Solving Asymptotic Series: A Guide for Understanding

[hanson]

Asymptotic methods...

hi all...
i am really frustrated with the asymptotic methods.
I just don't understand where those asymptotic expansions come from...
While I know the definition of a asymptotic series, but the author of the books seems always assume a series and eventually claim that is an asymptotic series to a function without going through the definition:
The remainder is much smaller than the last retained term in the series...

can someone explain the standard procedure of obtaining a asymptotic series solution to the simplest equation...?

 

[Navin A S]

Asymptotic methods are a type of approximation technique used to find approximate solutions to complex problems. The basic idea is to use a series of successive approximations to obtain an increasingly accurate solution. In most cases, this approach is used when exact solutions are difficult to obtain, or when the exact solutions are too time-consuming to calculate. Asymptotic methods involve the use of asymptotic expansions, which are essentially a series of terms that express the solution as a function of an independent variable. This expansion is then used to approximate the solution in the vicinity of the independent variable. The standard procedure for obtaining an asymptotic series solution to the simplest equation is as follows:1. Determine the form of the equation to be solved and its associated boundary conditions.2. Choose an appropriate ansatz function that satisfies the boundary conditions. 3. Substitute the ansatz into the equation and determine the coefficients of the resulting equation. 4. Solve the equation for the desired coefficients.5. Calculate the corresponding asymptotic series solution by substituting the determined coefficients into the ansatz. 6. Check the accuracy of the solution by comparing the series solution to the exact solution or to numerical simulations.

 

[ravenprp]

Asymptotic series are a powerful tool in mathematics for approximating functions and solving equations. They are particularly useful for problems that involve small or large parameters, where traditional methods may not be effective. However, they can be quite confusing and frustrating for those who are not familiar with them. In this guide, we will provide a step-by-step procedure for obtaining an asymptotic series solution to a simple equation.

Step 1: Identify the small or large parameter

The first step in solving an asymptotic series is to identify the small or large parameter in the equation. This parameter will be used as the basis for the asymptotic expansion.

Step 2: Expand the function in a series

Next, we expand the function in a series around the small or large parameter. This can be done using Taylor series or other methods, depending on the function and the problem at hand.

Step 3: Determine the order of the series

Once the series has been expanded, we need to determine the order of the series. This is the highest power of the small or large parameter that appears in the series.

Step 4: Retain the first few terms

In order to obtain an asymptotic solution, we only need to retain the first few terms of the series. These terms will provide a good approximation to the function, as long as the small or large parameter is sufficiently small or large.

Step 5: Check for convergence

It is important to check for convergence of the series. If the series does not converge, then it cannot be used to obtain an asymptotic solution.

Step 6: Calculate the remainder term

The remainder term is the difference between the exact solution and the asymptotic solution obtained from the first few terms of the series. It is important to note that this remainder term must be much smaller than the last retained term in order for the asymptotic solution to be valid.

Step 7: Improve the approximation

If the remainder term is not small enough, we can improve the approximation by retaining more terms in the series. This will provide a better estimate of the exact solution.

Step 8: Compare with other methods

Finally, it is important to compare the asymptotic solution with other methods, such as numerical methods or other analytical methods, to ensure its accuracy and validity.

In conclusion, asymptotic series can be a powerful tool for solving equations and approximating functions. By following this step-by-step guide, you can better understand and use asymptotic methods in your