Numerical Solution of Linear and Nonlinear Integral Equations Via Improved Block-Pulse Functions

Mahmoud Hamed Taha; Mohamed Abdel-Latif Ramadan; Galal Mahrous Moatimid

doi:doi:10.11648/j.ajmcm.20210602.11

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Numerical Solution of Linear and Nonlinear Integral Equations Via Improved Block-Pulse Functions

Mahmoud Hamed Taha, Mohamed Abdel-Latif Ramadan, Galal Mahrous Moatimid

Published in American Journal of Mathematical and Computer Modelling (Volume 6, Issue 2)

Received: 6 March 2021 Accepted: 22 March 2021 Published: 26 May 2021

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Abstract

This paper is concerned with a numerical method based on the improved block-pulse basis functions (IBPFs). It is done mainly to solve linear and nonlinear Volterra and Fredholm integral equations of the second kind. These equations can be simplified into a linear system of algebraic equations by using IBPFs and their operational matrix of integration. After that, the system can be programmed and solved using Mathematica. The changes made to the method obviously improved - as it will be shown in the numerical examples - the time taken by the program to solve the system of algebraic equations. Also, it is reflected in the accuracy of the solution. This modification works perfectly and improved the accuracy over the regular block–pulse basis functions (BPF). A slight change in the intervals of the BPF changes the whole technique to a new easier and more accurate technique. This change has worked well while solving different types of integral equations. The accompanied theorems of the IBPF technique and error estimation are stated and proved. The paper also dealt with the uniqueness and convergence theorems of the solution. Numerical examples are presented to illustrate the efficiency and accuracy of the method. The tables and required graphs are also shown to prove and demonstrate the efficiency.

Published in	American Journal of Mathematical and Computer Modelling (Volume 6, Issue 2)
DOI	10.11648/j.ajmcm.20210602.11
Page(s)	19-34
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Linear Integral Equations, Nonlinear Integral Equations, Improved Block-Pulse Functions, Operational Matrix, Vector Forms, Error Analysis

References

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[2]	E. Yusufoglu, Numerical expansion methods for solving systems of linear integral equations using interpolation and quadrature rules, International Journal of Computer Mathematics, 84 (1), 133-149 (2007).
[3]	M. A. Ramadan, G. M. Moatimid, M. H. Taha, A powerful method for obtaining exact solutions of Volterra integral equation's types, Global Journal of Pure and Applied Mathematics, 16 (2), 325-339 (2020).
[4]	K. Maleknejad and F. Mirzaee, Numerical solution of linear Fredholm integral equations system by rationalized Haar function method, International Journal of Computer Mathematics, 80 (11), 1397-1405 (2003).
[5]	K. Maleknejad, F. Mirzaee, and S. Abbasbandy, Solving linear integrodifferential equations system by using rationalized Haar function method, Applied Mathematics and Computation, 155 (2), 317-328 (2004).
[6]	R. R. Rizkalla, S. Sh. Tantawy, M. H. Taha, Application on differential transform method for some non-linear functions and for solving Volterra integral equations involving fresnel's integral, Journal of Fractional Calculus and Applications, 5 (23), 1-14 (2014).
[7]	J. Pour-Mahmoud, M. Y. Rahimi-Ardabili, and S. Shahmorad, Numerical solution of the system of Fredholm integrodifferential equations by the Tau method, Applied Mathematics and Computation, 168 (1), 465-478 (2005).
[8]	JH. He, Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering, 62, 178-257 (1999).
[9]	JH. He, A coupling method of homotopy technique and perturbation technique for nonlinear problems. International Journal of Non-Linear Mechanics, 35 (1), 37–43 (2000).
[10]	JH. He, Application of homotopy perturbation method to nonlinear wave equations, Chaos, Solitons and Fractals, 26 (3), 695–700 (2005).
[11]	A. M. Wazwaz., Linear and Non-linear Integral equations (methods and applications), Saint Xavier University, Chicago (2011).
[12]	F. Mirzaee, Numerical solution of system of linear integral equations via improvement of block-pulse functions, Journal of Mathematical Modeling, 4 (2), 133-159 (2016).
[13]	Z. H. Jiang and W. Schaufelberger, Block Pulse functions and their applications in control systems, Spriger-Verlag, Berlin, (1992).
[14]	W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, Singapore (1976).
[15]	K. Maleknejad, S. Sohrabi and B. Baranji, Application of 2D-BPFs to nonlinear integral equations, Communications in Nonlinear Science and Numerical Simulation, 15 (3), 527-535 (2010).
[16]	K. Maleknejad, Y. Mahmoudi, Numerical solution of linear Fredholm integral equation by using hybrid Taylor and block-pulse functions, Applied Mathematics and Computation, 149 (3), 799-806 (2004).
[17]	K. Maleknejad, K. Mahdiani, Iterated block-pulse method for solving Volterra integral equations, Applied Mathematics, 2 (1), 17-20 (2012).
[18]	A. Shahsavaran, Computational method to solve nonlinear integral equations using block pulse functions by collocation method, AppliedMathematical Sciences, 5 (65), 3211 – 3220 (2011).
[19]	Mohamed Ramadan, Heba S. Osheba, A new hybrid orthonormal Bernstein and improved block-pulse functions method for solving mathematical physics and engineering problems, Alexandria Engineering Journal, 5 (59), 3643-3652, (2020).

