Comparison Between Adomain Decomposition Method and Numerical Solutions of Linear Volterra Integral Equations of the Second Kind by Using the Fifth Order of Non-Polynomial Spline Functions

Elgaili Abdalla Elhassan Ibrahim; Abdel Radi Abdel Rahman Abdel Gadir Abdel Rahman; Neama Yahia Mohammed; Nageeb Abdallah Hamed Haroun

doi:doi:10.11648/j.ijamtp.20210703.12

Comparison Between Adomain Decomposition Method and Numerical Solutions of Linear Volterra Integral Equations of the Second Kind by Using the Fifth Order of Non-Polynomial Spline Functions

Elgaili Abdalla Elhassan Ibrahim, Abdel Radi Abdel Rahman Abdel Gadir Abdel Rahman, Neama Yahia Mohammed, Nageeb Abdallah Hamed Haroun

Published in International Journal of Applied Mathematics and Theoretical Physics (Volume 7, Issue 3)

Received: 11 August 2021 Accepted: 23 August 2021 Published: 31 August 2021

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Abstract

Volterra integral equations are a special type of integrative equations; they are divided into two categories referred to as the first and second type. This paper will deal with the second type which has wide range of the applications in physics and engineering problems. Spline functions are piece-wise polynomials of degree n joined together at the break points with n-1 continuous derivatives. The break points of splines are called Knot, spline function can be integrated and differentiated due to being piece wise polynomials and can easily store and implemented on digital computer, non-polynomial spline function apiece wise is a blend of trigonometric, as well as, polynomial basis function, which form a complete extended Chebyshev space. Matlab is a powerful computing system for handling the calculations involved scientific and engineering problems. The aim of this paper is to compare between Adomain decomposition method and numerical solution to solve Volterra Integral Equations of second kind using the fifth order non-polynomial Spline functions by Matlab. We followed the applied mathematical method numerically by Matlab. Numerical examples are presented to illustrate the applications of this methods and to compare the computed results with analytical solutions. Finally by comparison of numerical results, Simplicity and efficiency of this method be shown.

Published in	International Journal of Applied Mathematics and Theoretical Physics (Volume 7, Issue 3)
DOI	10.11648/j.ijamtp.20210703.12
Page(s)	68-79
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 Volterra Integral Equations, Second Kind, Non-Polynomial Spline Functions, Fifth Order, Adomain Decomposition Method

References

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[2]	Haq, F. I. (2009). Numerical Solution of Bounded Value Problem and Initial Value Problems Using Spline Function. ph.d, thesis, GhulamIshaq Institute of Engineering Science and Technology, Pakistan.
[3]	HARBI, S, MURAD, M and MAJEED, M 2015-A solution of second integral equation using third order Non-polynomial Spline function. College of Science for Women Baghdad University.
[4]	H. Brunner, theory and numerical solution of Volterra functional integral equations, Hong Kong Baptist University, (2010).
[5]	HOSSINPOUR, A 2012- The Solve of Integral Differential Equation by Non-polynomial Spline Function and Quadrature Formula. International Conference on Applied Mathematics and Pharmaceutical science Jan, pp 7-8, pp. 595-597.
[6]	LIMA, P and DIAGO, T 1997- An extrapolation method for a Volterra integral equation with weakly singular kernel Appl. Numer. Math, vol. 24, pp. 131-148.
[7]	Najwa, S and Mohammed, S presented a numerical solution for linear Voltera integral equations with weakly singular kernel by using non-polynomial spline function from the fifth degree, pp. 109-110.
[8]	Majeed. S. N (2014) Solution Of Second Kind Volterra Integro Equation Using Linear Non Polynomial Spline Equation. Mathematical Theory and Modeling.
[9]	M. S. Islam, M. Z. I. Bangalee, A. K.. Khan, and, A, Halder," approximate solution of systems of VIE's of second kind by Adomain decomposition method," Dhaka university journal of science, vol. 63, no. 1, pp. 15-18, 2007.
[10]	Rahman, M. M, Hakim M. A., Hassan M. K., Alam M. K. and Nowsher, L., 2012, Numerical Solution of Volterra Integral Equations of Second kind with the Help of Chebyshev Polynomials, Annals of Pure and Applied Mathematics, 1 (2): 158-167.
[11]	Ramadan, M. A.; EL-Danaf, T.; and E. I. Abdaal, F (2007). Application of the Non- Polynomial Spline Approach to the Solution of the Burgers Equation. The Open Applied Mathematics Journal (1): 15-20.
[12]	Rice, J. R. (1985). Numerical Method software and Analysis. Software and analysis, Mcgra Hill.
[13]	Sara. H. H, (2013), Algorithms for Solving Volterra Integral Equations Using Non-Polynomial Notch Functions, College of Science for Women Baghdad University.
[14]	Tahmasbi, A. (2008). New Approach numerical solution of Linear Volterra Integral Equations of Second kind. 3 (32), 1607-1610.
[15]	Taqi, A; Jumaa, B (2016). Non- polynomial spline functions to solve a system of two non- linear volterra integral equations. Kirkuk University, Scientific Studies.
[16]	WAZWAZ, A 2011-Linear and Non Linear Integral Equation Method and Application. Higher Education Press, Beijing, p. 66-69.
[17]	Zarebnia, M.; Hoshyar, M.; and Sedahti, M. (2011). Non-Polynomial Spline Method for the Solution of Problem in Calculus of Variations. word Academy Engendering Technology (51): 986-991.

