Can someone help me to correct the code ​​for this problem using ode45 solver?

Question is:

Governing equations:

Boundary conditions:

f'(10)= 0, (assume )

As tried here, I want to plot graphs for velocity profile using parameters like Pr, beta, and N.

I tried giving 'for' loop, but it failed.

How to enter the code command to plot a graph for velocity profile with different beta parameter?

Trial code

function dy=vj_Casson()

clc; clear all;

betava=[0.5 1 2 5];

for i=1:4

beta=betava(i);

tspan = [0 10];

S = 0.5;

y0 = [S 0 0 1 1];

% y0 = [S 1 0 1 0];

[eta,y] = ode45(@fun,tspan,y0);

plot(eta, y(:,1));

xlabel('\bf\eta','FontSize',20,'FontWeight','bold');

ylabel('f(\eta)','FontSize',10,'FontWeight','bold');

legend('\beta=0.5','\beta=1','\beta=2','\beta=5')

hold on

end

end

function dy = fun(eta,y)

dy = zeros(5,1);

Pr=0.3; N=0.1; beta=0.5;

dy(1) = y(2);

dy(2)=y(3);

dy(3)=(((2*y(2))^(2))-y(1)*y(3))./((1+(1./beta)));

dy(4) =y(5);

dy(5)=-(((3*Pr)*(y(1)*y(5)-y(2)*y(4)))/(4*N+3));

end

[Alex Sha]

The results below are what you want?

1: f' = df/dt = f'
2: f'' = df'/dt = f''
3: theta' = dtheta/dt = theta'
4: f''' = df''/dt = (2*(f')^2-f*f'')/(1+1/0.5)
5: theta'' = dtheta'/dt = 0.3*(f'*theta-f*theta')/(1+4/3*0.1)

Objective Function: 1.41854343905794E-26
Boundary Values Estimated:
	f''(t=0): -0.887446621710334
	theta'(t=0): -0.36495059823582

Answer 1

[Torsten]

This is a boundary value problem, not an initial value problem since conditions on f and theta are given on both ends of the integration interval (eta = 0 and eta = 10). You have to use bvp4c or bvp5c instead of ode45 to solve.

[VAIDEHI]

Thank you for the response.

Actually, this is an initial value problem for which I assumed the unknown boundary coniditions

i.e., f'(10)= 0, theta(10) = 0.

The following are the actual governing equations with the bc's where the second set of boundaries for the neta is assumed for solving using the ode45 solver.

Can you help me to frame a code for the initial value problem using ode 45 solver?

[Torsten]

You cannot use conditions at eta = 10 if you use ode45. You must assume two further conditions at eta = 0 to make ode45 work and try to adjust these conditions in several runs such that you arrive at your two conditions at eta = 10. Look up "shooting method" for more details.

I can assure you: the simpler way for you to go is to use bvp4c or bvp5c.

[VAIDEHI]

Okay, Thank you.