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An Analytic Approach to Weakly-Singular Integro-Dynamic Equation on Time Scales

Received: 12 December 2016     Accepted: 22 December 2016     Published: 16 January 2017
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Abstract

In this paper, we present a new and simple approach to resolve linear and nonlinear weakly-singular Volterra integro-dynamic equations of first and second order on any time scales. In order to eliminate the singularity of the equation, nabla derivative is used and then transforming the given first-order integro-dynamic equations onto an first-order dynamic equations on time scales. The validity of the method is illustrated with some examples.

Published in International Journal of Discrete Mathematics (Volume 1, Issue 1)
DOI 10.11648/j.dmath.20160101.14
Page(s) 20-29
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), 2017. Published by Science Publishing Group

Keywords

Time Scales, Integro-Dynamic Equations, Volterra Integro-Differential Equation

References
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[3] M. Bohner, A. Peterson, Dynamic Equations on Time Scales, Birkhauser Boston, 2001.
[4] M. Bohner, A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser Boston, 2003.
[5] L. Bougoffa, R. C. Rach and A. Mennouni, An approximate method for solving a class of weakly-singular Volterra integro-differential equations, Appl. Math. Comput., 217 (22) (2011), 8907-8913.
[6] H. Brunner, On the numerical solution of nonlinear Voltera integro-differential equations, BIT Numerical Mathematics, 13 (1973) 381-390
[7] H. Brunner, On the numerical solution of nonlinear Volterra-Fredholm integral equations by collocation methods, SIAM J. Numer. Anal. , 27 (1990) 987-1000.
[8] G. Ebadi, M. Rahimi-Ardabili, S. Shahmorad, Numerical solution of nonlinear Voltera integro-diferential equations by the Tau method, Appl. Math. Comput. 188 (2007) 1580-1586.
[9] A. L. Jensen, Dynamics of populations with nonoverlapping generations, continuous mortality, and discrete reproductive periods. Ecol. Modelling , 74 (1994), 305–309.
[10] O. Lepik, Hear wavelet method for nonlinear integro-diferential equations, Appl. Math. comput. 176 (2006) 324-333
[11] K. Maleknejad, Y. Mahmoudi, Taylor polynomial solution of higher-order nonlinear Volterra-Fredholm integro-differential equations, Appl. Math. Comput. 145 (2003) 641-653.
[12] Y. Mahmoudi, Waveled Galerkin methodfor numerical solution of integral equations, Appl. Math. Comput. 167 (2005) 1119-1129.
[13] L. V. Nedorezova, B. N. Nedorezova, Correlation between models of population dynamics in continuous and discrete time. Ecol. Modelling , 82 (1995), 93–97.
[14] A. Mısır, S. Öğrekçi, On approximate solution of first-order weakly-singular Volterra integro-dynamic equation on time scales, Gazi University Journal of Science, 28 (4) (2015) 651–658.
[15] D. B. Pachpatte, On approximate solutions of a Volterra type integrodifferential equation on time scales, Int. Journal of Math. Analysis, 4 (34) (2010) 1651-1659.
[16] J. Saberi-Nadjafi, A. Ghorbani, He’s homotopy perturbation method : An effective tool for solving nonlinear integral and integro-differential equations, Comput. Math. Appl., 58 (2009) 2379-2390.
[17] A. M. Wazwaz, A reliable algorithm for solving boundary value problems for higher-order integro-differential equations, App. Math. Comput. 188 (2001) 327-342.
[18] A. M. Wazwaz, The combined Laplace transforms-Adomain decomposition method for handling nonlinear Voltera integro-diferential equations, Appl. Math. Comput. 216 (2010) 1304-1309.
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    Adil Mısır. (2017). An Analytic Approach to Weakly-Singular Integro-Dynamic Equation on Time Scales. International Journal of Discrete Mathematics, 1(1), 20-29. https://doi.org/10.11648/j.dmath.20160101.14

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    ACS Style

    Adil Mısır. An Analytic Approach to Weakly-Singular Integro-Dynamic Equation on Time Scales. Int. J. Discrete Math. 2017, 1(1), 20-29. doi: 10.11648/j.dmath.20160101.14

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    AMA Style

    Adil Mısır. An Analytic Approach to Weakly-Singular Integro-Dynamic Equation on Time Scales. Int J Discrete Math. 2017;1(1):20-29. doi: 10.11648/j.dmath.20160101.14

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  • @article{10.11648/j.dmath.20160101.14,
      author = {Adil Mısır},
      title = {An Analytic Approach to Weakly-Singular Integro-Dynamic Equation on Time Scales},
      journal = {International Journal of Discrete Mathematics},
      volume = {1},
      number = {1},
      pages = {20-29},
      doi = {10.11648/j.dmath.20160101.14},
      url = {https://doi.org/10.11648/j.dmath.20160101.14},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20160101.14},
      abstract = {In this paper, we present a new and simple approach to resolve linear and nonlinear weakly-singular Volterra integro-dynamic equations of first and second order on any time scales. In order to eliminate the singularity of the equation, nabla derivative is used and then transforming the given first-order integro-dynamic equations onto an first-order dynamic equations on time scales. The validity of the method is illustrated with some examples.},
     year = {2017}
    }
    

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    T2  - International Journal of Discrete Mathematics
    JF  - International Journal of Discrete Mathematics
    JO  - International Journal of Discrete Mathematics
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    AB  - In this paper, we present a new and simple approach to resolve linear and nonlinear weakly-singular Volterra integro-dynamic equations of first and second order on any time scales. In order to eliminate the singularity of the equation, nabla derivative is used and then transforming the given first-order integro-dynamic equations onto an first-order dynamic equations on time scales. The validity of the method is illustrated with some examples.
    VL  - 1
    IS  - 1
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