A Coupling Method of Regularization and Adomian Decomposition for Solving a Class of the Fredholm Integral Equations Within Local Fractional Operators

Hassan Kamil Jassim

doi:doi:10.11648/j.dmath.20170203.16

| Peer-Reviewed

A Coupling Method of Regularization and Adomian Decomposition for Solving a Class of the Fredholm Integral Equations Within Local Fractional Operators

Hassan Kamil Jassim

Published in International Journal of Discrete Mathematics (Volume 2, Issue 3)

Received: 9 February 2017 Accepted: 13 March 2017 Published: 29 March 2017

Views: Downloads:

Download PDF

Share This Article

Twitter
Linked In
Facebook

Abstract

In this paper, we will apply the combined regularization-Adomian decomposition method within local fractional differential operators to handle local fractional Fredholm integral equation of the first kind. Theoretical considerations are being discussed. To illustrate the ability and simplicity of the method, some examples are provided. The iteration procedure is based on local fractional derivative. The obtained results reveal that the proposed methods are very efficient and simple tools for solving local fractional integral equations.

Published in	International Journal of Discrete Mathematics (Volume 2, Issue 3)
DOI	10.11648/j.dmath.20170203.16
Page(s)	95-99
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

Local Fractional Fredholm Integral Equation, Local Fractional Adomian Decomposition Method, Local Fractional Operator

