# Integration RUNGE_KUTTA in complex plane.

• nitish
In summary, Integration RUNGE_KUTTA in complex plane is a numerical method used to solve complex differential equations. It works by breaking down the equation into a system of real equations and using the standard RUNGE_KUTTA method. Its advantages include being able to handle a wider range of equations and being more accurate for complex equations with large imaginary components. However, it can be computationally expensive and produce inaccurate results for certain equations. It is commonly used in fields such as quantum mechanics, electromagnetics, and fluid dynamics, as well as in numerical simulations and modeling.
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

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.

## 1. What is Integration RUNGE_KUTTA in complex plane?

Integration RUNGE_KUTTA in complex plane is a numerical method used to solve differential equations involving complex numbers. It is an extension of the standard RUNGE_KUTTA method, which is used for solving differential equations with real numbers.

## 2. How does Integration RUNGE_KUTTA work in the complex plane?

Integration RUNGE_KUTTA in complex plane works by breaking down the complex differential equation into a system of real equations and then using the standard RUNGE_KUTTA method to solve it. The results are then combined to obtain the solution to the original complex equation.

## 3. What are the advantages of using Integration RUNGE_KUTTA in complex plane?

Integration RUNGE_KUTTA in complex plane is advantageous because it can handle a wider range of differential equations compared to other numerical methods. It is also more accurate for complex equations with large imaginary components.

## 4. What are the limitations of Integration RUNGE_KUTTA in complex plane?

One limitation of Integration RUNGE_KUTTA in complex plane is that it can be computationally expensive for equations with high order derivatives. It may also produce inaccurate results if the step size is too large or if the equation has multiple solutions.

## 5. In what fields is Integration RUNGE_KUTTA in complex plane commonly used?

Integration RUNGE_KUTTA in complex plane is commonly used in fields such as quantum mechanics, electromagnetics, and fluid dynamics, where complex differential equations are prevalent. It is also used in numerical simulations and modeling in engineering and physics.

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