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Elgaili Abdalla Elhassan Ibrahim, Abdel Radi Abdel Rahman Abdel Gadir Abdel Rahman, Neama Yahia Mohammed, Nageeb Abdallah Hamed Haroun. (2021). Comparison Between Adomain Decomposition Method and Numerical Solutions of Linear Volterra Integral Equations of the Second Kind by Using the Fifth Order of Non-Polynomial Spline Functions. International Journal of Applied Mathematics and Theoretical Physics, 7(3), 68-79. https://doi.org/10.11648/j.ijamtp.20210703.12

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ACS Style

Elgaili Abdalla Elhassan Ibrahim; Abdel Radi Abdel Rahman Abdel Gadir Abdel Rahman; Neama Yahia Mohammed; Nageeb Abdallah Hamed Haroun. Comparison Between Adomain Decomposition Method and Numerical Solutions of Linear Volterra Integral Equations of the Second Kind by Using the Fifth Order of Non-Polynomial Spline Functions. Int. J. Appl. Math. Theor. Phys. 2021, 7(3), 68-79. doi: 10.11648/j.ijamtp.20210703.12

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AMA Style

Elgaili Abdalla Elhassan Ibrahim, Abdel Radi Abdel Rahman Abdel Gadir Abdel Rahman, Neama Yahia Mohammed, Nageeb Abdallah Hamed Haroun. Comparison Between Adomain Decomposition Method and Numerical Solutions of Linear Volterra Integral Equations of the Second Kind by Using the Fifth Order of Non-Polynomial Spline Functions. Int J Appl Math Theor Phys. 2021;7(3):68-79. doi: 10.11648/j.ijamtp.20210703.12

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@article{10.11648/j.ijamtp.20210703.12,
  author = {Elgaili Abdalla Elhassan Ibrahim and Abdel Radi Abdel Rahman Abdel Gadir Abdel Rahman and Neama Yahia Mohammed and Nageeb Abdallah Hamed Haroun},
  title = {Comparison Between Adomain Decomposition Method and Numerical Solutions of Linear Volterra Integral Equations of the Second Kind by Using the Fifth Order of Non-Polynomial Spline Functions},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {7},
  number = {3},
  pages = {68-79},
  doi = {10.11648/j.ijamtp.20210703.12},
  url = {https://doi.org/10.11648/j.ijamtp.20210703.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20210703.12},
  abstract = {Volterra integral equations are a special type of integrative equations; they are divided into two categories referred to as the first and second type. This paper will deal with the second type which has wide range of the applications in physics and engineering problems. Spline functions are piece-wise polynomials of degree n joined together at the break points with n-1 continuous derivatives. The break points of splines are called Knot, spline function can be integrated and differentiated due to being piece wise polynomials and can easily store and implemented on digital computer, non-polynomial spline function apiece wise is a blend of trigonometric, as well as, polynomial basis function, which form a complete extended Chebyshev space. Matlab is a powerful computing system for handling the calculations involved scientific and engineering problems. The aim of this paper is to compare between Adomain decomposition method and numerical solution to solve Volterra Integral Equations of second kind using the fifth order non-polynomial Spline functions by Matlab. We followed the applied mathematical method numerically by Matlab. Numerical examples are presented to illustrate the applications of this methods and to compare the computed results with analytical solutions. Finally by comparison of numerical results, Simplicity and efficiency of this method be shown.},
 year = {2021}
}

