# Numerical Solution of Volterra Integral Equation and its Error Estimates Via Spectral Method

## Keywords:

Spectral method, Chebyshev basis function, Coefficients, Volterra Integral equations, Error estimates.## Abstract

In this article, numerical solution of Volterra integral equations is considered. A new approach in the application of spectral method is proposed, wherein Chebyshev polynomial of the first kind serves as the basis function. Essentially, the method is based on the approach of series solution where coefficients of in the residual equations are correspondingly equated to yield system of equations. Expression for error estimates which effectively serves as upper bound for accrued errors is arrived at. To illustrate the accuracy and effectiveness of the method and its error estimates, numerical examples on some standard integral equations are given.

## References

[2] B. S. Grewal. Numerical Methods in Engineering and Science. Delhi: Khanna Publishers, 2007.

[3] J. C. Mason and D. C. Handscomb. Chebyshev Polynomials. Boca Raton, London, New York, Washington DC: Chapman & Hall/CRC, 2003.

[4] A. S. Olagunju. “Chebyshev Series representation for Product of Chebyshev Polynomials and some notable functions”. IOSR Journal of Mathematics Vol. 2(2), pp 9-13, 2012.

[5] A. O. Taiwo and A. S Olagunju. “Chebyshev methods for the numerical solution of 4th order differential equations”. International Journal of Physical science. Vol. 7(13), pp 2032-2037, 2012.

[6] C. W. Clenshaw. “A note on the summation of Chebyshev series”. Maths Tab. Wash. Vol. 9, pp 118-120, 1955.

[7] E. W. Cheney. Introduction to Approximation Theory. Chelsea, New York: McGraw-Hill, 1982 2nd ed.

[8] T. J. Rivlin. Chebyshev Polynomials; From Approximation Theory to Algebra and number Theory. New York: John Wiley, 1990.

[9] A. M. Wazwaz. Linear and Nonlinear Integral equations, methods and application. Beijing: Higher Education Press, and Berlin Heidelbeerg: Springer-verlag, 2011.

[10] L. N. Trefethen. Spectral Methods in MATLAB. Philadelphia: SIAM, 2000.

## Downloads

## Published

## How to Cite

*American Academic Scientific Research Journal for Engineering, Technology, and Sciences*,

*30*(1), 47–56. Retrieved from https://asrjetsjournal.org/index.php/American_Scientific_Journal/article/view/2507

## Issue

## Section

## License

Authors who submit papers with this journal agree to the following terms.