On analytical solutions of the fractional differential equation with uncertainty: Application to the basset problem

Entropy

Volumn 17, Issue 2, 2015, Pages 885-902

  Salahshour, Soheil ^a   Ahmadian, Ali ^b   Senu, Norazak ^b   Baleanu, Dumitru ^c,d   Agarwal, Praveen ^e  

^a ISLAMIC AZAD UNIVERSITY   (Iran)

^b UNIVERSITY PUTRA MALAYSIA   (Malaysia)

^c CANKAYA UNIVERSITY   (Turkey)

^d INSTITUTE OF SPACE SCIENCES   (Romania)

^e ANAND INTERNATIONAL COLLEGE OF ENGINEERING   (India)

Author keywords

Basset problem; Caputo differentiability; Dynamical systems; Fuzzy fractional differential equation; Fuzzy laplace transform

Indexed keywords

EID: 84923359923     PISSN: None     EISSN: 10994300     Source Type: Journal    
DOI: 10.3390/e17020885     Document Type: Article

References (54)

1. Arara, A.; Benchohra, M.; Hamidi, N.; Nieto, J. Fractional order differential equations on an unbounded domain. Nonlinear Anal. Theory Methods Appl. 2010, 72, 580-586.
2. Bagley, R.L. On the fractional order initial value problem and its engineering applications. Fract. Calc. Appl. 1990, 12-20.
3. Beyer, H.; Kempfle, S. Definition of physically consistent damping laws with fractional derivatives. Zeitschrift für Angewandte Mathematik und Mechanik 1995, 75, 623-635.
4. Diethelm, K.; Ford, N. Analysis of fractional differential equations. J. Math. Anal. App. 2002, 265, 229-248.
5. Mainardi, F. Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 1996, 7, 1461-1477.
6. Ingo, C.; Magin, R.L.; Parrish, T.B. New Insights into the Fractional Order Diffusion Equation Using Entropy and Kurtosis. Entropy 2014, 16, 5838-5852.
7. Baleanu, D.; Güvenc, Z.; Tenreiro Machado, J. New Trends in Nanotechnology and Fractional Calculus Applications; Springer-Verlag: Berlin, Germany, 2010.
8. Dorčák, L.; Valsa, J.; Gonzalez, E.; Terpák, J.; Petráš, I.; Pivka, L. Analogue Realization of Fractional-Order Dynamical Systems. Entropy 2013, 15, 4199-4214.
9. Lakshmikantham, V.; Leela, S.; Devi, J. Theory of Fractional Dynamic Systems; Cambridge Scientific Publishers: Cambridge, UK, 2009.
10. Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications; Elsevier Science: Amsterdam, The Netherlands, 1998.
11. Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier Science: Amsterdam, The Netherlands, 2006.
12. Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J. Fractional Calculus: Models and Numerical Methods; World Scientific: Singapore, Singapore, 2012.
13. Khan, Y.; Wu, Q.; Faraz, N.; Yildirim, A.; Madani, M. A new fractional analytical approach via a modified Riemann-Liouville derivative. Appl. Math. Lett. 2012, 25, 1340-1346.
14. Lakshmikantham, V.; Vatsala, A.S. General uniqueness and monotone iterative technique for fractional differential equations. Appl. Math. Lett. 2012, 25, 1340-1346.
15. El-Ajou, A.; Abu Arqub, O.; Al Zhour, Z.; Momani, S. New Results on Fractional Power Series: Theories and Applications. Fuzzy Sets Syst. 2013, 15, 5305-5323.
16. Momani, S.; Aslam Noor, M. Numerical methods for fourth-order fractional integro-differential equations. Appl. Math. Comput. 2006, 182, 754-760.
17. Bhrawy, A.; Alofi, A.; Ezz-Eldien, S. A quadrature tau method for fractional differential equations with variable coefficients. Appl. Math. Lett. 2011, 24, 2146-2152.
18. Bhrawy, A.H.; Zaky, M.A. A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations. J. Comput. Phys. 2015, 281, 876-895.
19. Bhrawy, A.H.; Doha, E.H.; Baleanu, D.; Ezz-Eldien, S.S. A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations. J. Comput. Phys. 2015, in press.
20. Bhrawy, A.H.; Zaky, M.A.; Baleanu, D. New Numerical Approximations for Space-Time Fractional Burgers' Equations via a Legendre Spectral-Collocation Method. Rom. Rep. Phys. 2015, 67, In press.
21. Bhrawy, A.H.; Tharwat, M.M.; Yildirim, A. A new formula for fractional integrals of Chebyshev polynomials: Application for solving multi-term fractional differential equations. Appl. Math. Model. 2013, 37, 4245-4252.
22. Zhang, L.; Ahmad, B.; Wang, G. The existence of an extremal solution to a nonlinear system with the right-handed Riemann-Liouville fractional derivative. Appl. Math. Lett. 2014, 31,1-6.
23. Lakshmikantham, V.; Vatsala, A. Basic theory of fractional differential equations. Nonlinear Anal. Theory Methods Appl. 2008, 69, 2677-2682.
24. Felice, D.; Cafaro, C.; Mancini, S. Information Geometric Complexity of a Trivariate Gaussian Statistical Model. Entropy 2014, 16, 2944-2958.
25. Sun, J.; Cafaro, C.; Bollt, E.M. Identifying the Coupling Structure in Complex Systems through the Optimal Causation Entropy Principle. Entropy 2014, 16, 3416-3433.
