i need to solve F'''+FF''-​(F')^2-(M^​2)F'=0 with bvp,s are F(0)=3,F'(0)=-1 and F(infinity)=0

3rd order ODE. solve in ode45 command

[naveed ibrahim]

first convert this bvp ti to ivp, by converting it in to system of first order equations. then apply the shooting method to calculate the unknown initial condition(missing condition)..after finding that initial condition you can apply bvp,s.

Answer 1

[Torsten]

Use bvp4c.

Best wishes

Torsten.

Answer 2

[naveed ibrahim]

% % equation f'''+ff''-f^2

% % f(0)=0,f'(0)=-1 and f(inf)=0

% % first convert the the above into system of first order ode,s by letting

% % u=f',

% % v=u',then v'=u^2-2fu and crossponding initial conditions for this

% % system of equations are f(0)=0,u(0)=-1 and missing condition is v(0)=s

% % now differinte above initial value system w.r t s and apply NR method

% % to get missing condition s. after finding this missing condition we can

% % apply bvp,s etc

function [x,f,u,v,F,U,V]=shooting(s)

f1=@(x,f,u,v,F,U,V)u;

f2=@(x,f,u,v,F,U,V)v;

f3=@(x,f,u,v,F,U,V)u^2-f*v;

f4=@(x,f,u,v,F,U,V)U;

f5=@(x,f,u,v,F,U,V)V;

f6=@(x,f,u,v,F,U,V)2*u*U-(v*F+f*V);%%U, V and F are obtain by differinating i.v .p with respect to s.

h=0.1;

f(1)=0;

u(1)=-1;

v(1)=s;

F(1)=0;

U(1)=0;

V(1)=1;

x(1)=0;

for i=1:100

k1=h*f1(x(i),f(i),u(i),v(i),F(i),U(i),V(i));%% apply rk method 4

l1=h*f2(x(i),f(i),u(i),v(i),F(i),U(i),V(i));

m1=h*f3(x(i),f(i),u(i),v(i),F(i),U(i),V(i));

n1=h*f4(x(i),f(i),u(i),v(i),F(i),U(i),V(i));

o1=h*f5(x(i),f(i),u(i),v(i),F(i),U(i),V(i));

p1=h*f6(x(i),f(i),u(i),v(i),F(i),U(i),V(i));

k2=h*f1(x(i)+h/2,f(i)+k1/2,u(i)+l1/2,v(i)+m1/2,F(i)+n1/2,U(i)+o1/2,V(i)+p1/2);

l2=h*f2(x(i)+h/2,f(i)+k1/2,u(i)+l1/2,v(i)+m1/2,F(i)+n1/2,U(i)+o1/2,V(i)+p1/2);

m2=h*f3(x(i)+h/2,f(i)+k1/2,u(i)+l1/2,v(i)+m1/2,F(i)+n1/2,U(i)+o1/2,V(i)+p1/2);

n2=h*f4(x(i)+h/2,f(i)+k1/2,u(i)+l1/2,v(i)+m1/2,F(i)+n1/2,U(i)+o1/2,V(i)+p1/2);

o2=h*f5(x(i)+h/2,f(i)+k1/2,u(i)+l1/2,v(i)+m1/2,F(i)+n1/2,U(i)+o1/2,V(i)+p1/2);

p2=h*f6(x(i)+h/2,f(i)+k1/2,u(i)+l1/2,v(i)+m1/2,F(i)+n1/2,U(i)+o1/2,V(i)+p1/2);

k3=h*f1(x(i)+h/2,f(i)+k2/2,u(i)+l2/2,v(i)+m2/2,F(i)+n2/2,U(i)+o2/2,V(i)+p2/2);

l3=h*f2(x(i)+h/2,f(i)+k2/2,u(i)+l2/2,v(i)+m2/2,F(i)+n2/2,U(i)+o2/2,V(i)+p2/2);

m3=h*f3(x(i)+h/2,f(i)+k2/2,u(i)+l2/2,v(i)+m2/2,F(i)+n2/2,U(i)+o2/2,V(i)+p2/2);

n3=h*f4(x(i)+h/2,f(i)+k2/2,u(i)+l2/2,v(i)+m2/2,F(i)+n2/2,U(i)+o2/2,V(i)+p2/2);

o3=h*f5(x(i)+h/2,f(i)+k2/2,u(i)+l2/2,v(i)+m2/2,F(i)+n2/2,U(i)+o2/2,V(i)+p2/2);

p3=h*f6(x(i)+h/2,f(i)+k2/2,u(i)+l2/2,v(i)+m2/2,F(i)+n2/2,U(i)+o2/2,V(i)+p2/2);

k4=h*f1(x(i)+h,f(i)+k3,u(i)+l3,v(i)+m3,F(i)+n3,U(i)+o3,V(i)+p3);

l4=h*f2(x(i)+h,f(i)+k3,u(i)+l3,v(i)+m3,F(i)+n3,U(i)+o3,V(i)+p3);

m4=h*f3(x(i)+h,f(i)+k3,u(i)+l3,v(i)+m3,F(i)+n3,U(i)+o3,V(i)+p3);

n4=h*f4(x(i)+h,f(i)+k3,u(i)+l3,v(i)+m3,F(i)+n3,U(i)+o3,V(i)+p3);

o4=h*f5(x(i)+h,f(i)+k3,u(i)+l3,v(i)+m3,F(i)+n3,U(i)+o3,V(i)+p3);

p4=h*f6(x(i)+h,f(i)+k3,u(i)+l3,v(i)+m3,F(i)+n3,U(i)+o3,V(i)+p3);

f(i+1)=f(i)+(k1+2*k2+2*k3+k4)/6;

u(i+1)=u(i)+(l1+2*l2+2*l3+l4)/6;

v(i+1)=v(i)+(m1+2*m2+2*m3+m4)/6;

F(i+1)=F(i)+(n1+2*n2+2*n3+n4)/6;

U(i+1)=U(i)+(o1+2*o2+2*o3+o4)/6;

V(i+1)=V(i)+(p1+2*p2+2*p3+p4)/6;

x(i+1)=x(i)+h;

end

plot(x,u)

Answer 3

[naveed ibrahim]

here is the shooting with RK and NR method