Is my code for bvp4c for a set of odes is correctly written

Question

naygarp 2018-4-6

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https://ww2.mathworks.cn/matlabcentral/answers/393188-is-my-code-for-bvp4c-for-a-set-of-odes-is-correctly-written

评论： Mayokun Ojediran 2019-7-3

I have the following set of equations:

 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)

'f', 'g', 't', 'p' are the dependent variables and η is the independent variable

And the boundary conditions are as follows:

     f=0, f'= delta, t=1, p=1 at y=0
     f'-->0, t-->0, p-->0 as y----> infinity

So solving these equations numerically I have used 'bvp4c' solver

So converting these higher order equations to first order equations, I have let

f = y1, df/dη = y2, g = y3, t = y4, dt/dη = y5, p = y6, dp/dη = y7

So the first order equations are

 y1' = y2,
y2' = {y3*(y3^2+ gamma^2)}/(y3^2+ lambda*gamma^2) ,
y3' = y2^2/3 − (2/3)*y1*{y3*(y3^2+ gamma^2)/(y3^2+ lambda*gamma^2)} + Mny2,
y4' = y5,
y5' = −(1/(1+Rd))*{(2*Pr*y1*y5)/3 + Nb*y5*y7 + Nt*y5^2} ,
y6' = y7,
y7' = −(2*Le*y1*y7)/3 + {Nt/Nb*(1+Rd)}{(2*Pr*y1*y5)/3 + Nb*y5*y7 + Nt*y5^2}

I would like to know if my written code is correct and if I have got the correct plots

    function sol= proj
clc;clf;clear;
global lambda gama Pr Rd Lew Nb Nt Mn m
gama=1;
Lew=1;
Nt=1;  
Rd=1;
Pr=5;
Mn=1;
pp=[0.5 1 1.5];
qq=[0.2 0.5 0.9];
%figure(1)
%plot(2,1);hold on
options=bvpset('stats','on','RelTol',1e-9);
m=linspace(0,10);
solinit= bvpinit(m,[1,0,0,0,0,0,0]);
for i=1:numel(pp)
    lambda=pp(i);
    for i=1:numel(qq);
        Nb=qq(i)
sol= bvp4c(@projfun,@projbc,solinit,options);
y1=deval(sol,0)
solinit= sol;
plot(sol.x,sol.y(6,:));hold on
end
end
end
function f= projfun(x,y)
global lambda gama Pr Rd Lew Nb Nt Mn
f= [y(2)
    y(3)*(y(3)^2+gama^2)/(y(3)^2+lambda*gama^2)
    y(2)^2/3-(2*y(1)*y(3)*(y(3)^2+gama^2))/(3*(y(3)^2+lambda*gama^2))+Mn*y(2)
    y(5)
    -(2*Pr*y(1)*y(5))/(3*(1+Rd)) - (Nb*y(5)*y(7))/(1+Rd) - (Nt*y(5)^2)/(1+Rd)
    y(7)
    -(2*Lew*y(1)*y(7))/3+ Nt*((2*Pr*y(1)*y(5))/(3*(1+Rd)) + (Nb*y(5)*y(7))/(1+Rd) + (Nt*y(5)^2)/(1+Rd))/Nb];
end
function res= projbc(ya,yb)
res= [ya(1); ya(2)-1; ya(4)-1.0; ya(6)-1.0; yb(2); yb(4); yb(6)];
end

2 个评论
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Walter Roberson 2018-4-9

If you have the Symbolic Toolbox, then I recommend that you look at odeFunction and go through the series of steps shown in the documentation to convert an ode right through to a function handle that can be used with the ode* routines.

Mayokun Ojediran 2019-7-3

在 MATLAB Online 中打开

Hello, I am trying to adapt your code to solve the following equations:

(2n+1/2(n+1))f*theta' = (1/Bo^(2/n+1))*theta''
1/2*f'^2 +(2n+1/2(n+1))*f*f'' = -(f''^(n-1)*f'' + f''^(n-1)*f''') + theta

'f' and 'theta' are the dependent variables and η is the independent variable and Bo is a constant

f= f' = 0 , theta=1 at y=0
  f'-->0, theta-->0 as y----> infinity

And the boundary conditions are as follows:

Converting to First order equations,

theta = y1, dtheta/dη = y2, f = y3, df/dη = y4, d^2f/d\η^2 = y5

The first order equations are

y(2);
Bo^(2/(n+1))*((2*n+1)/2*(n+1))*((2*n+1)/2*(n+1))*y(3)*y(2);
y(4);
y(5);
(-y(4)^2*n+2*y(3)*y(5)*n+2*y(1)*n-y(4)^2+y(3)*y(5)+2*y(1))/(2*y(5)^(n-1)*n*(n+1));

I have tried adapting your code but not completely sure how to implement the boundary conditions, please help.

function sol= proj
clc;clf;clear;
global n Bo
n  = 0.4;
Bo = 0.0003548133892;
pp=[0.5 1 1.5];
qq=[0.2 0.5 0.9];
%figure(1)
%plot(2,1);hold on
options=bvpset('stats','on','RelTol',1e-9);
m=linspace(0,10);
solinit= bvpinit(m,[1,0,0,0,0,0,0]);
for i=1:numel(pp)
    lambda=pp(i);
    for i=1:numel(qq);
        Nb=qq(i)
sol= bvp4c(@projfun,@projbc,solinit,options);
y1=deval(sol,0)
solinit= sol;
plot(sol.x,sol.y(4,:));hold on
end
end
end
function f= projfun(x,y)
global lambda gama Pr Rd Lew Nb Nt Mn
f= [y(2)
    Bo^(2/(n+1))*((2*n+1)/2*(n+1))*((2*n+1)/2*(n+1))*y(3)*y(2);
    y(4)
    y(5)
 (-y(4)^2*n+2*y(3)*y(5)*n+2*y(1)*n-y(4)^2+y(3)*y(5)+2*y(1))/(2*y(5)^(n-1)*n*(n+1))];
end
function res= projbc(ya,yb)
res= [ya(1); ya(2)-1; ya(4)-1.0; ya(6)-1.0; yb(2); yb(4); yb(6)];
end