknowns
wj(t(2)),wj(t(3)),...,wj(t(n)) (j=1,2,3,4)
which can be solved by fsolve, e.g.
No chance to use ODE45 in this case.
Another way might be to use ODE45 and iteratively adjust the value of the integral, but I'm not sure whether this method will converge.
Good luck !
Best wishes
Torsten.
  1 comentario
Hewa selman
Hewa selman el 22 de Dic. de 2021
Hello
Now are you sure that we can adjust the value of integral, then put it in system and solve it by ode45 or ode15s.

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Más respuestas (2)

Claudio Gelmi
Claudio Gelmi el 6 de En. de 2017
Take a look at this solver:
Article "IDSOLVER: A general purpose solver for nth-order integro-differential equations": http://dx.doi.org/10.1016/j.cpc.2013.09.008
Best wishes,
Claudio
  2 comentarios
Fernando Fernandes
Fernando Fernandes el 14 de En. de 2021
Gelmi help me! How can I use your method to solve this equation?
Fernando Fernandes
Fernando Fernandes el 14 de En. de 2021
I've downloaded your paper, but i'm a beginner in Matlab. Do I need the solver in http://cpc.cs.qub.ac.uk/summaries/AEQU_v1_0.html ???
How can I install this?

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ash
ash el 28 de Jun. de 2015
Thanks, sorry for the late reply
I tried to apply the technique you suggested (as much as I understood) using Euler 1st order, also my problem is a BVP (3 initial conditions and 1 BC) kindly find my code bellow The problem is that the code is too slow, and I can only solve small number of points
Is that what you advised me to do in our previous comment?, are there any enhancement for that code?
Thanks
syms a b
w=10;
T=30;
L=500;
F=1;
E=160e3;
I=2500;
A=T*w;
Npnts=11;
x=linspace(0,L/2,Npnts);
q1=sym(zeros(1,length(x)));
q2=sym(zeros(1,length(x)));
q3=sym(zeros(1,length(x)));
q4=sym(zeros(1,length(x)));
h=x(2)-x(1);
q1(1)=0;
q2(1)=0;
q3(1)=b;
q4(1)=-F/2/E/I;
for i=1:length(x)
q1(i+1)=q1(i)+h*q2(i);
q2(i+1)=q2(i)+h*q3(i);
q3(i+1)=q3(i)+h*q4(i);
q4(i+1)=q4(i)+h*q3(i)*a*A/2/L/I;
end
integ_a=sum(q2.^2)*h-a/2;
sol_ab=solve(integ_a==0,q2(i+1)==0,a,b);
sol_a=sol_ab.a;
sol_b=sol_ab.b;
sol_index=1;
q1=subs(q1,a,sol_a(sol_index));
q1=double(subs(q1,b,sol_b(sol_index)));
  4 comentarios
Torsten
Torsten el 30 de Jun. de 2015
Sorry, should read
(w4(t(i+1))-w4(t(i)))/h = [sum_{j=1}^{j=Npnts-1}(w2(t(j+1))+w2(t(j)))*h/2]*w3(t(i))
Best wishes
Torsten.
SOZHAESWARI P
SOZHAESWARI P el 5 de Sept. de 2021
How to solve the numerical solution of nonlinear parabolic integro differential equation for two grid finite element method example MATLAB codings

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