Identyfikatory
Warianty tytułu
Języki publikacji
Abstrakty
In this paper, an extension is introduced into Max-Min Improved Euler methods for solving initial value problems of fuzzy fractional differential equations (FFDEs). Two modified fractional Euler type methods have been proposed and investigated to obtain numerical solutions of linear and nonlinear FFDEs. The proposed algorithms are tested on various illustrative examples. Exact values are also simulated to compare and discuss the closeness and accuracy of approximations so obtained. Comparatively, tables and graphs results reveal the complete reliability, efficiency and accuracy of the proposed methods.
Słowa kluczowe
Wydawca
Czasopismo
Rocznik
Tom
Strony
53--83
Opis fizyczny
Bibliogr. 39 poz., rys., tab.
Twórcy
autor
- Department of Mathematical Sciences University of Karachi, Karachi 75270, Pakistan
autor
- Department of Mathematical Sciences University of Karachi, Karachi 75270, Pakistan
autor
- Department of Mathematical Sciences University of Karachi, Karachi 75270, Pakistan
Bibliografia
- 1. Kilbas AA, Srivastava HM, Trujillo JJ, 2006, Theory and applications of fractional differential equations, San Diego, Elsevier.
- 2. Bagley RL, 1990, On the fractional order initial value problem and its engineering applications, in Fractional Calculus and Its Applications, K. Nishimoto, Ed., College of Engineering, Nihon University, Tokyo, Japan 12–20.
- 3. Podlubny I, 1990, Fractional differential equations, Academic Press, San Diego.
- 4. Machado JT, Kiryakova V, Mainardi F, 2011, Recent history of fractional calculus. Communications in Nonlinear Science and Numerical Simulation 16: 1140– 1153.
- 5. Diethelm K, Ford NJ, 2002, Analysis of fractional differential equations. Journal of Mathematical Analysis and Applications 265: 265-322.
- 6. Das S, 2008, Functional fractional calculus for system identification and controls. New York: Springer.
- 7. Adda FB, Cresson J, 2005, Fractional differential equations and the Schrodinger equation. Applied Mathematics and Computation 161: 323–345.
- 8. Fellah ZEA, Depollier C, Fellah M, 2002, Application of fractional calculus to the sound waves propagation in rigid porous materials: validation via ultrasonic measurements. Acta Acustica united with Acustica, 88: 34–39.
- 9. Rahimy M, 2012, Applications of fractional differential equations. Applied Mathematical Sciences 4: 2453–2461.
- 10. Khan NA, Mahmood A, Ara A, Khan NA, 2012, Analytical study of nonlinear fractional order Integro-differential equation: Revisit Volterra’s population model. International Journal of Differential Equations: Article ID 845945.
- 11. Ahmad B, Nieto JJ, 2009, Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Computer and Mathematics with Applications 58: 1838-1843.
- 12.Jumarie G, 2009, Table of some basic fractional calculus formulae derived from a improved Riemann-Liouville derivative for non-differentiable functions. Applied Mathematics Letters 22: 378–385.
- 13. Khan NA, Khan NU, Ara A, Jamil M, 2012, Approximate analytical solutions of fractional reaction-diffusion equations. Journal of King Saud University – Science 24: 111–118.
- 14. Khan NA, Ara A, Mahmood A, 2012, Numerical solutions of time fractional Burger equations: a comparison between generalized transformation technique with homotopy perturbation method. International Journal of Numerical Method and Heat Fluid Flow 22: 175–93.
- 15.Jafari H, Tajadodi H, Matikolai SAH, 2011, Homotopy perturbation pade technique for solving fractional Riccati differential equations. The International Journal of Nonlinear Sciences and Numerical Simulation 11: 271–276.
- 16.Jafari H, Tajadodi H, Nazari H, Khalique CM, 2011, Numerical solution of nonlinear Riccati differential equations with fractional order. The International Journal of Nonlinear Sciencesand Numerical Simulation 11: 179–182.
- 17. Araya D, Lizama C, 2008, Almost automorphic mild solutions to fractional differential equations. Nonlinear Analysis 69: 3692-3705.