References

[1]	X. J. Yang, “Local fractional integral equations and their applications”, Advances in Computer Science and its Applications, 1 (2012) 234- 239.
[2]	A. M. Wazwaz, “Linear and Nonlinear Integral Equations: Methods and Applications”, Springer, New York, NY, USA, 2011.
[3]	S. S. Ray and P. K. Sahu, “Numerical Methods for Solving Fredholm Integral Equations of Second Kind”, Abstract and Applied Analysis, 2013 (2013) 1-17.
[4]	K. Kolwankar, A. D. Gangal, “Local fractional Fokker–Planck equation”, Phys. Rev. Lett., 80 (1998) 214- 217.
[5]	X. J. Yang and D. Baleanu, “Fractal heat conduction problem solved by local fractional variation iteration Method,” Thermal Science, 17 (2) (2013) 625-628.
[6]	H. K. Jassim, “The Approximate Solutions of Three-Dimensional Diffusion and Wave Equations within Local Fractional Derivative Operator, Abstract and Applied Analysis, 2016 (2016) 1-5.
[7]	X. J. Yang, D. Baleanu, and W. P. Zhong, “Approximation solutions for diffusion equation on Cantor time-space,” Proceeding of the Romanian Academy A, 14 (2) (2013) 127–133.
[8]	H. Jafari, and H. K. Jassim, “Local Fractional Series Expansion Method for Solving Laplace and Schrodinger Equations on Cantor Sets within Local Fractional Operators”, International Journal of Mathematics and Computer Research, 2 (11) (2014) 736-744.
[9]	S. P. Yan, H. Jafari, and H. K. Jassim, “Local Fractional Adomian Decomposition and Function Decomposition Methods for Solving Laplace Equation within Local Fractional Operators”, Advances in Mathematical Physics, 2014 (2014) 1-7.
[10]	X. J. Yang, Local fractional integral equations and their applications, ACSA, 1 (4) (2012) 234- 239.
[11]	H. K. Jassim, The Approximate Solutions of Fredholm Integral Equations on Cantor Sets within Local Fractional Operators, Sahand Communications in Mathematical Analysis, 16 (1) (2016) 13-20.
[12]	H. Jafari, H. K. Jassim, S. P. Moshokoa, V. M. Ariyan and F. Tchier, Reduced differential transform method for partial differential equations within local fractional derivative operators, Advances in Mechanical Engineering, 8 (4) (2016) 1-6.
[13]	H. Jafari, H. K. Jassim, Local Fractional Variational Iteration Method for Nonlinear Partial Differential Equations within Local Fractional Operators, Applications and Applied Mathematics, 10 (2) (2015) 1055-1065.
[14]	H. K. Jassim, Y. J. Yang, and S. Q. Wang, Local Fractional Function Decomposition Method for Solving Inhomogeneous Wave Equations with Local Fractional Derivative, Abstract and Applied Analysis, 2014 (2014) 1-6.
[15]	H. K. Jassim, Analytical Approximate Solution for Inhomogeneous Wave Equation on Cantor Sets by Local Fractional Variational Iteration Method, International Journal of Advances in Applied Mathematics and Mechanics, 3 (1) (2015) 57-61.
[16]	X. J. Yang, Y. Zhang, A new Adomian decomposition procedure scheme for solving local fractional Volterra integral equation, Advances in Information Technology and Management, 1 (4) (2012) 158-161.
[17]	H. K. Jassim, Hussein Khashan Kadhim, Application of Local Fractional Variational Iteration Method for Solving Fredholm Integral Equations Involving Local Fractional Operators, Journal of University of Thi-Qar, 11 (1) (2016) 12-18.
[18]	X. J. Yang, Picards approximation method for solving a class of local fractional Volterra integral equations, Advances in Intelligent Transportation Systems, 1 (2012) 67-70.
[19]	H. K. Jassim, On Analytical Methods for Solving Poisson Equation, Scholars Journal of Research in Mathematics and Computer Science, 1 (1) (2016) 26- 35.
[20]	H. K. Jassim, Application of Laplace Decomposition Method and Variational Iteration Transform Method to Solve Laplace Equation, Universal Journal of Mathematics, 1 (1) (2016) 16-23.
[21]	D. Baleanu, H. K. Jassim, and M. Al-Qurashi, Approximate Analytical Solutions of Goursat Problem within Local Fractional Operators, Journal of Nonlinear Science and Applications, 9 (6) (2016) 4829-4837.
[22]	X. J. Yang, J. A. Tenreiro Machadob, and H. M. Srivastava, Anew numerical technique for solving the local fractional diffusion equation: Two dimensional extended differential transform approach, Applied Mathematics and Computation, 274 (2016) 143-151.
[23]	A. M. Yang,, X. J. Yang, and C. Cattani, et al., Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets, Abstract and Applied Analysis, 2013 (2013) 1-5.
[24]	X. J. Yang, H. M. Srivastava and C. Cattani, Local fractional homotopy perturbation method for solving fractal partial differential equations arising in mathematical physics, Rom. Rep. Phys., 67 (3) (2015) 752-761.
[25]	Y. Zhang, X. J. Yang, and C. Cattani, Local fractional homotopy perturbation method for solving nonhomogeneous heat conduction equations in fractal domains, Entropy, 17 (2015) 6753-6764.
[26]	X. J. Yang, D. Baleanu, H. M. Srivastava, Local fractional similarity solution for the diffusion equation defined on Cantor sets. Appl. Math. Lett. 47 (2015) 54-60.
[27]	H. K. Jassim, New Approaches for Solving Fokker Planck Equation on Cantor Sets within Local Fractional Operators, Journal of Mathematics, 2015 (2015) 1-8.
[28]	H. Jafari, and H. K. Jassim, Local Fractional Laplace Variational Iteration Method for Solving Nonlinear Partial Differential Equations on Cantor Sets within Local Fractional Operators, Journal of Zankoy Sulaimani-Part A, 16 (4) (2014) 49-57.
[29]	M. S. Hu, R. P. Agarwal and X. J. Yang, Local fractional Fourier series with application to wave equation in fractal vibrating string, Abstract and Appl. Analysis, 2012 (2012) 1-15.
[30]	Y. J. Yang, D. Baleanu, and X. J. Yang, Analysis of fractal wave equations by local fractional Fourier series method, Advances in Mathematical Physics, 2013 (2013) 1-6,.
[31]	A. M. Yang, C. Cattani, C. Zhang, G. N. Xie, and X. J. Yang, Local fractional Fourier series solutions for nonhomogeneous heat equations arising in fractal heat with local fractional derivative, Advances in Mechanical Engineering, 2014 (2014) 1-5.
[32]	H. K. Jassim, Local Fractional variational iteration transform method to solve partial differential equations arising in mathematical physics, International Journal of Advances in Applied Mathematics and Mechanics, 3 (1) (2015) 71-76.
[33]	H. Jafari, H. K. Jassim, Local Fractional Variational Iteration Method for Nonlinear Partial Differential Equations within Local Fractional Operators, Applications and Applied Mathematics, 10 (2) (2015) 1055-1065.

Cite This Article

Plain Text BibTeX RIS

APA Style

Hassan Kamil Jassim. (2017). A Coupling Method of Regularization and Adomian Decomposition for Solving a Class of the Fredholm Integral Equations Within Local Fractional Operators. International Journal of Discrete Mathematics, 2(3), 95-99. https://doi.org/10.11648/j.dmath.20170203.16

Copy | Download

ACS Style

Hassan Kamil Jassim. A Coupling Method of Regularization and Adomian Decomposition for Solving a Class of the Fredholm Integral Equations Within Local Fractional Operators. Int. J. Discrete Math. 2017, 2(3), 95-99. doi: 10.11648/j.dmath.20170203.16

Copy | Download

AMA Style

Hassan Kamil Jassim. A Coupling Method of Regularization and Adomian Decomposition for Solving a Class of the Fredholm Integral Equations Within Local Fractional Operators. Int J Discrete Math. 2017;2(3):95-99. doi: 10.11648/j.dmath.20170203.16

