How Can the Variational Iteration Method Solve This Differential Equation?

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In summary, the person is seeking help with solving a differential equation using the variational iteration method. Their ultimate goal is to find an approximate relation between A and omega (frequency of oscillation) for the given equation. They mention that A is a Heaviside step function and provide initial conditions for the equation. They also clarify that omega is not explicitly mentioned in the equation but represents the frequency of oscillation. Additionally, they mention that as A increases, the frequency decreases and ultimately reaches zero at a particular value of A. They are seeking an approximate analytical form for this relationship.
  • #1
dekarman
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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
 
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  • #2
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.
 
  • #3
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.
 
  • #4
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.
 

FAQ: How Can the Variational Iteration Method Solve This Differential Equation?

What is the Variational Iteration Method?

The Variational Iteration Method (VIM) is a numerical technique used to solve differential equations. It is an iterative process that approximates the solution by expanding it in a series and using variational calculus to determine the coefficients of the series.

How does the Variational Iteration Method work?

The VIM involves constructing a correction functional, which is a function of the unknown solution, and using it to generate a series of approximations. The coefficients of the series are determined by minimizing the error between the approximate solution and the actual solution. This process is repeated until the desired accuracy is achieved.

What types of problems can the Variational Iteration Method solve?

The VIM is a versatile method that can be applied to a wide range of problems, including ordinary differential equations, partial differential equations, and integral equations. It has also been used in many different fields of science and engineering, such as physics, biology, and finance.

What are the advantages of using the Variational Iteration Method?

One of the main advantages of the VIM is that it does not require linearization of the problem, unlike other numerical methods. This allows it to handle nonlinear equations more effectively. Additionally, the VIM is a relatively simple and efficient method, making it a popular choice for solving differential equations.

Are there any limitations to the Variational Iteration Method?

The VIM may not be suitable for problems with highly oscillatory solutions or those with very steep gradients. It also requires the initial guess to be close to the actual solution, which can be challenging for some problems. Furthermore, the convergence of the method may be slow for some equations.

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