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The Adomian Decomposition Method of Volterra Integral Equation of Second Kind

Received: 23 August 2018     Accepted: 20 September 2018     Published: 1 November 2018
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Abstract

In this work, we consider linear and nonlinear Volterra integral equations of the second kind. Here, by converting integral equation of the first kind to a linear equation of the second kind and the ordinary differential equation to integral equation we are going to solve the equation easily. The Adomian decomposition method or shortly (ADM) is used to find a solution to these equations. The Adomian decomposition method converts the Volterra integral equations into determination of computable components. The existence and uniqueness of solutions of linear (or nonlinear) Volterra integral equations of the second kind are expressed by theorems. If an exact solution exists for the problem, then the obtained series convergence very rabidly to that solution. A nonlinear term F(u) in nonlinear volterra integral equations is Lipschitz continuous and has polynomial representation. Finally, the sufficient condition that guarantees a unique solution of Volterra (linear and nonlinear) integral equations with the choice of the initial data is obtained, and the solution is found in series form. Theoretical considerations are being discussed. To illustrate the ability and simplicity of the method. A few examples including linear and nonlinear are provided to show validity and applicability of this approach. The results are taken from the works mentioned in the reference.

Published in American Journal of Applied Mathematics (Volume 6, Issue 4)
DOI 10.11648/j.ajam.20180604.12
Page(s) 142-148
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), 2018. Published by Science Publishing Group

Keywords

Linear Integral Equations, Fredholm Integral Equations, Regularization Method, Direct Computation Method, Two - Dimensional Integral Equations

References
[1] G. Adomian, Nonlinear Stochastic Operator Equations, Academic Press, San Diego, CA, 1986.
[2] G. Adomian, Solution of physical problems by decomposition, Appl. Math. Comput. 27 (9/10) (1994) 145-154.
[3] K. E. Atkinson, A survey of numerical methods for solving nonlinear integral equations. Integral Equations Appl., 1992, 4, 15–40.
[4] J. Biazar, E. Babolian, Solution of a system of nonlinear Volterra integral equations of the second kind, Far East J. Math. Sci. 2 (6) (2000)935–945.
[5] E. Babolian, J. Biazar, A. R. Vahidi, The decomposition method applied to systems of Fredholm integral equations of the second kind, Appl. Math. Comput. 148 (2004) 443–452.
[6] I. L. El-Kalla, Convergence of the Adomian method applied to a class of nonlinear integral equations, Elsevier Ltd. All rights reserved, 21 (2008) 372–376.
[7] H. Sadeghi Goghary, Sh. Javadi, E. Babolian, Restarted Adomian method for system of nonlinear Volterra integral equations, Appl. Math. Comput. 161 (2005) 745–751.
[8] H. Jafari, E. Tayyebi1, S. Sadeghi, C. M. Khalique, A new modification of the Adomian decomposition method for nonlinear integral equations, Int. J. Adv. Appl. Math, and Mech. 1(4) (2014) 33 – 39.
[9] M. Rahman. (2007). Integral Equations and their Applications, 1st Ed; WIT Press.
[10] P. Linz. (1985). Analytical and Numerical Methods for Volterra Equations.
[11] Porter, David. (1990). Integral equations A practical treatment, from spectral theory to applications, first edition.
[12] F. Smithies (1958). Cambridge Tracts in Mathematics and Mathematical Physics.
[13] Peter J. Collins. (2006). Differential and Integral equations, WIT Press.
[14] T. A. Burton Eds. (2005). Volterra Integral and Differential Equations, 2nd ed; Academic Press, Elsevier.
[15] S. Abbasbandy, Numerical solution of the integral equations: Homotopy perturbation method and Adomian decomposition method, Appl. Math. Comput. 173 (2006) 393–500.
[16] Stephen M. Zemyan, the Classical Theory of Integral Equations, Springer, Pennsylvania State University Mont Alto Mont Alto, PA, USA, 2010.
[17] Svetlin G. Georgiev, Integral Equations on Time Scales, Atlantis Press and the author(s) 2016.
[18] A. Wazwaz. A First Course in Integral Equations -Solutions Manual, 2nd ed; World Scientific Publishing Co. Pte. Ltd.
[19] F. G. Trkloml, Integral Equations First published (1957) 49, Library of congress catalog number 57-1.
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  • APA Style

