How Can the Variational Iteration Method Solve This Differential Equation?

[dekarman]

Hi,

I am trying to solve the following differential equation using the variational iteration method:

u''+u=A/((1-u)^2) with initial conditions, u(0)=u'(0)=0.

My ultimate aim is to obtain the relation between A and w (i.e. omega).

A is a Heaviside step function i.e. A(t)=A*H(t).

Can anybody help me out in the process of applying Variational Iteration Method to this problem.

Thanks.
Manish

 

[HallsofIvy]

I have no idea what you are talking about. You say, "My ultimate aim is to obtain the relation between A and w (i.e. omega)" but there is no "omega" in the problem.

 

[dekarman]

Hi,
Thanks for the reply.

The term omega, although not explicitly found in the differential equation. represents the frequency of oscillation.

When A is zero, the RHS of the equation is zero and the frequency of oscillation is equal to 1, which indicates that the period is equal to 2*pi.

However, when A is increased, the frequency changes (reduces) and ultimately goes to zero for a particular value of A.

I wish to obtain this relationship in an approximate analytical form.

I guess now it is clear.

 

[dekarman]

Hi,

A is a constant.

I will also comment on the qualitative behavior of the system.

for extremely small values of A, the system has oscillation frequency equal to 1, which is evident from the Differential Equation.

As the value of A is increased oscillation frequency decreases

At a particular value of A, the frequency reduces to zero.

I need an approximate relation between A and omega which captures the aforementioned behavior.

Thank you.