Copy | Download

@article{10.11648/j.dmath.20170203.16,
  author = {Hassan Kamil Jassim},
  title = {A Coupling Method of Regularization and Adomian Decomposition for Solving a Class of the Fredholm Integral Equations Within Local Fractional Operators},
  journal = {International Journal of Discrete Mathematics},
  volume = {2},
  number = {3},
  pages = {95-99},
  doi = {10.11648/j.dmath.20170203.16},
  url = {https://doi.org/10.11648/j.dmath.20170203.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20170203.16},
  abstract = {In this paper, we will apply the combined regularization-Adomian decomposition method within local fractional differential operators to handle local fractional Fredholm integral equation of the first kind. Theoretical considerations are being discussed. To illustrate the ability and simplicity of the method, some examples are provided. The iteration procedure is based on local fractional derivative. The obtained results reveal that the proposed methods are very efficient and simple tools for solving local fractional integral equations.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - A Coupling Method of Regularization and Adomian Decomposition for Solving a Class of the Fredholm Integral Equations Within Local Fractional Operators
AU  - Hassan Kamil Jassim
Y1  - 2017/03/29
PY  - 2017
N1  - https://doi.org/10.11648/j.dmath.20170203.16
DO  - 10.11648/j.dmath.20170203.16
T2  - International Journal of Discrete Mathematics
JF  - International Journal of Discrete Mathematics
JO  - International Journal of Discrete Mathematics
SP  - 95
EP  - 99
PB  - Science Publishing Group
SN  - 2578-9252
UR  - https://doi.org/10.11648/j.dmath.20170203.16
AB  - In this paper, we will apply the combined regularization-Adomian decomposition method within local fractional differential operators to handle local fractional Fredholm integral equation of the first kind. Theoretical considerations are being discussed. To illustrate the ability and simplicity of the method, some examples are provided. The iteration procedure is based on local fractional derivative. The obtained results reveal that the proposed methods are very efficient and simple tools for solving local fractional integral equations.
VL  - 2
IS  - 3
ER  -

Copy | Download

Author Information

Hassan Kamil Jassim

Department of Mathematics, Faculty of Education for Pure Sciences, University of Thi-Qar, Nasiryah, Iraq

Download PDF

Sections

Plain Text BibTeX RIS

APA Style

Hassan Kamil Jassim. (2017). A Coupling Method of Regularization and Adomian Decomposition for Solving a Class of the Fredholm Integral Equations Within Local Fractional Operators. International Journal of Discrete Mathematics, 2(3), 95-99. https://doi.org/10.11648/j.dmath.20170203.16

Copy | Download

ACS Style

Hassan Kamil Jassim. A Coupling Method of Regularization and Adomian Decomposition for Solving a Class of the Fredholm Integral Equations Within Local Fractional Operators. Int. J. Discrete Math. 2017, 2(3), 95-99. doi: 10.11648/j.dmath.20170203.16

Copy | Download

AMA Style

Hassan Kamil Jassim. A Coupling Method of Regularization and Adomian Decomposition for Solving a Class of the Fredholm Integral Equations Within Local Fractional Operators. Int J Discrete Math. 2017;2(3):95-99. doi: 10.11648/j.dmath.20170203.16

Copy | Download

@article{10.11648/j.dmath.20170203.16,
  author = {Hassan Kamil Jassim},
  title = {A Coupling Method of Regularization and Adomian Decomposition for Solving a Class of the Fredholm Integral Equations Within Local Fractional Operators},
  journal = {International Journal of Discrete Mathematics},
  volume = {2},
  number = {3},
  pages = {95-99},
  doi = {10.11648/j.dmath.20170203.16},
  url = {https://doi.org/10.11648/j.dmath.20170203.16},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20170203.16},
  abstract = {In this paper, we will apply the combined regularization-Adomian decomposition method within local fractional differential operators to handle local fractional Fredholm integral equation of the first kind. Theoretical considerations are being discussed. To illustrate the ability and simplicity of the method, some examples are provided. The iteration procedure is based on local fractional derivative. The obtained results reveal that the proposed methods are very efficient and simple tools for solving local fractional integral equations.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - A Coupling Method of Regularization and Adomian Decomposition for Solving a Class of the Fredholm Integral Equations Within Local Fractional Operators
AU  - Hassan Kamil Jassim
Y1  - 2017/03/29
PY  - 2017
N1  - https://doi.org/10.11648/j.dmath.20170203.16
DO  - 10.11648/j.dmath.20170203.16
T2  - International Journal of Discrete Mathematics
JF  - International Journal of Discrete Mathematics
JO  - International Journal of Discrete Mathematics
SP  - 95
EP  - 99
PB  - Science Publishing Group
SN  - 2578-9252
UR  - https://doi.org/10.11648/j.dmath.20170203.16
AB  - In this paper, we will apply the combined regularization-Adomian decomposition method within local fractional differential operators to handle local fractional Fredholm integral equation of the first kind. Theoretical considerations are being discussed. To illustrate the ability and simplicity of the method, some examples are provided. The iteration procedure is based on local fractional derivative. The obtained results reveal that the proposed methods are very efficient and simple tools for solving local fractional integral equations.
VL  - 2
IS  - 3
ER  -

Copy | Download