Gaussian Quadrature on a Repeated Integral

In summary, the conversation discusses the difficulty in evaluating a repeated integral and the possibility of using Gaussian Quadrature to obtain a numerical result. The use of Cauchy's formula on repeated integrals is also mentioned, but there are uncertainties due to the presence of φ. It is suggested to use two different symbols for the differential variables and to split the repeated integral into two simple integrals for easier computation. The potential singularity of the integral is also brought up.
  • #1
olukelliot
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upload_2017-11-28_18-6-45.png

Hi there,
I am having some difficulty evaluating a repeated integral, which is the first of two shown in the image.
I had hoped to be able to use Gaussian Quadrature to provide a numerical result, however am unsure on if this is possible for a repeated integral?

I have attempted to use Cauchy' formula on repeated integrals to obtain a single integral, which is shown on the bottom in the image. However I am once again unsure on performing this due to the presence of φ.

Any ideas on what I'm doing wrong/ missing?
Thanks
 

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  • #2
Your first integral is too ambiguous. You use φ for both integrations as well as as a variable in the integrand. You need to use two different symbols for the differential variables, so there would be no ambiguity for the variable in the integrand.
 
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  • #3
Additionally, it seems like the integral could be singular, depending on the values of ##r_1##, ##m## and ##l##. It is not clear from your thread what the values of these quantities are.
 
  • #4
You have to split the repeated integral into min. two simple integrals (with different borders) and each of them could be numerical integrated by Gauss.
See:
COMPUTATION OF DEFINITE INTEGRAL OVER REPEATED INTEGRAL Katar´ina Tvrda´, Maria Minarova´
Tatra mountains matematical publications
 

1. What is Gaussian Quadrature on a Repeated Integral?

Gaussian Quadrature on a Repeated Integral is a numerical method used to approximate the value of repeated integrals. It involves using a set of predetermined nodes and weights to calculate the integral, resulting in a more accurate approximation compared to traditional methods such as the Trapezoidal Rule.

2. How does Gaussian Quadrature on a Repeated Integral work?

This method works by selecting a set of nodes and weights that are specifically designed to accurately calculate integrals of a certain degree. The nodes are the points at which the function is evaluated, and the weights are the coefficients used to calculate the integral. By using these predetermined values, the integral can be calculated with a higher degree of precision.

3. What are the advantages of using Gaussian Quadrature on a Repeated Integral?

The main advantage of this method is that it can provide more accurate approximations of repeated integrals compared to other numerical methods. It is also relatively simple to implement and can handle a wide range of integrals, making it a versatile tool for scientific research and calculations.

4. Are there any limitations to using Gaussian Quadrature on a Repeated Integral?

While this method is more accurate than other numerical methods, it does have some limitations. It may not work well for integrals with highly oscillatory or discontinuous functions, and the accuracy of the approximation can be affected by the number of nodes and weights used.

5. How is Gaussian Quadrature on a Repeated Integral used in scientific research?

This method is commonly used in scientific research for a variety of applications, such as calculating the energies of quantum mechanical systems, solving partial differential equations, and performing numerical simulations. It is also used in various fields such as physics, engineering, and economics to solve complex integrals and obtain accurate results.

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