International Journal of Computer Science & Network Security

International Journal of Computer Science & Network Security (국제컴퓨터통신보호논문지학회)
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A Computer Oriented Solution for the Fractional Boundary Value Problem with Fuzzy Parameters with Application to Singular Perturbed Problems

  • Asklany, Somia A. (Computers and information technology department, Faculty of science and art, Turif, Northern Border University) ;
  • Youssef, I.K. (Mathematics department, Islamic University)
  • Received : 2021.12.05
  • Published : 2021.12.30
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Abstract

A treatment based on the algebraic operations on fuzzy numbers is used to replace the fuzzy problem into an equivalent crisp one. The finite difference technique is used to replace the continuous boundary value problem (BVP) of arbitrary order 1<α≤2, with fuzzy boundary parameters into an equivalent crisp (algebraic or differential) system. Three numerical examples with different behaviors are considered to illustrate the treatment of the singular perturbed case with different fractional orders of the BVP (α=1.8, α=1.9) as well as the classical second order (α=2). The calculated fuzzy solutions are compared with the crisp solutions of the singular perturbed BVP using triangular membership function (r-cut representation in parametric form) for different values of the singular perturbed parameter (ε=0.8, ε=0.9, ε=1.0). Results are illustrated graphically for the different values of the included parameters.

Keywords

Acknowledgement

The authors gratefully acknowledge the approval and the support of this research study by the grant no. SAT - 2017-1-8-F-7374 from the Deanship of Scientific Research at Northern Border University, Arar, K.S.A.

References

  1. M. Friedman, M. Ming and A. Kandel, Fuzzy Linear Systems, Fuzzy Sets and Systems 96, pp. 201-209, 1998. https://doi.org/10.1016/S0165-0114(96)00270-9
  2. T. Allahviranloo, M. Ghanbari, 5379On the algebraic solution of fuzzy linear systems based on interval theory, Applied Mathematical Modelling 36, 5360- , 2012. https://doi.org/10.1016/j.apm.2012.01.002
  3. I.K.Youssef, R.A.Ibrahim, Boundary Value Problems, Fredholm integral Equations, SOR and KSOR Methods, Life Science Journal, Vol. 10, No.2, 304-312, 2013.
  4. Hewayda M.S. Lotfy, Azza A. Tahha, I.K. Youssef, Fuzzy Linear systems via boundary value problem, Soft Computing, pp. 9647-9655, 2019. https://doi.org/10.1007/s00500-018-3529-7
  5. I. Podlubny, Fractional Differential Equations, Camb. Academic Press, San Diego, CA, 1999.
  6. Martin Stynes, A finite difference method for a two-point boundary value problem with a caputo fractional derivative, IMA Journal of Numerical Analysis, doi:10.1093/imanum/dru011, 1-24, 2014.
  7. MM Alsuyuti, EH Doha, SS Ezz-Eldien, I.K.Youssef, Spectral Galerkin schemes for a class of multi-order fractional pantograph equations, Journal of Computational and Applied Mathematics, Vol. 10, No.2, 304-312, 2021.
  8. I. K. Youssef, A. M. Shukur, Precondition for discretized fractional Boundary Value Problems, Pure and Applied mathematics Journal, Life Science Journal, Vol. 3, No.1, 1-6, 2014.
  9. Yousef Jamali Mohammad Amirian Matlob, The Concepts and Applications of Fractional Order Differential Calculus in Modelling of Viscoelastic Systems: A primer 1,2* , physics.bio-ph, 2017.
  10. Asklany, S.A.; Elhelow, K.; Youssef, I.K.; El-wahab, M.A. Rainfall events prediction using rule-based fuzzy inference system. Atmos. Res. 2011, 101, 228-236. https://doi.org/10.1016/j.atmosres.2011.02.015
  11. S. A. Askalany, K. Elhelow and M. Abd El-wahab, "On using Adaptive Hybrid Intelligent System in PM10 Prediction" International journal of Soft Computing and Engineering (IJSCE), Vol.6, Issue 4, September 2016
  12. J-F Yin and K. Wang, Splitting Iterative Methods for Fuzzy System of Linear Equations, Computational Mathematics and Modeling, 20(3),2009.
  13. T. Allahviranloo, Successive over Relaxation Iterative Method for Fuzzy System of Linear Equations, Applied Mathematics and Computation 162, pp. 189-196, 2005. https://doi.org/10.1016/j.amc.2003.12.085
  14. Asklany, S., Mansouri, W., Othmen, S., Levenberg-Marquardt deep learning algorithm for sulfur dioxide prediction, IJCSNS 19 (2019) 7.
  15. W. A. Mansouri, S. Asklany and S. H. J. I. OTHMAN, "Priority based Fuzzy Decision Multi-RAT Scheduling Algorithm in Heterogeneous Wireless Networks", vol. 19, no. 12, pp. 69, 2019
  16. S. Othmen, S. Asklany, and A. Belghith, "A fuzzy-based delay and energy-aware routing protocol for multi-hop cellular networks," in Proc. Inter. Wireless Commun. & Mobile Computing Conf., Tangier, Morocco, June 2019, pp. 1952-1957.
  17. S. Othmen, S. Asklany, and W. Mansouri, "Fuzzy logic based on-demand routing protocol for multi-hop cellular networks (5G)," Inter. Journal of Computer Science and Network Security, vol. 19, no. 12, pp. 29-36, Dec. 2019 https://doi.org/10.22937/IJCSNS.2019.19.12.5
  18. J. Rashidinia, M. Ghasemi and Z. Mahmoodi, Spline approach to the solution of a singularly perturbed boundary value problems, Applied mathematics and computation 189, pp. 72-78, 2007. https://doi.org/10.1016/j.amc.2006.11.067
  19. Tan Jingjing, Xinguang Zhang , Lishan Liu ,and Yonghong Wu ,An Iterative Algorithm for Solving n-Order FractionalDifferential Equation with Mixed Integral and MultipointBoundary Conditions ,Hindawi,Complexity Volume 2021, Article ID 8898859, p10
  20. E. Sousa, how to approximate the fractional derivative of order 1 < α ≤ 2, Proceedings of FDA'10. The 4th IFAC Workshop Fractional Differentiation and its Applications.Badajoz, Spain, October 18-20, 2010.
  21. Diethelm, K. and Ford, N. J., Analysis of fractional differential equations, Mathematical Analysis and Applications, Vol. 265, 229-248, 2002. https://doi.org/10.1006/jmaa.2000.7194
  22. Ascher, U. M. Matheij, R.M. M. Russell, R. D., Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, 1995.
  23. Richard. L. Burden and J. Douglas. Faires, Numerical Analysis (2011), 9th ed.
  24. Goyal, V., On a second order nonlinear boundary value problem, Applied Mathematics Letters, Vol. 19, 1406-1408, 2006. https://doi.org/10.1016/j.aml.2006.02.012
  25. Gasilov N, Amrahov S E, Fatullayev A G (2011) Linear differential equations with fuzzy boundary values. Proceeding of the 5th int. conference" application of information and communication technologies" (AICT2011) 696- 700, Baku, Azerbaijan, October