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TY  - JOUR
T1  - Comparison Between Adomain Decomposition Method and Numerical Solutions of Linear Volterra Integral Equations of the Second Kind by Using the Fifth Order of Non-Polynomial Spline Functions
AU  - Elgaili Abdalla Elhassan Ibrahim
AU  - Abdel Radi Abdel Rahman Abdel Gadir Abdel Rahman
AU  - Neama Yahia Mohammed
AU  - Nageeb Abdallah Hamed Haroun
Y1  - 2021/08/31
PY  - 2021
N1  - https://doi.org/10.11648/j.ijamtp.20210703.12
DO  - 10.11648/j.ijamtp.20210703.12
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 68
EP  - 79
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20210703.12
AB  - Volterra integral equations are a special type of integrative equations; they are divided into two categories referred to as the first and second type. This paper will deal with the second type which has wide range of the applications in physics and engineering problems. Spline functions are piece-wise polynomials of degree n joined together at the break points with n-1 continuous derivatives. The break points of splines are called Knot, spline function can be integrated and differentiated due to being piece wise polynomials and can easily store and implemented on digital computer, non-polynomial spline function apiece wise is a blend of trigonometric, as well as, polynomial basis function, which form a complete extended Chebyshev space. Matlab is a powerful computing system for handling the calculations involved scientific and engineering problems. The aim of this paper is to compare between Adomain decomposition method and numerical solution to solve Volterra Integral Equations of second kind using the fifth order non-polynomial Spline functions by Matlab. We followed the applied mathematical method numerically by Matlab. Numerical examples are presented to illustrate the applications of this methods and to compare the computed results with analytical solutions. Finally by comparison of numerical results, Simplicity and efficiency of this method be shown.
VL  - 7
IS  - 3
ER  -

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Author Information

Elgaili Abdalla Elhassan Ibrahim

Department of Mathematics, Faculty of Science, Sudan University for Science and Technology, Khartoum, Sudan
Abdel Radi Abdel Rahman Abdel Gadir Abdel Rahman

Department of Mathematics, Faculty of Education, Omdurman Islamic University, Omdurman, Sudan
Neama Yahia Mohammed

Department of Mathematics, College of Science, Tabuk University, Tabuk, Saudi Arabia
Nageeb Abdallah Hamed Haroun

Department of Mathematics, Faculty of Education, Omdurman Islamic University, Omdurman, Sudan

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APA Style

Elgaili Abdalla Elhassan Ibrahim, Abdel Radi Abdel Rahman Abdel Gadir Abdel Rahman, Neama Yahia Mohammed, Nageeb Abdallah Hamed Haroun. (2021). Comparison Between Adomain Decomposition Method and Numerical Solutions of Linear Volterra Integral Equations of the Second Kind by Using the Fifth Order of Non-Polynomial Spline Functions. International Journal of Applied Mathematics and Theoretical Physics, 7(3), 68-79. https://doi.org/10.11648/j.ijamtp.20210703.12

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ACS Style

Elgaili Abdalla Elhassan Ibrahim; Abdel Radi Abdel Rahman Abdel Gadir Abdel Rahman; Neama Yahia Mohammed; Nageeb Abdallah Hamed Haroun. Comparison Between Adomain Decomposition Method and Numerical Solutions of Linear Volterra Integral Equations of the Second Kind by Using the Fifth Order of Non-Polynomial Spline Functions. Int. J. Appl. Math. Theor. Phys. 2021, 7(3), 68-79. doi: 10.11648/j.ijamtp.20210703.12

