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.

## UCL Discovery

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:.

Singular integral equations-I

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.

## Rendiconti Lincei - Matematica e Applicazioni

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.

### Tohoku Mathematical Journal

• Some Classes of Singular Equations | Siegfried Proessdorf;
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Zozulya, P. Gonzalez-Chi, Weakly singular, singular and hypersingular integrals in 3-D elasticity and fracture mechanics, J. Assari, H. Adibi, M. Dehghan, The numerical solution of weakly singular integral equations based on the meshless product integration MPI method with error analysis, Appl. Metzler, J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J.

A, 37 , — Bagley, P. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Samko, A. Kilbas, O. Hilfer Ed. Baleanu, K. Diethelm, E.

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### Some classes of singular equations

Momani, C. Khalique, Application of Legendre wavelets for solving fractional differential equations, Comput. Mirzaee, E. Hadadiyan, Numerical solution of linear Fredholm integral equations via two-dimensional modification of hat functions, Appl. Hadadiyan, Approximation solution of nonlinear Stratonovich Volterra integral equations by applying modification of hat functions, J. Hadadiyan, Numerical solution of Volterra-Fredholm integral equations via modification of hat functions, Appl.

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## Applied Singular Integral Equations - CRC Press Book

Zhao, Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Tripathi, V. Baranwal, R. Pandey, O. Singh, A new numerical algorithm to solve fractional differential equations based on operational matrix of generalized hat functions, Commun. Nonlinear Sci.