MATLAB: Does the code for shooting method using ‘ODE45’ or ‘ODE23s’ does not converge to the boundary value.

Question

I have used shooting method with ' ode45' or ' ode23s'.

But , the solution doesn't converge and it takes a lot of time.

The equations are

f"=g(g^2+gamma^2)/(g^2+lambda*gamma^2) ------ (1)
g'= (1/3)*f'^2-(2/3)*(f*f")+ Mn*f' ------------------------(2)
t"+Rd*t"+ 2*Pr*f*t'/3+ Nb*t'*p'+Nt*(t')^2= 0------(3)
p"+(2*Lew*f*p')/3+ Nt*t"/Nb= 0 ------------------------(4)

With the initial and boundary conditions

f(0)=0, f'(0)=1, t(0)=1, p(0)=1
f'(infinity)=0, t(infinity)=0, p(infinity)=0

The code for shooting method using ode45 is

function shooting_method
clc;clf;clear;
global lambda gama Pr Rd Lew Nb Nt Mn
gama=1;  
Mn=1;
Rd=0.1;
Pr=10;
Nb=0.2;
Lew=10;
Nt=0.2;
lambda=0.5;
x=[1 1 1];
options= optimset('Display','iter');
x1= fsolve(@solver,x);
end
function F=solver(x)
options= odeset('RelTol',1e-8,'AbsTol',[1e-8 1e-8 1e-8 1e-8 1e-8 1e-8 1e-8]);
[t,u]= ode45(@equation,[0 10],[0 1 x(1) 1 x(2) 1 x(3)],options)
s=length(t);
F= [u(s,2),u(s,4),u(s,6)];
%deval(0,u)
plot(t,u(:,2),t,u(:,4),t,u(:,6));
end
function dy=equation(t,y)
global lambda gama Pr Rd Lew Nb Nt Mn
dy= zeros(7,1);
dy(1)= y(2);
dy(2)= y(3)*(y(3)^2+gama^2)/(y(3)^2+lambda*gama^2);
dy(3)= y(2)^2/3-(2*y(1)*y(3)*(y(3)^2+gama^2))/(3*(y(3)^2+lambda*gama^2))+Mn*y(2);
dy(4)= y(5);
dy(5)= -(2*Pr*y(1)*y(5))/(3*(1+Rd)) - (Nb*y(5)*y(7))/(1+Rd) - (Nt*y(5)^2)/(1+Rd);
dy(6)= y(7);
dy(7)= -((2*Lew*y(1)*y(7))/3)+ (Nt/Nb)*((2*Pr*y(1)*y(5))/(3*(1+Rd)) + (Nb*y(5)*y(7))/(1+Rd) + (Nt*y(5)^2)/(1+Rd));
end

Answer 1

Try to start with the solution you get from "bvp4c" for the vector x.

Best wishes

Torsten.