How to apply asymptotic matching condition in bvp4c?

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Day Rosli 2016-2-4

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/266660-how-to-apply-asymptotic-matching-condition-in-bvp4c

Hi, I want to know how to write the code to apply asymptotic matching condition? In the paper, it is stated that they apply asymptotic matching condition for the range of 0.5<n=<1. The asymptotic matching condition is,

(U'(x),V'(x))=(n/(x*(n-1)))*(U(x),V(x)) as x->infinity

This is the code without the asymptotic condition.

function ScriptAA
n=0.5;
infinity=20;
solinit = bvpinit(linspace(0,infinity,30),[0 1 0 -1 0]);
sol = bvp4c(@(x,y)VK(x,y,n),@VKbc,solinit);
x = [0:2:20];
y = deval(sol,x);
figure
plot(sol.x,sol.y(1,:));      %Plot U,V,W,Theta
title('Plots of U versus x')
ylabel('U(x)')
xlabel('x')
drawnow
hold on
for i=2:6
  n = n+0.1;
  sol = bvp4c(@(x,y)VK(x,y,n),@VKbc,sol);
   lines = {'b-','c-','r-','g-','k-','y-'};
   plot(sol.x,sol.y(1,:),lines{i});            %Plot U,V,W,Theta
end
legend('n = 0.5, n = 0.6','n = 0.7','n = 0.8','n = 0.9','n = 1.0',1);
hold off
function yprime = VK(x,y,n)
a = ((1-n)/(n+1))*x;
c = (y(4)^2+y(5)^2)^((1-n)/(n+1));
b = ((c*y(4))^2+(c*y(5))^2)^((n-1)/2);
yprime = [ c*y(4)        % U'
           c*y(5)        % V'
           -2*y(1) - a*c*y(4)   % W'
           y(1)^2 - (y(2)+1)^2 + (y(3)+a*y(1))*c*y(4)
           2*y(1)*(y(2)+1) + (y(3)+a*y(1))*c*y(5)
           ];
end
function res = VKbc(ya,yb)
    res = [ya(1)
           ya(2)
           ya(3)
           yb(1)
           yb(2)+1
           ];
end
end