How to solve boundary value problem using shooting method (ode45)?

Question

Rajesh 2016-9-22

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编辑： Rajesh 2016-9-23

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Hi,

I am trying to solve a differential equation using shooting method. I have a differential equation of the form:

dydx = 6/h^3*(1/e/r*(V-U)-V*h);

r and h are defined in the form of vectors in the main function. The value of 'e' can vary from 0.1 to 0.001 and is constant for a given run. The BCs are y(-1) = 1 and y(1) = 0. U and V are the parameters which I am trying to determine. I am using ode45 to integrate the differential equation from the left boundary y(-1) = 1 and then using fsolve to set the value at the right boundary to 0. This is my code:

global x_range H R Pf0 Pb0 e 
xf = 1;
xb = -1;
Pf0 = 0;
Pb0 = 1;
U0 = 1;
V0 = 1;
xf0 = [U0;V0];
e = 0.01; %can vary from 0.1 to 0.001
x_range = linspace(xb,xf,100);
H = ones(1,100);
R = 1-e*H;
fun = @(x) ode_solver(x);
[xf_opt,fval,exitflag] = fsolve(fun,xf0);
function y = ode_solver(x0)
global x_range U V Pb0
U = x0(1);
V = x0(2);
[xf,yf] = ode45(@diff_eqn,x_range,Pb0);
y = yf(1,end)
figure(1), plot(xf,yf)
function dydx = diff_eqn(x,~)
global x_range R H U V e
r = spline(x_range,R,x);
h = spline(x_range,H,x);
dydx = 6/h^3*(1/e/r*(V-U)-V*h);

The code returns that no solution was found. The value of y at the right boundary goes below 0. Am I doing something wrong here? Can I approach the problem differently?