    Ali Elhrary Abaoub, Abejela Salem Shkheam, Suad Mawloud Zali. (2018). The Adomian Decomposition Method of Volterra Integral Equation of Second Kind. American Journal of Applied Mathematics, 6(4), 142-148. https://doi.org/10.11648/j.ajam.20180604.12

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    ACS Style

    Ali Elhrary Abaoub; Abejela Salem Shkheam; Suad Mawloud Zali. The Adomian Decomposition Method of Volterra Integral Equation of Second Kind. Am. J. Appl. Math. 2018, 6(4), 142-148. doi: 10.11648/j.ajam.20180604.12

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    AMA Style

    Ali Elhrary Abaoub, Abejela Salem Shkheam, Suad Mawloud Zali. The Adomian Decomposition Method of Volterra Integral Equation of Second Kind. Am J Appl Math. 2018;6(4):142-148. doi: 10.11648/j.ajam.20180604.12

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  • @article{10.11648/j.ajam.20180604.12,
      author = {Ali Elhrary Abaoub and Abejela Salem Shkheam and Suad Mawloud Zali},
      title = {The Adomian Decomposition Method of Volterra Integral Equation of Second Kind},
      journal = {American Journal of Applied Mathematics},
      volume = {6},
      number = {4},
      pages = {142-148},
      doi = {10.11648/j.ajam.20180604.12},
      url = {https://doi.org/10.11648/j.ajam.20180604.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.20180604.12},
      abstract = {In this work, we consider linear and nonlinear Volterra integral equations of the second kind. Here, by converting integral equation of the first kind to a linear equation of the second kind and the ordinary differential equation to integral equation we are going to solve the equation easily. The Adomian decomposition method or shortly (ADM) is used to find a solution to these equations. The Adomian decomposition method converts the Volterra integral equations into determination of computable components. The existence and uniqueness of solutions of linear (or nonlinear) Volterra integral equations of the second kind are expressed by theorems. If an exact solution exists for the problem, then the obtained series convergence very rabidly to that solution. A nonlinear term F(u) in nonlinear volterra integral equations is Lipschitz continuous and has polynomial representation. Finally, the sufficient condition that guarantees a unique solution of Volterra (linear and nonlinear) integral equations with the choice of the initial data is obtained, and the solution is found in series form. Theoretical considerations are being discussed. To illustrate the ability and simplicity of the method. A few examples including linear and nonlinear are provided to show validity and applicability of this approach. The results are taken from the works mentioned in the reference.},
     year = {2018}
    }
    

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    T1  - The Adomian Decomposition Method of Volterra Integral Equation of Second Kind
    AU  - Ali Elhrary Abaoub
    AU  - Abejela Salem Shkheam
    AU  - Suad Mawloud Zali
    Y1  - 2018/11/01
    PY  - 2018
    N1  - https://doi.org/10.11648/j.ajam.20180604.12
    DO  - 10.11648/j.ajam.20180604.12
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
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    EP  - 148
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.20180604.12
    AB  - In this work, we consider linear and nonlinear Volterra integral equations of the second kind. Here, by converting integral equation of the first kind to a linear equation of the second kind and the ordinary differential equation to integral equation we are going to solve the equation easily. The Adomian decomposition method or shortly (ADM) is used to find a solution to these equations. The Adomian decomposition method converts the Volterra integral equations into determination of computable components. The existence and uniqueness of solutions of linear (or nonlinear) Volterra integral equations of the second kind are expressed by theorems. If an exact solution exists for the problem, then the obtained series convergence very rabidly to that solution. A nonlinear term F(u) in nonlinear volterra integral equations is Lipschitz continuous and has polynomial representation. Finally, the sufficient condition that guarantees a unique solution of Volterra (linear and nonlinear) integral equations with the choice of the initial data is obtained, and the solution is found in series form. Theoretical considerations are being discussed. To illustrate the ability and simplicity of the method. A few examples including linear and nonlinear are provided to show validity and applicability of this approach. The results are taken from the works mentioned in the reference.
    VL  - 6
    IS  - 4
    ER  - 

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Author Information
  • Mathematics Department, Science Faculty, Sabratha University, Sabratha, Libya

  • Mathematics Department, Science Faculty, Sabratha University, Sabratha, Libya

  • Mathematics Department, Science Faculty, Sabratha University, Sabratha, Libya

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