# Approximate Solution Technique for Singular Fredholm Integral Equations of the First Kind with Oscillatory Kernels

## Main Article Content

## Abstract

An efficient quadrature formula was developed for evaluating numerically certain singular Fredholm integral equations of the first kind with oscillatory trigonometric kernels. The method is based on the Lagrange interpolation formula and the orthogonal polynomial considered are the Legendre polynomials whose zeros served as interpolation nodes. A test example was provided for the verification and validation of the rule developed. The results showed the convergence of the solution and can be improved by increasing *n*.

Keywords:

Singular kernel, oscillatory kernel, lagrange interpolation, orthogonal polynomial, legendre polynomial.

## Article Details

How to Cite

*Journal of Advances in Mathematics and Computer Science*,

*32*(6), 1-9. https://doi.org/10.9734/jamcs/2019/v32i630163

Section

Original Research Article

## References

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Iserles A, Nørsett SP. On quadrature methods for highly oscillatory integrals and their implementation. BIT Numerical Mathematics. 2004;44(4):755-772.

Evans GA, Chung K. Some theoretical aspects of generalised quadrature methods. Journal of Complexity. 2003;19(3):272-285.

Xiang S, Cho YJ, Wang H, Brunner H. Clenshaw–curtis–filon-type methods for highly oscillatory bessel transforms and applications. IMA Journal of Numerical Analysis. 2011;31(4):1281-1314.

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Okecha GE, Onwukwe CE. On the solution ofintegral equations of the first kind with singular kernels of Cauchy-type. International Journal of Mathematics andComputer Science. 2012;7(2):129-140.

Seifi A, Lotfi T, Allahviranloo T, Paripour M. An effective collocation technique to solve the singular Fredholm integral equations with Cauchy kernel. Advances in Difference Equations; 2017.

Barnett A. The calculation of spherical bessel and coulomb functions computational atomic physics. In Bartschat Berlin K, Hinze J. (Eds.). Computational Atomic Physics: Electrons and Positron Collisions with Atoms and Ions. Berlin: Springer. 1996;181-202.

Filon LNG. On a quadrature formula for trigonometric integrals. Proceedings of the Royal Society of Edinburgh. 1929;49:38-47.

Flinn E. A modification of Filon's method of numerical integration. Journal of the ACM (JACM). 1960;7(2):181-184.

Stetter HJ. Numerical approximation of Fourier-transforms. Numerische Mathematik. 1966;8(3):235-249.

Mikloško J. Numerical integration with weight functions $cos kx $, $sin kx $ on $[0, 2pi/t] $, $ t= 1, 2,dots$. Aplikace matematiky. 1969;14(3):179-194.

Piessens R, Poleunis F. A numerical method for the integration of oscillatory functions. BIT Numerical Mathematics. 1971;11(3):317-327.

Ting BY, Luke YL. Computation of integrals with oscillatory and singular integrands. Mathematics of Computation. 1981;37(155):169-183.

Brunner H. On the numerical solution of first-kind Volterra integral equations with highly oscillatory kernels. Isaac Newton Institute. 2010;HOP:13-17.

Graham IG. Galerkin methods for second kind integral equations with singularities. Mathematics of Computation. 1982;39(160):519-533.

Iserles A, Nørsett SP. On quadrature methods for highly oscillatory integrals and their implementation. BIT Numerical Mathematics. 2004;44(4):755-772.

Evans GA, Chung K. Some theoretical aspects of generalised quadrature methods. Journal of Complexity. 2003;19(3):272-285.

Xiang S, Cho YJ, Wang H, Brunner H. Clenshaw–curtis–filon-type methods for highly oscillatory bessel transforms and applications. IMA Journal of Numerical Analysis. 2011;31(4):1281-1314.

Abramowitz M, Stegun IA. Handbook of mathematical functions: With formulas graphs, and mathematical tables: New York, Dover Publicatiom Courier Corporation; 1970.

Polyanin AD, Manzhirov AV. Handbook of integral equations: CRC Press LLC; 1998.

Okecha GE, Onwukwe CE. On the solution ofintegral equations of the first kind with singular kernels of Cauchy-type. International Journal of Mathematics andComputer Science. 2012;7(2):129-140.

Seifi A, Lotfi T, Allahviranloo T, Paripour M. An effective collocation technique to solve the singular Fredholm integral equations with Cauchy kernel. Advances in Difference Equations; 2017.

Barnett A. The calculation of spherical bessel and coulomb functions computational atomic physics. In Bartschat Berlin K, Hinze J. (Eds.). Computational Atomic Physics: Electrons and Positron Collisions with Atoms and Ions. Berlin: Springer. 1996;181-202.