Solution of Fredholm Integral Equations by Collocation

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Solution of Fredholm Integral Equations by Collocation In this document the solution of Fredholm integral equations1 by the method of collocation is outlined. The method is applied to an example of a Fredholm equation of the second kind. Collocation In this section the method of collocation is applied to the general Fredholm Integral Equation of the second kind:



          

The first stage in the development of the collocation method is to represent (and thereby approximate) the functions and using a linear sum of basis functions2. The most simple functional approximations are piecewise constant, piecewise linear or piecewise quadratic3.





 and its approximation are ‘matched’ at

For polynomial basis functions, the function a set of points and the basis functions χ 1 ,

 

and

χ 2 ,…, χ n

are usually chosen so that

 ( )  {     in



this

case

   ∑    where



  

The piecewise polynomial representation of

  is then substituted into the integral equation:









      ∑ ∑          It follows that





    ∑             

In the collocation method the equation above is enforced at the points

also termed the collocation points.

Integral Equations Basis Functions 3 Piecewise Polynomial Approximation 1 2

 , which are


This gives the following set of n equations





( ()   ∑  (  )    ()    

The integral equation can now be written as a linear system of equations, as follows;



 ∑     

  ∫ (  )   and   ()   In most practical examples the  are computed by a numerical integration method. The set of

where

4

n

equations can then be written concisely using matrices and vectors5 as follows

      which can be solved using standard methods for solving linear systems of equations6. The method of collocation is applied to an integral equation in the following example and the solution of the integral equation is implemented on an accompanying Excel spreadsheet 7. Example In this example the method of collocation is applied to the following integral equation

                           

 on [0,1]. The analytic solution

for which we are required to find an approximation to to the integral equation is

      

Using the most simple piecewise constant basis functions, the integral equation can be written in discrete form

 ⁄      ∑                 ⁄       . where  is to represent the function  on    ⁄ for , we By letting  take the value of the collocation point (      obtain the following set of  equations  ⁄        ∑                   ⁄    Numerical Integration or Quadrature Matrix Definitions 6 Solution of Linear Systems of Equations 7 Solution of Fredholm Integral Equations by Collocation.xlsx 4 5


The system of equations can now be written more concisely in matrix-vector form, as follows:

       where

⁄     ⁄       and

           Let the integrals be computed using a quadrature rule with m points and weights, defined on [-1,1]; and .

    ⁄ , ⁄ ] is transformed In order to compute  , first the domain of the integral [ onto the standard domain [-1, 1] using the substitution                      Applying the quadrature rule gives

        ∑    On the spreadsheet the solution is developed on two separate sheets, one with four collocation points and one with eight collocation points. The integration rule that is used on both sheets is the 2-point Gaussian quadrature rule [ref Gaussian Quadrature ] and is coloured in blue. The results for the 4 collocation point method is shown in the following table





     

error

0.125

0.878922828

0.882496903

0.003574075

0.375

0.682939979

0.687289279

0.004349299

0.625

0.530159048

0.535261429

0.005102381

0.875

0.411029118

0.41686202

0.005832902


The results for the 8 collocation point method is shown in the following table



    



error

0.0625

0.938567799

0.939413063

0.000845263

0.1875

0.828085612

0.829029118

0.000943506

0.3125

0.730575116

0.731615629

0.001040513

0.4375

0.644512315

0.645648526

0.001136211

0.5625

0.568552255

0.569782825

0.00123057

0.6875

0.501507991

0.502831578

0.001323587

0.8125

0.442332027

0.44374731

0.001415283

0.9375

0.390099927

0.391605627

0.0015057

It can be observed that doubling the number of collocation points reduces the error to about ¼ ⨉. The method appears to be [ref Big [ref Big O notation in Mathematics ], where



  ⁄ the length of each element of the approximating function. ‘

’

Solution of Fredholm Integral Equations by Collocation

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