# MATLAB: How to draw Fig. 4

how to draw fig. 4 from the attached pdf

function mainPr=1; L=-1;D=1;R=0.1;Sc=1;% D=input('D=');   %%D=0.5, 1, 1.5xa=0;xb=6;solinit=bvpinit(linspace(xa,xb,101),[0 1 0 1 0 1 0]);sol=bvp4c(@ode,@bc,solinit);xint=linspace(xa,xb,101);sxint=deval(sol,xint);figure(1)plot(xint,sxint(2,:));    xlabel('\eta'); ylabel('f^\prime');hold onfunction res=bc(ya,yb)res=[ya(1); ya(2)-L-D*ya(3); ya(4)-1; ya(7)-1; yb(2)-1; yb(4);yb(6)];  end function dydx=ode(x,y)    dydx=[y(2);  y(3);  y(2)^2-y(3)*y(1)-1;  y(5);  -3*Pr*y(1)*y(5)/(3+4*R);  y(7); -Sc*y(1)*y(7)];              endend
Actually I dont know the concept of DUAL solution.

[sol1, sol2] = mainfunction [sol1, sol2] = mainPr=1;L=-1;D=[0.5, 1, 1.5];R=0.1;Sc=1;for k = 1:numel(D)    xa=0;xb=6;    solinit1=bvpinit(linspace(xa,xb,101),[0 1 0 1 0 1 0]);    solinit2=bvpinit(linspace(xa,xb,101),[-1 -1 -1 -1 -1 -1 0]);    sol1=bvp4c(@ode,@bc,solinit1);    sol2=bvp4c(@ode,@bc,solinit2);            figure(1)    plot(sol1.x,sol1.yp(1,:),'-r')    xlabel('\eta');    ylabel('f^\prime');    hold on    plot(sol2.x,sol2.yp(1,:),'-b')endhold off    function res=bc(ya,yb)        res=[ya(1); ya(2)-L-D(k)*ya(3); ya(4)-1; ya(7)-1; yb(2)-1; yb(4);yb(6)];    end    function dydx=ode(x,y)        dydx=[y(2);  y(3);  y(2)^2-y(3)*y(1)-1;  y(5);  -3*Pr*y(1)*y(5)/(3+4*R);  y(7); -Sc*y(1)*y(7)];    endend