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Mahmoud Hamed Taha, Mohamed Abdel-Latif Ramadan, Galal Mahrous Moatimid. (2021). Numerical Solution of Linear and Nonlinear Integral Equations Via Improved Block-Pulse Functions. American Journal of Mathematical and Computer Modelling, 6(2), 19-34. https://doi.org/10.11648/j.ajmcm.20210602.11

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ACS Style

Mahmoud Hamed Taha; Mohamed Abdel-Latif Ramadan; Galal Mahrous Moatimid. Numerical Solution of Linear and Nonlinear Integral Equations Via Improved Block-Pulse Functions. Am. J. Math. Comput. Model. 2021, 6(2), 19-34. doi: 10.11648/j.ajmcm.20210602.11

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AMA Style

Mahmoud Hamed Taha, Mohamed Abdel-Latif Ramadan, Galal Mahrous Moatimid. Numerical Solution of Linear and Nonlinear Integral Equations Via Improved Block-Pulse Functions. Am J Math Comput Model. 2021;6(2):19-34. doi: 10.11648/j.ajmcm.20210602.11

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@article{10.11648/j.ajmcm.20210602.11,
  author = {Mahmoud Hamed Taha and Mohamed Abdel-Latif Ramadan and Galal Mahrous Moatimid},
  title = {Numerical Solution of Linear and Nonlinear Integral Equations Via Improved Block-Pulse Functions},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {6},
  number = {2},
  pages = {19-34},
  doi = {10.11648/j.ajmcm.20210602.11},
  url = {https://doi.org/10.11648/j.ajmcm.20210602.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20210602.11},
  abstract = {This paper is concerned with a numerical method based on the improved block-pulse basis functions (IBPFs). It is done mainly to solve linear and nonlinear Volterra and Fredholm integral equations of the second kind. These equations can be simplified into a linear system of algebraic equations by using IBPFs and their operational matrix of integration. After that, the system can be programmed and solved using Mathematica. The changes made to the method obviously improved - as it will be shown in the numerical examples - the time taken by the program to solve the system of algebraic equations. Also, it is reflected in the accuracy of the solution. This modification works perfectly and improved the accuracy over the regular block–pulse basis functions (BPF). A slight change in the intervals of the BPF changes the whole technique to a new easier and more accurate technique. This change has worked well while solving different types of integral equations. The accompanied theorems of the IBPF technique and error estimation are stated and proved. The paper also dealt with the uniqueness and convergence theorems of the solution. Numerical examples are presented to illustrate the efficiency and accuracy of the method. The tables and required graphs are also shown to prove and demonstrate the efficiency.},
 year = {2021}
}