26. Agarwal, R.; Lakshmikantham, V.; Nieto, J. On the concept of solution for fractional differential equations with uncertainty. Nonlinear Anal. Theory Methods Appl. 2000, 72, 2859-2862.
27. Allahviranloo, T.; Salahshour, S.; Abbasbandy, S. Explicit solutions of fractional differential equations with uncertainty. Soft Comput. 2012, 16, 297-302.
28. Mazandarani, M.; Vahidian Kamyad, A. Modified fractional Euler method for solving Fuzzy Fractional Initial Value Problem. Commun. Nonlinear Sci. Numer. Simulat. 2013, 18, 12-21.
29. Salahshour, S.; Allahviranloo, T.; Abbasbandy, S. Solving fuzzy fractional differential equations by fuzzy Laplace transforms. Commun. Nonlinear Sci. Numer. Simul. 2012, 17, 1372-1381.
30. Allahviranloo, T.; Ahmadi, M. Fuzzy laplace transforms. Soft Comput. 2010, 14, 235-243.
31. Mazandarani, M.; Najariyan, M. Type-2 fuzzy fractional derivatives. Commun. Nonlinear Sci. Numer. Simul. 2014, 19, 2354-2372.
32. Bede, B.; Gal, S. Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets Syst. 2005, 151, 581-599.
33. Bede, B.; Rudas, I.; Bencsik, A. First order linear fuzzy differential equations under generalized differentiability. Inf. Sci. 2007, 177, 1648-1662.
34. Ahmadian, A.; Suleiman, M.; Salahshour, S.; Baleanu, D. A Jacobi operational matrix for solving a fuzzy linear fractional differential equation. Adv. Diff. Equ. 2013, 104, doi:10.1186/1687-1847-2013-104.
35. Ahmadian, A.; Suleiman, M.; Salahshour, S. An operational matrix based on Legendre polynomials for solving fuzzy fractional-order differential equations. Abstr. Appl. Anal. 2013, 2013, doi:10.1155/2013/505903.
36. Balooch Shahriyar, M.R.; Ismail, F.; Aghabeigi, S.; Ahmadian, A.; Salahshour, S. An Eigenvalue-Eigenvector Method for Solving a System of Fractional Differential Equations with Uncertainty. Math. Probl. Eng. 2013, 2013, doi:10.1155/2013/579761.
37. Salahshour, S.; Allahviranloo, T.; Abbasbandy, S.; Baleanu, D. Existence and uniqueness results for fractional differential equations with uncertainty. Adv. Diff. Equ. 2012, 112, doi:10.1186/1687-1847-2012-112.
38. Allahviranloo, T.; Abbasbandy, S.; Balooch Shahryari, M.R.; Salahshour, S.; Baleanu, D. On Solutions of Linear Fractional Differential Equations with Uncertainty. Abstr. Appl. Anal. 2013, 2013, doi:10.1155/2013/178378.
39. Diamond, P. Theory and applications of fuzzy Volterra integral equations. IEEE Trans. Fuzzy Syst. 2002, 157, 97-102.
40. Perfilieva, I. Cauchy problem with fuzzy initial condition and its approximate solution with the help of fuzzy transform. In Proceedings of IEEE International Conference on Fuzzy Systems, 2008 (IEEE World Congress on Computational Intelligence), Hong Kong, China, 1-6 June 2008; pp. 2285-2290.
41. Jafarian, A.; Golmankhaneh, A.R.; Baleanu, D. On Fuzzy Fractional Laplace Transformation. Inf. Control 2014, 2014, doi:10.1155/2014/295432.
42. Diamond, P.; Kloeden, P. Metric Spaces of Fuzzy Sets: Theory and Applications; World Scientific: Singapore, Singapore, 1994.
43. Zimmermann, H. Fuzzy Set Theory And Its Applications; Springer-Verlag: Berlin, Germany, 2001.
44. Friedman, M.; Ma, M.; Kandel, A. Numerical solutions of fuzzy differential and integral equations. Fuzzy Sets Syst. 1999, 106, 35-48.
45. Puri, M.; Ralescu, D. Fuzzy random variables. J. Math. Anal. Appl. 1986, 114, 409-422.
46. Wu, H-C. The improper fuzzy Riemann integral and its numerical integration. Inf. Sci. 1998, 111, 109-137.
47. Chalco-Cano, Y.; Román-Flores, H. On new solutions of fuzzy differential equations. Chaos Solitons Fractals 2008, 38, 112-119.
48. Alikhani, R.; Bahrami, F. Global solutions for nonlinear fuzzy fractional integral and integrodifferential equations. Commun. Nonlinear Sci. Numer. Simul. 2013, 18, 2007-2017.
49. Salahshour, S.; Abbasbandy, S. A comment on "Global solutions for nonlinear fuzzy fractional integral and integrodifferential equations". Commun. Nonlinear Sci. Numer. Simul. 2014, 19,1256-1258.
50. Lupulescu, V. Fractional calculus for interval-valued functions. Fuzzy Sets Syst. 2014, doi:10.1016/j.fss.2014.04.005.
51. Basset, A. A Treatise on Hydrodynamics: With Numerous Examples; Deighton, Bell and Co.: Cambridge, UK, 1888.
52. Basset, A. The Descent of a Sphere in a Viscous Liquid. Nature 1910, 83, 521.
53. Mainardi, F. Fractional relaxation-oscillation and fractional diffusion-wave phenomena. Chaos Solitons Fractals 1996, 7, 1461-1477.
54. Mainardi, F.; Pironi, P.; Tampieri, F. On a generalization of the Basset problem via Fractional Calculus. In Proceedings of the 15th Canadian Congress of Applied Mechanics, CANCAM'95, University of Victoria, Victoria, Canada, 28 May-2 June 1995; pp. 836-837.