On the Modified Block-Pulse Function for Volterra Integral Equation of The First ‎Kind‎

Subject Areas : International Journal of Industrial Mathematics

مهناز محمدی ¹ , علیرضا وحیدی ² , طیبه دمیرچلی ³ , سعید خضرلو ⁴ , مصطفی نوری ⁵

1 - گروه ریاضی، واحد تهران جنوب، دانشگاه آزاد اسلامی، تهران، ایران.
2 - گروه ریاضی، واحد شهر ری، دانشگاه آزاد اسلامی، تهران، ایران.
3 - گروه ریاضی، واحد شهر ری، دانشگاه آزاد اسلامی، تهران، ایران.
4 - گروه ریاضی، واحد تهران جنوب، دانشگاه آزاد اسلامی، تهران، ایران.
5 - گروه ریاضی، واحد تهران جنوب، دانشگاه آزاد اسلامی، تهران، ایران.

Received: 2021-06-26 Accepted : 2021-10-02 Published : 2022-09-01

Keywords: Volterra integral equation of the first kind, Block-pulse functions, Operational matrix, Function expansion, ‎Convergence,

Abstract :

In this work, we present a simple efficient modified Block-Pulse functions (MBPFs) for numerical solution of class of linear Volterra integral equation of first kind. The peresent method is based on converting Volterra integral equation of the first kind into Volterra integral equation of the second kind. Some theorems are included to show the convergence and advantage of this method. Numerical results show that the approximate solutions have a good degree of ‎accuracy.‎

References:

 [1] S. Hatamzadeh-Varmazyar, M. NaserMoghadasi, Z. Masouri, A moment method simulation of electromagnetic scattering from coducting bodis, Progress in Electromagnetics Research 81 (2008) 99-119.
[2] S. Hatamzadeh-Varmazyar, M. NaserMoghadasi, E. Babolian, Z. Masouri, Numerical approach to survery the problem of electromagnetic scattering from resistive strips based on using a set of orthogonal basis functions, Progress in Electromagnetics Research 81 (2008) 393-412.
[3] S. Aggarwal, R. Chauhan, N. Sharma, Application of elzaki transform for solving linear Volterra integral equation first kind, International Journul of Research in Advent Technology 6 (2018) 3687-3692.
[4] A. R. Vahidi, T. Damercheli, A modified ADM for solving systems of linear Fredholm integral equations of the second kind, Applied Mathematical Sciences 6 (2012) 1267-1273.
[5] A. Shoja, A. R. Vahidi, E. Babolian, A spectral iterative method for solving nonlinear singular Volterra integral equations of Abel type, Applied Mathematical Sciences 112 (2017) 79-90.
[6] M. A. Alzhrani, H. O. Bakodah, M. Al-mazmumy, A 3/8 Simpson’s numerical scheme for the classes of Volterra integral equations of first kind, Nonlinear Analysis and Differential Equations 7 (2019) 99-113. http://dx.doi.org/10.12988/nade.2019.9812/
[7] K. Maleknejad, R. Mollapourasl, M. Alizadeh, Numerical solution of Volterra type integral equation of the first kind with wavelet basis,Appl. Math. Comput. 194 (2007) 400-405.
[8] K. Maleknejad, E. Hashemizadeh, R. Ezzati, A new approach to the numerical solution of Volterra integral equations by using Bernstein’s approximation, Commun. Nonlin. Sci. Numer. Silmulat. 16 (2011) 647-655.
[9] M. Bahmanpour, M. A. Fariborzi Araghi, Numerical solution of Fredholm and Volterra integral equations of the first kind using wavelets bases, The Journul of Mathematics and Computer Science 5 (2012) 337-345.
[10] I. O. Isaac, A. O. Unanam, Quadrature method for one-dimensional Volterra integral equations of the first kind, Journul of Modern Mathematics and Statistics 4 (2010) 109-114.
[11] N. Khan, M. S. Hashmi, S. Iqbal, T. Mahmood, Optimal homotopy asympotic method for solving Volterra integral equations of first kind, Alexandria Engineering Journal 5 (2014) 751-755.
[12] E. Babolian, Z. Masouri, Direct method to solve Volterra integral equation of the first kind using operational matrix with BlockPulse functions, Journal of Computational and Applied Mathematics 220 (2008) 51-57.
[13] K. Maleknejad, B. Rahimi, Modification of block{pulse functions and their application to solve numbering Volterra integral equation of the first kind, Commuh. Nonlin. SCi. Numer. Simulat. 16 (2011) 2469-2477.
[14] Z. Masouri, E. Babolian, S. H. Varmazyar, An expansion-iterative method for numeically solving Volterra integral equation of the first kind, Comput. Math. APPl. 59 (2010) 1491-149.
[15] ZH. Jiang, W. Schaufelberger, Block-pulse functions and their applications in control systems, Springer-Verlag, Berlin, (1992).
[16] A. M. Wazwaz, Linear and Nonlinear Integral Equations, Methods and Applications, Springer Heidelberg Dordrecht London NewYork, ISBN 978-3-642-21448-6.
[17] F. Mirzaee, E. Haddadiyan, Approximate solutions for mixed nonlinear VolterraFredholm type integral equations via modified block-pulse functions, Journul of the Association of Arab Universities for Basic and Applied Sciences 12 (2012) 65-73.
[18] S. Aggarwal, N. Sharma, Laplace transform for the solution of first kind linear Volterra integral equation, Journul of Advanced Researchin Applied Mathematics and Statistics 4 (2019) 16-23.