- 18. Khan NA, Ara A, Jamil M, 2011, An efficient approach for solving the Riccati equation with fractional orders. Computer and Mathematics with Applications 61: 2683-2689.
- 19. Abbasbandy S, Shirzadi A, 2010, Homotopy analysis method for multiple solutions of the fractional Sturm–Liouville problems. Numerical Algorithms 54: 521– 532.
- 20. Khan NA, Jamil M, Ara A, Khan NU, 2011, On efficient method for system of fractional differential equations. Advances in Difference Equations: Article ADE/303472
- 21. Zadeh LA, 1965, Fuzzy sets. Information and Control 8: 338-353.
- 22. Zadeh LA, 1978, Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems 1: 3-28.
- 23. Agarwal RP, Lakshmikantham V, Nieto JJ, 2010, On the concept of solution for fractional differential equations with uncertainty. Nonlinear Analysis 72: 2859- 2862.
- 24. Ahmad MZ, Hasan MK, Abbasbandy S, 2013, Solving fuzzy fractional differential equations using Zadeh’s extension principle. The ScientificWorld Journal 2013: Article ID 454969.
- 25. Arshad S, Lupulescu V, 2011, On the fractional differential equations with uncertainty. Nonlinear Analysis: Theory, Methods &Applications 74: 3685–3693.
- 26. Allahviranloo T, Salahshour S, Abbasbandy S, 2012, Explicit solutions of fractional differential equations with uncertainty. Soft Computing 16: 297–302.
- 27. Salahshour S, Allahviranloo T, Abbasbandy S, 2012, Solving fuzzy fractional differential equations by fuzzy Laplace transforms. Communications in Nonlinear Science and Numerical Simulation 17: 1372–1381.
- 28. Alikhani R, Bahrami F, 2013, Global solutions for nonlinear fuzzy fractional integral and integro differential equations. Communications in Nonlinear Science and Numerical Simulation 18: 2007–2017.
- 29. Tapaswini S, Chakraverty S, 2012, A new approach to fuzzy initial value problem by improved Euler method. Fuzzy Information and Engineering 3: 293-312.
- 30. Ochoche A, 2008, Improving the improved Euler method for better performance on autonomous initial value problems. Leonardo Journal of Sciences 12: 57 – 66.
- 31. Ma M, Friedman M, Kandel A, 1999, Numerical solutions of fuzzy differential equations. Fuzzy Sets and Systems 105: 133–138.
- 32. Mazandarani M, Kamyad AV, 2013, Improved fractional Euler method for solving fuzzy fractional initial value problem. Communications in Nonlinear Science and Numerical Simulation 18: 12–21.
- 33. Ahmad MZ, Hasan MK, Baets BD, 2013, Analytical and numerical solutions of fuzzy differential equations. Information Sciences 236: 156–167.
- 34. Allahviranloo T, Abbasbandy S, Salahshour S, Hakimzadeh A, 2011, A new method for solving fuzzy linear differential equations. Computing 92: 181–197.
- 35. Shokri J, 2007, Numerical solution of fuzzy differential equations. Applied Mathematical Sciences 1: 2231-2246.
- 36. Duraisamy C, Usha B, 2010, Another approach to solution of fuzzy differential equations by improved Euler’s method. Proceedings of the International conference on Communication and Computational Intelligence-kongu Engineering College, Perundurai, Erode, TN, India: 52-55.
- 37. Odibat Z, Momani S, 2008, An algorithm for the numerical solution of differential equations of fractional order. Journal of Applied Mathematics & Informatics 26: 15--27.
- 38. Aminikhah H, Hemmatnezhad M, 2010, An efficient method for quadratic Riccati differential equation. Communication in Nonlinear Science and Numerical Simulation 15: 835--839.
- 39. Bede B, Gal SG, 2005, Generalizations of the differentiability of fuzzy-numbervalued functions with applications to fuzzy differential equations. Fuzzy Sets System 151: 581--599.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-288478d0-74a5-4bf5-a582-3bc10b3e5bb6
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