Let the unknown function in Equation 1. Substituting the approximate solution 2. In above Equation 2. Then 2. Let , be the zeros of and , respectively. Substituting the collocation points into 2.
Solving the system of Equation 2. In this section, we consider some problems to illustrate the above method. So, one gets. Firstly, let us consider in detail the case I , , for. This results in. It is easy to estimate the values and. By choosing the collocation points , for we obtain the following system of linear equations:.
By solving this system for the unknown coefficients that produces. From 3. Which coincides with the exact solution. The error of approximate solution 3. Secondly, let us consider in detail the case II , , for. By choosing the collocation points , for we obtain the following system of linear equations :. Thirdly, let us consider in detail the case III , , for. Table 1.
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Illustrates errors of approximate solutions of Equation 3. Fourthly, In case IV , , for. So one gets.
By choosing the collocation points , for we obtain the following. Table 2. Thirdly, In case IV , , for. Firstly, let us consider in detail the case II , , for. By solving the system 3. Secondly, in case III , , for. Table 3. Numerical results Tables show that the errors of approximate solutions of Examples in different Cases with small value of n are very small. These show that the methods developed are very accurate and in fact for a linear function give the exact solution.
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All rights reserved. Chakrabarti, A. ZAMM, 69, Springer, Berlin. Ladopoulous, E. Zhang, Solving Cauchy singular integral equations by using general quadrature-collocation nodes, Comput.
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