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TY - JOUR
T1 - Numerical Solution of Linear and Nonlinear Integral Equations Via Improved Block-Pulse Functions
AU - Mahmoud Hamed Taha
AU - Mohamed Abdel-Latif Ramadan
AU - Galal Mahrous Moatimid
Y1 - 2021/05/26
PY - 2021
N1 - https://doi.org/10.11648/j.ajmcm.20210602.11
DO - 10.11648/j.ajmcm.20210602.11
T2 - American Journal of Mathematical and Computer Modelling
JF - American Journal of Mathematical and Computer Modelling
JO - American Journal of Mathematical and Computer Modelling
SP - 19
EP - 34
PB - Science Publishing Group
SN - 2578-8280
UR - https://doi.org/10.11648/j.ajmcm.20210602.11
AB - This paper is concerned with a numerical method based on the improved block-pulse basis functions (IBPFs). It is done mainly to solve linear and nonlinear Volterra and Fredholm integral equations of the second kind. These equations can be simplified into a linear system of algebraic equations by using IBPFs and their operational matrix of integration. After that, the system can be programmed and solved using Mathematica. The changes made to the method obviously improved - as it will be shown in the numerical examples - the time taken by the program to solve the system of algebraic equations. Also, it is reflected in the accuracy of the solution. This modification works perfectly and improved the accuracy over the regular block–pulse basis functions (BPF). A slight change in the intervals of the BPF changes the whole technique to a new easier and more accurate technique. This change has worked well while solving different types of integral equations. The accompanied theorems of the IBPF technique and error estimation are stated and proved. The paper also dealt with the uniqueness and convergence theorems of the solution. Numerical examples are presented to illustrate the efficiency and accuracy of the method. The tables and required graphs are also shown to prove and demonstrate the efficiency.
VL - 6
IS - 2
ER -

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Author Information

Mahmoud Hamed Taha

Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt
Mohamed Abdel-Latif Ramadan

Department of Mathematics and Computer Science, Faculty of Science, Menoufia University, Menoufia, Egypt
Galal Mahrous Moatimid

Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt

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APA Style

Mahmoud Hamed Taha, Mohamed Abdel-Latif Ramadan, Galal Mahrous Moatimid. (2021). Numerical Solution of Linear and Nonlinear Integral Equations Via Improved Block-Pulse Functions. American Journal of Mathematical and Computer Modelling, 6(2), 19-34. https://doi.org/10.11648/j.ajmcm.20210602.11

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ACS Style

Mahmoud Hamed Taha; Mohamed Abdel-Latif Ramadan; Galal Mahrous Moatimid. Numerical Solution of Linear and Nonlinear Integral Equations Via Improved Block-Pulse Functions. Am. J. Math. Comput. Model. 2021, 6(2), 19-34. doi: 10.11648/j.ajmcm.20210602.11

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AMA Style

Mahmoud Hamed Taha, Mohamed Abdel-Latif Ramadan, Galal Mahrous Moatimid. Numerical Solution of Linear and Nonlinear Integral Equations Via Improved Block-Pulse Functions. Am J Math Comput Model. 2021;6(2):19-34. doi: 10.11648/j.ajmcm.20210602.11

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@article{10.11648/j.ajmcm.20210602.11,
  author = {Mahmoud Hamed Taha and Mohamed Abdel-Latif Ramadan and Galal Mahrous Moatimid},
  title = {Numerical Solution of Linear and Nonlinear Integral Equations Via Improved Block-Pulse Functions},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {6},
  number = {2},
  pages = {19-34},
  doi = {10.11648/j.ajmcm.20210602.11},
  url = {https://doi.org/10.11648/j.ajmcm.20210602.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20210602.11},
  abstract = {This paper is concerned with a numerical method based on the improved block-pulse basis functions (IBPFs). It is done mainly to solve linear and nonlinear Volterra and Fredholm integral equations of the second kind. These equations can be simplified into a linear system of algebraic equations by using IBPFs and their operational matrix of integration. After that, the system can be programmed and solved using Mathematica. The changes made to the method obviously improved - as it will be shown in the numerical examples - the time taken by the program to solve the system of algebraic equations. Also, it is reflected in the accuracy of the solution. This modification works perfectly and improved the accuracy over the regular block–pulse basis functions (BPF). A slight change in the intervals of the BPF changes the whole technique to a new easier and more accurate technique. This change has worked well while solving different types of integral equations. The accompanied theorems of the IBPF technique and error estimation are stated and proved. The paper also dealt with the uniqueness and convergence theorems of the solution. Numerical examples are presented to illustrate the efficiency and accuracy of the method. The tables and required graphs are also shown to prove and demonstrate the efficiency.},
 year = {2021}
}

Copy | Download