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A note on the convergence of fuzzy transformed finite difference methods

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Abstract

In this paper, we develop numerical methods based on the fuzzy transform methods (FTMs). In this approach we apply fuzzy transforms on discrete version of the derivatives and use it to derive FTMs. We also establish convergence of the proposed FTMs. To test the efficiency of the proposed FTMs, we apply the FTM schemes on the second order nonlinear singular boundary value problems and fourth order BVPs. We allow the source term of the differential equation to have jump discontinuity and study the effect of jump on FTMs and finite difference methods. The work shows that FTMs are better for both class of BVPs considered in this paper, having singularity, nonlinearity and jump discontinuity.

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References

  1. Arqub, O.A.: Adaptation of reproducing kernel algorithm for solving fuzzy Fredholm–Volterra integro differential equations. Neural Comput. Appl. 28(7), 1591–1610 (2017)

    Article  Google Scholar 

  2. Arqub, O.A., Al-Smadi, M., Momani, S., Hayat, T.: Numerical solutions of fuzzy differential equations using reproducing kernel hilbert space method. Soft. Comput. 20(8), 3283–3302 (2016)

    Article  MATH  Google Scholar 

  3. Arqub, O.A., Al-Smadi, M., Momani, S., Hayat, T.: Application of reproducing kernel algorithm for solving second-order, two-point fuzzy boundary value problems. Soft. Comput. 21(23), 7191–7206 (2017)

    Article  MATH  Google Scholar 

  4. Chandrasekhar, S.: Introduction to the Study of Stellar Structure. Dover, New York (1967)

    Google Scholar 

  5. Chen, W., Shen, Y.H.: Approximate solution for a class of second-order ordinary differential equations by the fuzzy transform. J. Intell. Fuzzy Syst. 27, 73–82 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Coco, A., Currenti, G., Negro, C., Russo, G.: A second order finite-difference ghost-point method for elasticity problems on unbounded domains with applications to volcanology. Commun. Comput. Phys. 16, 983–1009 (2014). 10

    Article  MathSciNet  MATH  Google Scholar 

  7. Holcapek, M., Valášek, R.: Numerical solution of partial differential equations with the help of fuzzy transform technique. In: 2017 IEEE International Conference on Fuzzy Systems, pp. 1–6 (2017)

  8. Jain, M.K.: Numerical Solutions of Differential Equations. New Age International, New Delhi (2018)

    Google Scholar 

  9. Jain, M.K., Iyengar, S.R.K., Jain, R.K.: Numerical methods for scientific and engineering computation. New Age International (P) Limited, Chennai (2012)

    MATH  Google Scholar 

  10. Keskin, Ali Umit: Boundary Value Problems for Engineers: With MATLAB Solutions. Springer, New York (2019)

    Book  MATH  Google Scholar 

  11. Khastan, A., Perfilieva, I., Alijani, Z.: A new fuzzy approximation method to cauchy problems by fuzzy transform. Fuzzy Sets Syst. 288, 75–95 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Khastana, A., Alijania, Z., Perfilieva, I.: Fuzzy transform to approximate solution of two-point boundary value problems. Math. Methods Appl. Sci. 40, 6147–6154 (2015)

    Article  MathSciNet  Google Scholar 

  13. Pandey, R.K., Verma, A.K.: A note on derivative dependent singular boundary value problems arising in physiology. ar**v (2019)

  14. Pandey, R.K.: A finite difference method for a class of singular two-point boundary value problems arising in physiology. Int. J. Comput. Math. 65, 131–140 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Pandey, R.K., Verma, A.K.: Existence-uniqueness results for a class of singular boundary value problems arising in physiology. Nonlinear Anal. Real World Appl. 9(1), 40–52 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Pandey, R.K., Verma, A.K.: Existence-uniqueness results for a class of singular boundary value problems-ii. J. Math. Anal. Appl. 338(2), 1387–1396 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pandey, R.K., Verma, A.K.: A note on existence-uniqueness results for a class of doubly singular boundary value problems. Nonlinear Anal. Theory Methods Appl. 71(7), 3477–3487 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Pandey, R.K., Verma, A.K.: Monotone method for singular bvp in the presence of upper and lower solutions. Appl. Math. Comput. 215(11), 3860–3867 (2010)

