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How to obtain the Jacobian matrix from the finite difference method as far as possible,when the shooting method is used?

Asked by Xuan Ling Zhang on 17 Oct 2019
Latest activity Edited by Xuan Ling Zhang on 17 Oct 2019
Hi, I have a problem when i use the shooting method. That is the Jacobian matrix calculated approximately from the finite difference method is too expensive to obtain. Not only that, i should run the Ode45 solver many times. Anyone can give some suggestions?In addition, how to obtain the sufficiently reasonable values of initial guesses(i.e., d_1, d_2, d_3, d_4) ?
Thank you in advance!
The specific description of this problem is given as follows:
Function main %
d_1=8.5513e-011;d_2=3.4673e-004;d_3=3.4673e-004;d_4=-0.0034;%The current guesses
L=0.5;
span=[0 L];
options = odeset('RelTol',1e-5,'AbsTol',1e-6);
delta=1e-12;
init_0=[0 d_1 0 d_2 0 0 d_3 d_4];
[x,y0]=ode45(@Comp,span,init0,options);
Init_1=[0 d_1+delta 0 d_2 0 0 d_3 d_4]; Init_2=[0 d_1 0 d_2+delta 0 0 d_3 d_4];
Init_3=[0 d_1 0 d_2 0 0 d_3+delta d_4]; Init_4=[0 d_1 0 d_2 0 0 d_3 d_4+delta];
[x,y1]=ode45(@Comp,span,Init1,options);[x,y2]=ode45(@Comp,span,Init2,options);
[x,y3]=ode45(@Comp,span,Init3,options);[x,y4]=ode45(@Comp,span,Init4,options);
%% the following:the first column
Df(1,1)=(y1(end,1)-y0(end,1))/delta; Df(2,1)=(y1(end,3)-y0(end,3))/delta; Df(3,1)=(y1(end,5)-y0(end,5))/delta;
Df(4,1)=(y1(end,6)-y0(end,6))/delta; % DF is
%% the following:the tecond column
Df(1,2)=(y2(end,1)-y0(end,1))/delta; Df(2,2)=(y2(end,3)-y0(end,3))/delta; Df(3,2)=(y2(end,5)-y0(end,5))/delta;
Df(4,2)=(y2(end,6)-y0(end,6))/delta;
%% the following:the third column
Df(1,3)=(y3(end,1)-y0(end,1))/delta; Df(2,3)=(y3(end,3)-y0(end,3))/delta; Df(3,3)=(y3(end,5)-y0(end,5))/delta;
Df(4,3)=(y3(end,6)-y0(end,6))/delta;
%% the following:the forth column
Df(1,4)=(y4(end,1)-y0(end,1))/delta; Df(2,4)=(y4(end,3)-y0(end,3))/delta; Df(3,4)=(y4(end,5)-y0(end,5))/delta;
Df(4,4)=(y4(end,6)-y0(end,6))/delta;
B=Df;
End
% The init_j (j=0,1,2...) is the initial condition. d_i (j=1,2,3) are the unknown initial parameters and are evaluated using those current guesses, which is actually the so-called process of shooting. Df is the Jacobian matrix mentioned above.
Function dy= Comp (x,y)
G1=-9.3799e-016; G2 =-3.2988e-020; G3=8.8425e+003; G4=0.3121; G5=1.1243e+006;
G6=4.4751e-012; G7=-5.1691e-015; G8=4.2188e+007; G9=0.1536; f0=23; L=0.5;
dy=zeros(8,1);
dy(1) = y(2);
dy(2) = G1*(y(3)+y(6))+G2*y(8)-y(6)*y(7);
dy(3) = y(4);
dy(4) = G3*(y(3)+y(6))+G4*y(8);
dy(5) = y(6);
dy(6) = y(7);
dy(7) = y(8);
dy(8)=G5*(y(4)+y(7))+G6*y(4)*y(7)+G7*y(7)*y(7)+G8*y(2)*y(7)+G8*y(6)*y(6)*y(7)/2+G9*f0*sin(pi*x/L);
end
% The boundary conditions of the nonlinear equations described by the code 'dy=Comp (x,y)' is: y(1)= y(3) = y(5) = y(6)=0at x=0 and L

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