The resolution of the systems of differential equations using backward differentiation formula method

The resolution of the systems of differential equations using backward differentiation formula method

                                 dY(t)/dt = AY + B  (*)
                                  Y(t0)    = Y0.

 Input: :      A          : size matrix  (n,n).
                  B          : size matrix  (n,s).

                  Y0          : size matrix  (n,s).
                     tf , t0    : temps.

 Output:     Y(1,:)    : solution Y_{1,1}(t) of  (*)  .

   Author                 :  LAKHLIFA SADEK.

    Last modification     :  10/08/2019
 

 E-mail: lakhlifasdek@gmail.com; sadek.l@ucd.ac.ma 

From the test, this method was found to be faster than the normal method known in Matlab (ode23s) in terms of time to calculate the approximate solution.

test

close all

clear all

clc

n=16; s=2; t0=0;tf=1;

A=randn(n);

B=randn(n,s); B=B/norm(B,'fro');

Y0=randn(n,s);

disp('BDF method')

tic

[t2,Y] = BDFSDE(A,B,Y0,t0,tf);

toc

odefun = @(t,y) fonction(t,y,A,B);

disp('ode23s')

tic

[T,y1] = ode23s(odefun,[t0 tf],Y0);

 toc

hold on

plot(t2,Y(1,:))

plot(T,y1(:,1))

xlabel('Temps')

ylabel('The solution Y_{1,1}')

title(['n=',num2str(n),', s=',num2str(s)])

legend('BDF method','ode23s')

 hold off