Bvp4c Error; cannot solve collocation equations- singular Jacobian

The eqns to be solved: f'''+ff''+1-f'^2-A(f'-1+(n/2)f'')+M(1-f')=0 (1/Pr)Y''+fY'-2f'Y-A(2Y+(n/2)Y')+Ec(M(1-f')^2+f''^2)=0 w/BC's: f(0)=0,f'(0)=lambda,Y(0)=1,f'(inf)=>1,Y(inf)=>0 it is known that there are two solutions where one can be found at infinity=10 and will add infinity=40 when I can fix error. Later will also be running for different values of lambda. I have checked and rechecked equation, I have also played with the initial guesses. I would appreciate any guidance.

global Pr A M Ec lambda
Pr=1; A=0.1; M=0.05; Ec=1; lambda=-1.33;
%
rlow=0;
rhigh=10;
N=50;
options=bvpset('stats','on','RelTol',1e-9);
solinit=bvpinit(linspace(rlow,rhigh,N),[0, -1, 0, 1 ,0]);
%
sol=bvp4c(@projode,@mybcs,solinit,options);
function dy=projode(n,y)
global Pr A M Ec
dy=[y(2); y(3); A*y(2)+A*n*y(3)/2+M*y(2)+y(2)^2-y(1)*y(3)-1-A-M; y(4);...
  Pr*(2*y(2)*y(4)+A*2*y(4)+A*n*y(5)/2-y(1)*y(5)-Ec*M*(1-y(2))^2-Ec*y(3)^2)];

function dy=projode(n,y)
global Pr A M Ec
dy=[y(2); y(3); A*y(2)+A*n*y(3)/2+M*y(2)+y(2)^2-y(1)*y(3)-1-A-M; y(4);...
  Pr*(2*y(2)*y(4)+A*2*y(4)+A*n*y(5)/2-y(1)*y(5)-Ec*M*(1-y(2))^2-Ec*y(3)^2)];

function res=mybcs(ya,yb)
global lambda
res=[ya(1) ya(2)-lambda  ya(4)-1 yb(2)-1  yb(4)];

Answer 1

[Torsten]

Replace

function dy=projode(n,y)
global Pr A M Ec
dy=[y(2); y(3); A*y(2)+A*n*y(3)/2+M*y(2)+y(2)^2-y(1)*y(3)-1-A-M; y(4);...
  Pr*(2*y(2)*y(4)+A*2*y(4)+A*n*y(5)/2-y(1)*y(5)-Ec*M*(1-y(2))^2-Ec*y(3)^2)];

by

function dy=projode(n,y)
global Pr A M Ec
dy=[y(2); y(3); A*y(2)+A*n*y(3)/2+M*y(2)+y(2)^2-y(1)*y(3)-1-A-M; y(5);...
  Pr*(2*y(2)*y(4)+A*2*y(4)+A*n*y(5)/2-y(1)*y(5)-Ec*M*(1-y(2))^2-Ec*y(3)^2)];

Best wishes

Torsten.

[Shelly]

Thank You! I feel silly now. That worked. I then just had to tweak the initial guesses a bit more to get the result I was looking for. Thanks again.

[naygarp]

Hey Shelly I have a similar problem as yours

How do I modify your code in mine, please help

Here's my equations(coupled) need to use shooting method with RK-4

(1+2M*eta)f''' + 2M*f"+ f*f" -f'^2 - k1*f' + lambda*theta=0 ---------- (1)

(1+2M*eta)theta" + 2M*theta' + Pr(f*theta'-f'*theta)=0 -------(2)

'f' and 'theta' are functions of 'eta', eta is an independent variable

3 initial conditions are given: eta=0, f(0)=0, f'(0)=1,theta(0)=1

Say I reduce these equations (1) and (2) to five ode (shooting method)

f'=z ; f(0)=0 -----(3)

z'=p ; z(0)=1 ------(4)

p'= (-2M*f"-f*f"+f'^2+k1*f'-lamda*theta)/(1+2*M*eta) ;p(0)= (guess value) -----(5)

theta'= q ; theta(0)=1 -----(6)

q' = (-2M*theta'-Pr(f*theta'-f'*theta))/(1+2*M*eta) ; q(0)= (guess value) ------(7)

The boundary conditions that needs to be satisfied are: f'(eta=10)=> 0 and theta(eta=10)=>0 as eta=>10

Given:

M= 1

k1= 0.1

lamda= 0.1

Pr= 0.7

taking step length: h= 0.01