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Solution of Volterra and Fredholm Classes of Equations via Triangular Orthogonal Function (A Combination of Right Hand Triangular Function and Left Hand Triangular Function) and Hybrid Orthogonal Function (A Combination of Sample Hold Function and Right Hand Triangular Function)

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Abstract

In this paper the authors have dealt with seven kinds of non-linear Volterra and Fredholm classes of equations. The authors have formulated an algorithm for solving the aforementioned equation types via Hybrid Function (HF) and Triangular Function (TF) piecewise-linear orthogonal approach. In this approach the authors have reduced integral equation or integro-differential equation into equivalent system of simultaneous non-linear equation and have employed either Newton’s method or Broyden’s method to solve the simultaneous non-linear equations. The authors have calculated the L2-norm error and the max-norm error for both HF and TF method for each kind of equations. Through the illustrated examples, the authors have shown that the HF based algorithm produces stable result, on the contrary TF-computational method yields either stable, anomalous or unstable results.

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References

  1. A.-M. Wazwaz, Linear and Nonlinear Integral Equations: Methods and Applications (Springer, Berlin, 2011). ISBN 978-3-642-21448-6

    Book  MATH  Google Scholar 

  2. H. Brunner, Implicitly linear collocation method for nonlinear Volterra equations. Appl. Numer. Math. 9(3–5), 235–247 (1982)

    MathSciNet  MATH  Google Scholar 

  3. M.G. Porshokouhi et al., Numerical solution for non-linear Fredholm integral equations by Newton–Kantorovich method and comparison with HPM and ADM. Int. J. Pure Appl. Sci. Technol. 3(1), 44–49 (2011)

    Google Scholar 

  4. Jafar. Biazar, Mostafa. Eslami, Homotopy perturbation and Taylor series for Volterra integral equations of the second kind. Middle-East J. Sci. Res. 7(4), 604–609 (2011)

    Google Scholar 

  5. K. Kumar, I.H. Sloan, A new collocation-type method for Hammerstein integral equations. Math. Comp. 48(178), 585–593 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  6. G.Q. Han, Asymptotic error expansion of a collocation-type method for Volterra–Hammerstein integral equations. Appl. Numer. Math. 13(5), 357–369 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Deb, A. Dasgupta, G. Sarkar, A new set of orthogonal functions and its application to the analysis of dynamic systems. J. Frankl. Inst. 343, 1–26 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Deb, G. Sarkar, P. Mandal, A. Biswas, A. Ganguly, D. Biswas, Transfer function identification from impulse response via a new set of orthogonal hybrid functions (HF). Appl. Math. Comput. 218, 4760–4787 (2012)

    MathSciNet  MATH  Google Scholar 

  9. A. Micke, M. Bulow, Application of Volterra integral equations to the modelling of the sorption kinetics of multi-component mixtures in porous media: I. Fundam. Gas Sep. Purif. 4(3), 158–164 (1990)

    Article  Google Scholar 

  10. O. Diekmann, Thresholds and travelling waves for the geographical spread of infection. J. Math. Biol. 6, 109–130 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  11. K. Maleknejad, M. Hadizadeh, A note on a nonlinear problem in cell membrane theory, nonlinear analysis. Theory Methods Appl. 30(6), 3947–3950 (1997)

    Article  MATH  Google Scholar 

  12. M. Valipour, M.E. Banihabib, S.M.R. Behbahni, Comparison of the ARMA, ARIMA, and the autoregressive artificial neural network models in forecasting the monthly inflow of Dez dam reservoir. J. Hydrol. 476(7), 433–441 (2013)

    Article  Google Scholar 

  13. D.P. Viero, Modeling anisotropy in free-surface overland and shallow inundation flows. Adv. Water Resour. 104, 1–14 (2017)

    Article  Google Scholar 

  14. M. Valipour, Global experience on irrigation management under different scenarios. J. Water Land Dev. 32(1), 95–102 (2017)

    Article  MathSciNet  Google Scholar 

  15. M. Valipour, Variations of land use and irrigation for next decades under different scenarios. Irriga. Braz. J. Irrig. Drain. 1(1), 262–288 (2016)

    Google Scholar 

  16. M. Valipour, How much meteorological information is necessary to achieve reliable accuracy for rainfall estimations? Agriculture 6(4), 1–9 (2016)

    Article  Google Scholar 

  17. M. Valipour, M.A.G. Sefidkouhi, M.R. Sarjaz, Selecting the best model to estimate potential evapotranspiration with respect to climate change and magnitudes of extreme events. Agric. Water Manag. 180(Part A), 50–60 (2017)

    Article  Google Scholar 

  18. A. Deb, S. Roychoudhury, G. Sarkar, Analysis and Identification of Time-Invariant, Time-Varying and Multi-delay Systems Using Orthogonal Hybrid Functions: Theory and Algorithms with MATLAB ® , Studies in Systems, Decision and Control, vol. 46. Springer, Switzerland (2016)

  19. A. Deb, S. Gautam, A. Sengupta, Triangular Orthogonal Functions for the Analysis of Continuous Time Systems (Anthem Press, London, 2011)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Defence Institute of Advanced Technology, Pune, 411025, Maharashtra, India

    Anirban Mukhopadhyay

  2. St. Thomas’ College of Engineering and Technology, Kolkata, 700023, West Bengal, India

    Anindita Ganguly & Saumya Deep Chatterjee

Authors
  1. Anirban Mukhopadhyay
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  2. Anindita Ganguly
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  3. Saumya Deep Chatterjee
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Correspondence to Saumya Deep Chatterjee.

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Mukhopadhyay, A., Ganguly, A. & Chatterjee, S.D. Solution of Volterra and Fredholm Classes of Equations via Triangular Orthogonal Function (A Combination of Right Hand Triangular Function and Left Hand Triangular Function) and Hybrid Orthogonal Function (A Combination of Sample Hold Function and Right Hand Triangular Function). J. Inst. Eng. India Ser. B 99, 181–209 (2018). https://doi.org/10.1007/s40031-017-0297-1

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  • DOI: https://doi.org/10.1007/s40031-017-0297-1

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