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AMA Style

Elgaili Abdalla Elhassan Ibrahim, Abdel Radi Abdel Rahman Abdel Gadir Abdel Rahman, Neama Yahia Mohammed, Nageeb Abdallah Hamed Haroun. Comparison Between Adomain Decomposition Method and Numerical Solutions of Linear Volterra Integral Equations of the Second Kind by Using the Fifth Order of Non-Polynomial Spline Functions. Int J Appl Math Theor Phys. 2021;7(3):68-79. doi: 10.11648/j.ijamtp.20210703.12

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@article{10.11648/j.ijamtp.20210703.12,
  author = {Elgaili Abdalla Elhassan Ibrahim and Abdel Radi Abdel Rahman Abdel Gadir Abdel Rahman and Neama Yahia Mohammed and Nageeb Abdallah Hamed Haroun},
  title = {Comparison Between Adomain Decomposition Method and Numerical Solutions of Linear Volterra Integral Equations of the Second Kind by Using the Fifth Order of Non-Polynomial Spline Functions},
  journal = {International Journal of Applied Mathematics and Theoretical Physics},
  volume = {7},
  number = {3},
  pages = {68-79},
  doi = {10.11648/j.ijamtp.20210703.12},
  url = {https://doi.org/10.11648/j.ijamtp.20210703.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijamtp.20210703.12},
  abstract = {Volterra integral equations are a special type of integrative equations; they are divided into two categories referred to as the first and second type. This paper will deal with the second type which has wide range of the applications in physics and engineering problems. Spline functions are piece-wise polynomials of degree n joined together at the break points with n-1 continuous derivatives. The break points of splines are called Knot, spline function can be integrated and differentiated due to being piece wise polynomials and can easily store and implemented on digital computer, non-polynomial spline function apiece wise is a blend of trigonometric, as well as, polynomial basis function, which form a complete extended Chebyshev space. Matlab is a powerful computing system for handling the calculations involved scientific and engineering problems. The aim of this paper is to compare between Adomain decomposition method and numerical solution to solve Volterra Integral Equations of second kind using the fifth order non-polynomial Spline functions by Matlab. We followed the applied mathematical method numerically by Matlab. Numerical examples are presented to illustrate the applications of this methods and to compare the computed results with analytical solutions. Finally by comparison of numerical results, Simplicity and efficiency of this method be shown.},
 year = {2021}
}

Copy | Download

TY  - JOUR
T1  - Comparison Between Adomain Decomposition Method and Numerical Solutions of Linear Volterra Integral Equations of the Second Kind by Using the Fifth Order of Non-Polynomial Spline Functions
AU  - Elgaili Abdalla Elhassan Ibrahim
AU  - Abdel Radi Abdel Rahman Abdel Gadir Abdel Rahman
AU  - Neama Yahia Mohammed
AU  - Nageeb Abdallah Hamed Haroun
Y1  - 2021/08/31
PY  - 2021
N1  - https://doi.org/10.11648/j.ijamtp.20210703.12
DO  - 10.11648/j.ijamtp.20210703.12
T2  - International Journal of Applied Mathematics and Theoretical Physics
JF  - International Journal of Applied Mathematics and Theoretical Physics
JO  - International Journal of Applied Mathematics and Theoretical Physics
SP  - 68
EP  - 79
PB  - Science Publishing Group
SN  - 2575-5927
UR  - https://doi.org/10.11648/j.ijamtp.20210703.12
AB  - Volterra integral equations are a special type of integrative equations; they are divided into two categories referred to as the first and second type. This paper will deal with the second type which has wide range of the applications in physics and engineering problems. Spline functions are piece-wise polynomials of degree n joined together at the break points with n-1 continuous derivatives. The break points of splines are called Knot, spline function can be integrated and differentiated due to being piece wise polynomials and can easily store and implemented on digital computer, non-polynomial spline function apiece wise is a blend of trigonometric, as well as, polynomial basis function, which form a complete extended Chebyshev space. Matlab is a powerful computing system for handling the calculations involved scientific and engineering problems. The aim of this paper is to compare between Adomain decomposition method and numerical solution to solve Volterra Integral Equations of second kind using the fifth order non-polynomial Spline functions by Matlab. We followed the applied mathematical method numerically by Matlab. Numerical examples are presented to illustrate the applications of this methods and to compare the computed results with analytical solutions. Finally by comparison of numerical results, Simplicity and efficiency of this method be shown.
VL  - 7
IS  - 3
ER  -

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