Integration RUNGE_KUTTA in complex plane.

[nitish]

Hi,
I have one serious problem while solving rayleigh equation using blasius profile, in which so as to remove the singularity the intergration contour is defined in a complex plane.
4th order runge kutta is used but if the step size( h) is is in complex, it is giving some error. would anyone please help me solving it?
here is the MATLAB code. thank you...reply as soon as possible.

nstep=100;
pi=3.14;
error=1e-4;
ycr=1.35;
yci=1.35;
ymax=8;


nstep2=5*nstep-1;
ncirc=nstep;
ncirc2=3*ncirc+1;

for j=1:ncirc2
theta=pi*(j-1)/(ncirc2-1);
yr=ycr*(1-cos(theta));
yi=-yci*sin(theta);
Y(j)= (yr+i*yi);
end

for j=ncirc2:nstep2
yr=ycr*2+ymax*(j-ncirc2)/(nstep2-ncirc2);
Y(j)=(yr+i*0);
end

%y=zeros(1,61);
%z=zeros(1,61);
%p=zeros(1,61);
%u=zeros(1,61);

F(1)=0+0*i;
U(1)=0+0*i;
UP(1)=1.660287+0*i;
UPP(1)=0+0*i;
[y,z,p,u]=blasius1(nstep2,Y,F,U,UP,UPP);
pnew=p(1);
znew=z(nstep2);

UP(1)=1.05*UP(1);
[y,z,p,u]=blasius1(nstep2,Y,F,U,UP,UPP);
pold=p(1);
zold=z(nstep2);

residue=1e30;

while(residue>error)
U(1),UPP(1),F(1),Y,nstep2;
UP(1)=pnew-(((znew)-1)*((pold)-(pnew)))/((zold)-(znew));
[y,z,p,u]=blasius1(nstep2,Y,F,U,UP,UPP);
zold=znew;
pold=pnew;
znew=z(nstep2);
pnew=p(1);
residue=abs((z(nstep2))-1.0);
end
%plot(real(Y),real(z));
hold on
plot(real(Y),abs(p));
hold on
%plot(real(Y),real(y));
hold on
%plot(real(Y),real(u));
================================================== ===============
blasius.m(separate file):
function[y,z,p,u]=blasius1(num,Y,F,U,UP,UPP)

y(1)=F(1);
z(1)=U(1);
p(1)=UP(1);
u(1)=UPP(1);

for j=1num-1)

h=((Y(j+1))-(Y(j)))*0.5;
h=real(h);

A11=(h)*(z(j));
A12=(h)*(p(j));
A13=(-1*(h)*((y(j))*(p(j))));
A14=(-(1.4806)*(h)*(((y(j))*(u(j))+(z(j))*(p(j)))));

A21=(h)*((z(j))+(A12)*0.5);
A22=(h)*((p(j))+(A13)*0.5);
A23=(-1*(h)*(((y(j))+(A11)*0.5)*((p(j))+(A13)*0.5)));
A24=(-(1.4806)*(h)*((((y(j))+(A11)*0.5)*((u(j))+(A14)*0. 5)+((z(j))+(A12)*0.5)*((p(j))+(A13)*0.5))));

A31=(h)*((z(j))+(A22)*0.5);
A32=(h)*((p(j))+(A23)*0.5);
A33=(-1*(h)*(((y(j))+(A21)*0.5)*((p(j))+(A23)*0.5)));
A34=(-(1.4806)*(h)*((((y(j))+(A21)*0.5)*((u(j))+(A24)*0. 5)+((z(j))+(A22)*0.5)*((p(j))+(A23)*0.5))));

A41=(h)*((z(j))+(A32));
A42=(h)*((p(j))+(A33));
A43=(-1*(h)*(((y(j))+(A31))*((p(j))+(A33))));
A44=(-(1.4806)*(h)*((((y(j))+(A31))*((u(j))+(A34))+((z(j ))+(A32))*((p(j))+(A33)))));

y(j+1)=((y(j))+(((A11)+2*(A21)+2*(A31)+(A41))/6));
z(j+1)=((z(j))+(((A12)+2*(A22)+2*(A32)+(A42))/6));
p(j+1)=((p(j))+(((A13)+2*(A23)+2*(A33)+(A43))/6));
u(j+1)=((u(j))+(((A14)+2*(A24)+2*(A34)+(A44))/6));

end

 

[gtoguy97]

end</code>you can try to use a numerical integration method such as Euler's method or Runge-Kutta methods. The idea is that you start with an initial approximation of the solution and then iteratively update the approximation until the error reduces to a predefined tolerance. You can also use a symbolic integration approach if you are using MATLAB. It should be able to handle the complex values and solve for the Rayleigh equation. Hope this helps.