second order differential systems of a non linear ODE

Considering theth differential system equations to be solved are:

f''(t) = 3*f(t)*g(t) + 5 g''(t) = 4*g(t)*f(t) + 7 with initial conditions: f(0) = 1.5, g'(0) = 0 and boundary constraints tf = 1: g(1) = 3, f'(1) =q* f(1)

With q is natural number

how we can plot the solution satysfying the boundary conditions

[Jan]

This sound like a job fpr bvp4c .

[Lewis Fer]

how we can do it Mr Jan?

[Jan]

Start with reading the corresponding documentation:

doc bvp4c

Then modify the exmples given there to your problem. If you have a specific problem with this, post the details here.

The information, that q is a natural number does not allow to implement this. The readers of your question cannot guess, if you want to run the code for a certain number of inputs, if you should find a specific value of q.

[Lewis Fer]

The problem is : The differential system equations to be solved are: f''(t) = 3*f(t)*g(t) + 5 g''(t) = 4*g(t)*f(t) + 7 with initial conditions: f(0) = 1.5, g'(0) = 0 and boundary constraints tf = 1: g(1) = 3, f'(1) =2* f(1) ( for example q=2 )

[Lewis Fer]

I have already solve the problem with q=1 but for q different to 1 I have not, I used the following code in

https://www.mathworks.com/matlabcentral/answers/853120-how-to-solve-a-system-of-second-order-nonlinear-differential-equations-with-boundary-conditions?s_tid=srchtitle

:

%%%%%%%%%%

main function

xmesh = linspace(0,1,10);

sol = bvpinit(xmesh, @iguess);

sol = bvp5c(@odefcn,@bcfcn,sol,bvpset('RelTol ' ,1e-13 ,'AbsTol ',1e-13,'Nmax ',6000))

Y0 = sol.y(:,1)

solde45 = ode45(@odefcn,[0,1],Y0,odeset('RelTol',1e-13,'AbsTol',1e-13))

close all;

figure;

ylblstr = ['f','g', 'h=fp ','j = gp '];

for i = 1:4

subplot(4,1,i); hold on

plot(sol.x,sol.y(i,:))

plot(solde45.x,solde45.y(i,:),'--')

ylabel(ylblstr(i))

end

xlabel('t')

function res = bcfcn(ya,yb)

%

% f(0) = 1.5

% g'(0) = 0 -> j(0) = 0

% g(2) = 3

% f'(2) = f(2) -> h(2) = f(2)

res = [

ya(1) - 1.5;

ya(4) - 0;

yb(2) - 3;

yb(1) - yb(3);

];

end

function Y = iguess(x)

f = sin(x);

g = cos(x);

h = cos(x);

j = -sin(x);

Y = [f;g;h;j];

end

function dY = odefcn(t,Y)

% f'' = 3 f g + 5

% g'' = 4 f g + 7

% let h = f'

% let j = g'

% Y = [f;g;h;j]

f = Y(1,:);

g = Y(2,:);

h = Y(3,:);

j = Y(4,:);

dY = [

h;

j;

3*f*g + 5;

4*f*g + 7;

];

end

[Jan]

What is your problem with solving the problem for different q? There is a huge number of different natural number, so what is the actual problem you want to solve?

[Lewis Fer]

I want solve this problem for each natural number that we want to test some results about the Effect when we have q <10 and q>10 fo rexample ?

Can you show me how to do it for a general case of a given natural number?

in mathematic can we replace : f'(1) =2* f(1) by f'(1)=2*exp(2) ?

[Lewis Fer]

in mathematic can we replace : f'(1) =q* f(1) by f'(1)=q*exp(q) ? because the solution of the differential equation

f'(x) =q* f(x) has the forme f(x)=C*exp(qx) and I know tha C in our case equal=1

Answer 1

[Sulaymon Eshkabilov]

Small typo err in your code:

xmesh = linspace(0,1,10);
sol = bvpinit(xmesh, @iguess);
sol = bvp5c(@odefcn,@bcfcn,sol,bvpset('RelTol ' ,1e-13 ,'AbsTol ',1e-13,'Nmax ',6000)); % ERR: No space after: 'RelTol', 'AbsTol', 'Nmax' 

[Lewis Fer]

I fix it this err thank you Sulaymon Eshkabilov , and the main question how to do it if q=2 or 3 for example ?