    MathSciNet  MATH  Google Scholar 

  19. Pandey, R.K., Verma, A.K.: On solvability of derivative dependent doubly singular boundary value problems. J. Appl. Math. Comput. 33(1), 489–511 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Perfilieva, I.: Chapter 9: fuzzy transform—application to the reef growth problem. In: Demicco, R.V., Klir, G.J. (eds.) Fuzzy Logic in Geology, pp. 275–300. Academic Press, Burlington (2004)

    Chapter  Google Scholar 

  21. Perfilieva, I.: Fuzzy transforms: theory and applications. Fuzzy Sets Syst. 157(8), 993–1023 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Perfilieva, I., Chaldeeva, E.: Fuzzy transformation and its applications. In: Proceedings of the 4th Czech—Japan Seminar on Data Analysis and Decision Making under Uncertainity, pp. 116–124 (2001)

  23. Perfilieva, I., Danková, M., Bede, B.: Towards a higher degree f-transform. Fuzzy Sets Syst. 180(1), 3–19 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Perfilieva, I., Kreinovich, V.: Fuzzy transforms of higher order approximate derivatives: a theorem. Fuzzy Sets Syst. 180(1), 55–68 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Perfilieva, I., Kreinovich, V.: Why fuzzy transform is efficient in large-scale prediction problems: a theoretical explanation. Adv. Fuzzy Syst. 2011, 5 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Perfilieva, I., Stevuliakova, P., Valasek, R.: F-transform for numerical solution of two point boundary value problem. Iran. J. Fuzzy Syst. 14(6), 1–13 (2017)

    MathSciNet  MATH  Google Scholar 

  27. Rashidinia, J., Golbabaee, A.: Convergence of numerical solution of a fourth-order boundary value problem. Appl. Math. Comput. 171(2), 1296–1305 (2005)

    MathSciNet  MATH  Google Scholar 

  28. Singh, M., Verma, A.K., Agarwal, R.P.: On an iterative method for a class of 2 point and 3 point nonlinear sbvps. J. Appl. Anal. Comput. 9(4), 1–19 (2019). 01

    MathSciNet  Google Scholar 

  29. Stepnicka, M., Valásek, R.: Fuzzy transforms and their application on wave equation. J. Electr. Eng. 55, 7 (2004)

    MATH  Google Scholar 

  30. Stepnicka, M., Valásek, R.: Numerical solution of partial differential equations with help of fuzzy transform. In: The 14th IEEE International Conference on Fuzzy Systems, pp. 1104–1109 (2005)

  31. Wang, C., Qiu, Z.P.: Fuzzy finite difference method for heat conduction analysis with uncertain parameters. Acta. Mech. Sin. 30(3), 383–390 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  32. Zeinali, M., Alikhani, R., Shahmorad, S., Bahrami, F., Perfilieva, I.: On the structural properties of \(f^m\)-transform with applications. Fuzzy Sets Syst. 342, 32–52 (2018)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgements

We are thankful to anonymous reviewers for their valuable comments, that has helped to improve the paper. Thanks a lot for their time and effort.

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Authors and Affiliations

  1. Department of Mathematics, IIT Patna, Bihta, Bihar, 801103, India

    Amit K. Verma, Sheerin Kayenat & Gopal Jee Jha

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  1. Amit K. Verma
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  2. Sheerin Kayenat
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  3. Gopal Jee Jha
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Correspondence to Amit K. Verma.

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Sheerin Kayenat: Dedicate to my beloved father.

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Verma, A.K., Kayenat, S. & Jha, G.J. A note on the convergence of fuzzy transformed finite difference methods. J. Appl. Math. Comput. 63, 143–170 (2020). https://doi.org/10.1007/s12190-019-01312-8

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  • DOI: https://doi.org/10.1007/s12190-019-01312-8

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