On Analytical Solutions of the Fractional Differential Equation with Uncertainty: Application to the Basset Problem

In this paper, we apply the concept of Caputo’s H-differentiability, constructed based on the generalized Hukuhara difference, to solve the fuzzy fractional differential equation (FFDE) with uncertainty. This is in contrast to conventional solutions that either require a quantity of fractional derivatives of unknown solution at the initial point (Riemann–Liouville) or a solution with increasing length of their support (Hukuhara difference). Then, in order to solve the FFDE analytically, we introduce the fuzzy Laplace transform of the Caputo H-derivative. To the best of our knowledge, there is limited research devoted to the analytical methods to solve the FFDE under the fuzzy Caputo fractional differentiability. An analytical solution is presented to confirm the capability of the proposed method.

[1]  Tofigh Allahviranloo,et al.  SOLVING FUZZY FRACTIONAL DIFFERENTIAL EQUATIONS BY FUZZY LAPLACE TRANSFORMS , 2012 .

[2]  Tofigh Allahviranloo,et al.  Fuzzy Laplace transforms , 2009, Soft Comput..

[3]  Barnabás Bede,et al.  Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations , 2005, Fuzzy Sets Syst..

[4]  Hsien-Chung Wu,et al.  The Improper Fuzzy Riemann Integral and its Numerical Integration , 1998, Inf. Sci..

[5]  Richard L. Magin,et al.  New Insights into the Fractional Order Diffusion Equation Using Entropy and Kurtosis , 2014, Entropy.

[6]  Ali Ahmadian,et al.  An Operational Matrix Based on Legendre Polynomials for Solving Fuzzy Fractional-Order Differential Equations , 2013 .

[7]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[8]  Vasile Lupulescu,et al.  Fractional calculus for interval-valued functions , 2015, Fuzzy Sets Syst..

[9]  Yong Zhou.  Basic Theory of Fractional Differential Equations , 2014 .

[10]  Muhammad Aslam Noor,et al.  Numerical methods for fourth-order fractional integro-differential equations , 2006, Appl. Math. Comput..

[11]  Omar Abu Arqub,et al.  New Results on Fractional Power Series: Theory and Applications , 2013, Entropy.

[12]  Stefano Mancini,et al.  Information Geometric Complexity of a Trivariate Gaussian Statistical Model , 2014, Entropy.

[13]  A. B. BASSET,et al.  The Descent of a Sphere in a Viscous Liquid , 1910, Nature.

[14]  Phil Diamond.  Theory and applications of fuzzy Volterra integral equations , 2002, IEEE Trans. Fuzzy Syst..

[15]  Abraham Kandel,et al.  Numerical solutions of fuzzy differential and integral equations , 1999, Fuzzy Sets Syst..

[16]  Mehran Mazandarani,et al.  Type-2 fuzzy fractional derivatives , 2014, Commun. Nonlinear Sci. Numer. Simul..

[17]  Bernard De Baets,et al.  Cauchy problem with fuzzy initial condition and its approximate solution with the help of fuzzy transform , 2008, 2008 IEEE International Conference on Fuzzy Systems (IEEE World Congress on Computational Intelligence).

[18]  V. Lakshmikantham,et al.  General uniqueness and monotone iterative technique for fractional differential equations , 2008, Appl. Math. Lett..

[19]  Dumitru Baleanu,et al.  A Jacobi operational matrix for solving a fuzzy linear fractional differential equation , 2013 .

[20]  Dumitru Baleanu,et al.  On Fuzzy Fractional Laplace Transformation , 2014 .

[21]  Jie Sun,et al.  Identifying the Coupling Structure in Complex Systems through the Optimal Causation Entropy Principle , 2014, Entropy.

[22]  Ali H. Bhrawy,et al.  A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations , 2015, J. Comput. Phys..

[23]  Saeid Abbasbandy,et al.  A comment on "Global solutions for nonlinear fuzzy fractional integral and integrodifferential equations" , 2014, Commun. Nonlinear Sci. Numer. Simul..

[24]  Bashir Ahmad,et al.  The existence of an extremal solution to a nonlinear system with the right-handed Riemann-Liouville fractional derivative , 2014, Appl. Math. Lett..

[25]  J. A. Tenreiro Machado,et al.  New Trends in Nanotechnology and Fractional Calculus Applications , 2010 .

[26]  A. Bhrawy,et al.  A new formula for fractional integrals of Chebyshev polynomials: Application for solving multi-term fractional differential equations , 2013 .

[27]  Ali Ahmadian,et al.  An Eigenvalue-Eigenvector Method for Solving a System of Fractional Differential Equations with Uncertainty , 2013 .

[28]  Dumitru Baleanu,et al.  A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diffusion-wave equations , 2015, J. Comput. Phys..

[29]  Ali H. Bhrawy,et al.  A quadrature tau method for fractional differential equations with variable coefficients , 2011, Appl. Math. Lett..

[30]  H. Román-Flores,et al.  On new solutions of fuzzy differential equations , 2008 .

[31]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[32]  H. Zimmermann,et al.  Fuzzy Set Theory and Its Applications , 1993 .

[33]  Juan J. Nieto,et al.  Fractional order differential equations on an unbounded domain , 2010 .

[34]  A. B. Basset,et al.  A treatise on hydrodynamics with numerous examples , 1888 .

[35]  Ravi P. Agarwal,et al.  On the concept of solution for fractional differential equations with uncertainty , 2010 .

[36]  P. Kloeden,et al.  Metric Spaces Of Fuzzy Sets Theory And Applications , 1975 .

[37]  Ali Vahidian Kamyad,et al.  Modified fractional Euler method for solving Fuzzy Fractional Initial Value Problem , 2013, Commun. Nonlinear Sci. Numer. Simul..

[38]  Tofigh Allahviranloo,et al.  Explicit solutions of fractional differential equations with uncertainty , 2012, Soft Comput..

[39]  Tofigh Allahviranloo,et al.  On Solutions of Linear Fractional Differential Equations with Uncertainty , 2013 .

[40]  V. Lakshmikantham,et al.  Basic theory of fractional differential equations , 2008 .

[41]  Saudi Arabia,et al.  NEW NUMERICAL APPROXIMATIONS FOR SPACE-TIME FRACTIONAL BURGERS' EQUATIONS VIA A LEGENDRE SPECTRAL-COLLOCATION METHOD , 2014 .

[42]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[43]  K. Diethelm,et al.  Fractional Calculus: Models and Numerical Methods , 2012 .

[44]  Imre J. Rudas,et al.  First order linear fuzzy differential equations under generalized differentiability , 2007, Inf. Sci..

[45]  María Angeles Gil,et al.  Fuzzy random variables , 2001, Inf. Sci..

[46]  Tofigh Allahviranloo,et al.  Existence and uniqueness results for fractional differential equations with uncertainty , 2012, Advances in Difference Equations.

[47]  F. Mainardi.  Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena , 1996 .

[48]  V. Lakshmikantham,et al.  Theory of Fractional Dynamic Systems , 2009 .

[49]  Yasir Khan,et al.  A new fractional analytical approach via a modified Riemann-Liouville derivative , 2012, Appl. Math. Lett..

[50]  Horst R. Beyer,et al.  Definition of physically consistent damping laws with fractional derivatives , 1995 .

[51]  Juraj Valsa,et al.  Analogue Realization of Fractional-Order Dynamical Systems , 2013, Entropy.