How can draw this kind of graph?

how can we draw a velocity profile of the 3rd-order non-linear boundary value problem that converges for different values? give me code for u'''+u*u''+s*[Gr*G+Gm*H-M*u']=0 where s=0.02, Gm=10, Gr=4 and M=0.1 with boundary condition u=1,u'=0.001; G=0.001; H=0.001 when t=0 and u'=0; G-0; H=0 when t= infinity, where H=2 and G=4 within one graph for Different values of Gm

[Rik]

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[Sam Chak]

@gkb

When does H jump from 0.001 to 2?

When does G jump from 0.001 to 4?

If H switches from 0.001 to 2 at t = ∞, does it mean H stays at 0.001 forever?

[gkb]

yes

[gkb]

[Torsten]

Use bvp4c or bvp5c to set up the problem.

[Sam Chak]

@gkb, I'm confused.

You only introduced ONE differential equation in the question.

where the parameters other than the state variables u, , , are constants.

Why do F, G, and H magically appear in the context? Can we settle the first and then post another question for the system of differential equations F, G, H?

Answer 1

[Syed Sohaib Zafar]

Here's your required code

code_gkb_Matlab

%%%%%%%%%%%%%%%
function code_gkb_Matlab 
    global eps Gr Gm M Pr Jh Sc S0
    
    %eq1
    eps=0.001; Gr=0.5; M=0.5;
    % eq2
    Pr=1.4; Jh=0.3; %eps M
    % eq3
    Sc=0.7; S0=0.2;
    
    val=[0.1 0.2 0.3 0.4];
    for i = 1:1:4;
        Gm = val(i);
        
        solinit = bvpinit(linspace(0,6),[0 1 0 1 0 1 0]);
        sol= bvp4c(@shootode,@shootbc,solinit);
        eta = sol.x;
        f = sol.y;
        
        figure (1)
        plot(eta,f(2,:));
        xlabel( '\eta');
        ylabel('\bf f'' (\eta)');
        hold on
    end
    lgd=legend('Gm = 0.1','Gm = 0.2','Gm = 0.3','Gm = 0.4');
end


function dydx  = shootode(eta,f);
    global eps Gr Gm M Pr Jh Sc S0
    dydx = [f(2)
        f(3)
        -f(1)*f(3)-eps*(Gr*f(4)+Gm*f(6)-M*f(2))
        f(5)   
        -Pr*f(1)*f(5)+Jh*Pr*((1/eps)*f(3)^2+M*f(2)^2)
        f(7)    
        2*Sc*f(2)*f(6)-Sc*f(1)*f(7)-S0*Sc*(-Pr*f(1)*f(5)+Jh*Pr*((1/eps)*f(3)^2+M*f(2)^2))
        ];
end

function res = shootbc(fa,fb)
    global eps
    res = [fa(1)-1; fa(2)-eps; fa(4)-eps; fa(6)-eps; fb(2)-0; fb(4)-0; fb(6)-0];
end

[DGM]

Edited to add formatting and to make it run.