Numerical solutions for systems of fractional order differential equations with Bernoulli wavelets

References

• R. C. Koeller, Applications of fractional calculus to the theory of viscoelasticity, J. Appl. Mech. 51 (1984), pp. 299–307. doi: 10.1115/1.3167616
   Web of Science ®Google Scholar
• L. Debnath, Recent applications of fractional calculus to science and engineering, Int. J. Math. Math. Sci. 2003 (2003), pp. 3413–3442. doi: 10.1155/S0161171203301486
   Google Scholar
• J. G. Lu and G. Chen, A note on the fractional-order Chen system, Chaos Solitons Fractals 27 (2006), pp. 685–688. doi: 10.1016/j.chaos.2005.04.037
   Web of Science ®Google Scholar
• C. Li and G. Chen, Chaos in the fractional order Chen system and its control, Chaos Solitons Fractals 22 (2004), pp. 549–554. doi: 10.1016/j.chaos.2004.02.035
   Web of Science ®Google Scholar
• D. Cafagna and G. Grassi, On the simplest fractional-order Memristor-based chaotic system, Nonlinear Dyn. 70 (2012), pp. 1185–1197. doi: 10.1007/s11071-012-0522-z
   Web of Science ®Google Scholar
• M. F. Tolba, A. M. AbdelAty, N. S. Soliman, L. A. Said, A. H. Madian, A. T. Azar and A. G. Radwan, FPGA implementation of two fractional order chaotic systems, Int. J. Electr. Commun. (AEU) 78 (2017), pp. 162–172. doi: 10.1016/j.aeue.2017.04.028
   Web of Science ®Google Scholar
• K. Diethelm, N.J. Ford and A.D. Freed, A predictor–corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn. 29 (2002), pp. 3–22. doi: 10.1023/A:1016592219341
   Web of Science ®Google Scholar
• J. Huang, Y. Tang and L. Vazquez, Convergence analysis of a block-by-block method for fractional differential equations, Numer. Math. Theory Methods Appl. 5 (2012), pp. 229–241. doi: 10.4208/nmtma.2012.m1038
   Web of Science ®Google Scholar
• A. Elsaid, Homotopy analysis method for solving a class of fractional partial differential equations, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), pp. 3655–3664. doi: 10.1016/j.cnsns.2010.12.040
   Web of Science ®Google Scholar
• I. Hashim, O. Abdulaziz and S. Momani, Homotopy analysis method for fractional IVPs, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), pp. 674–684. doi: 10.1016/j.cnsns.2007.09.014
   Web of Science ®Google Scholar
• D.W. Brzezinski and P. Ostalczyk, Numerical calculations accuracy comparison of the Inverse Laplace Transform algorithms for solutions of fractional order differential equations, Nonlinear Dyn. 84 (2016), pp. 65–77. doi: 10.1007/s11071-015-2225-8
   Web of Science ®Google Scholar
• E. Babolian, A.R. Vahidi and A. Shoja, An efficient method for nonlinear fractional differential equations: Combination of the Adomian decomposition method and Spectral method, Indian J. Pure Appl. Math. 45 (2014), pp. 1017–1028. doi: 10.1007/s13226-014-0102-7
   Web of Science ®Google Scholar
• C. Li and Y. Wang, Numerical algorithm based on Adomian decomposition for fractional differential equations, Comput. Math. Appl. 57 (2009), pp. 1672–1681. doi: 10.1016/j.camwa.2009.03.079
   Web of Science ®Google Scholar
• A. Elsaid, The variational iteration method for solving Riesz fractional partial differential equations, Comput. Math. Appl. 60 (2010), pp. 1940–1947. doi: 10.1016/j.camwa.2010.07.027
   Web of Science ®Google Scholar
• S. Yang, A. Xiao and H. Su, Convergence of the variational iteration method for solving multi-order fractional differential equations, Comput. Math. Appl. 60 (2010), pp. 2871–2879. doi: 10.1016/j.camwa.2010.09.044
   Web of Science ®Google Scholar
• K. Aruna and A.S.V. Ravi Kanth, Two-dimensional differential transform method and modified differential transform method for solving nonlinear fractional Klein–Gordon equation, Natl. Acad. Sci. Lett. 37 (2014), pp. 163–171. doi: 10.1007/s40009-013-0209-0
   Web of Science ®Google Scholar
• J. Liu and G. Hou, Numerical solutions of the space-and time-fractional coupled Burgers equations by generalized differential transform method, Appl. Math. Comput. 217 (2011), pp. 7001–7008. doi: 10.1016/j.amc.2011.01.111
   Web of Science ®Google Scholar
• N.H. Sweilam, M.M. Khader and A.M. Nagy, Numerical solution of two-sided space-fractional wave equation using finite difference method, J. Comput. Appl. Math. 235 (2011), pp. 2832–2841. doi: 10.1016/j.cam.2010.12.002
   Web of Science ®Google Scholar
• T. Allahviranloo, Z. Gouyandeh and A. Armand, Numerical solutions for fractional differential equations by Tau-Collocation method, Appl. Math. Comput. 27 (2015), pp. 979–990.
   Google Scholar
• Y.-M. Chen, Y.-Q. Wei, D.-Y. Liu and H. Yu, Numerical solution for a class of nonlinear variable order fractional differential equations with Legendre wavelets, Appl. Math. Lett. 46 (2015), pp. 83–88. doi: 10.1016/j.aml.2015.02.010
   Web of Science ®Google Scholar
• M. Yi and J. Huang, Wavelets operational matrix method for solving fractional differential equations with variable coefficients, Appl. Math. Comput. 230 (2014), pp. 383–394. doi: 10.1016/j.amc.2013.06.102
   Web of Science ®Google Scholar
• S. Aljoudi, B. Ahmad, J.J. Nieto and A. Alsaedi, A coupled system of Hadamard type sequential fractional differential equations with coupled strip conditions, Chaos Solitons Fractals 91 (2016), pp. 39–46. doi: 10.1016/j.chaos.2016.05.005
   Web of Science ®Google Scholar
• C.-Z. Bai and J.-X. Fang, The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations, Appl. Math. Comput. 150 (2004), pp. 611–621.
   Web of Science ®Google Scholar
• X. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett. 22 (2009), pp. 64–69. doi: 10.1016/j.aml.2008.03.001
   Web of Science ®Google Scholar
• M.P. Lazarevic, Finite time stability analysis of PDα fractional control of robotic time-delay systems, Mech. Res. Commun. 33 (2006), pp. 269–279. doi: 10.1016/j.mechrescom.2005.08.010
   Web of Science ®Google Scholar
• S. Gupta, D. Kumar and J. Singh, Numerical study for systems of fractional differential equations via Laplace transform, J. Egypt. Math. Soc. 23 (2015), pp. 256–262. doi: 10.1016/j.joems.2014.04.003
   Google Scholar
• Y. Chen and H.-L. An, Numerical solutions of coupled Burgers equations with time- and space-fractional derivatives, J. Egypt. Math. Soc. 200 (2008), pp. 87–95.
   Google Scholar
• E. Keshavarz, Y. Ordokhani and M. Razzaghi, Bernoulli wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations, Appl. Math. Model. 38 (2014), pp. 6038–6051. doi: 10.1016/j.apm.2014.04.064
   Web of Science ®Google Scholar
• P.K. Sahu and S. Saha Ray, A new Bernoulli wavelet method for accurate solutions of nonlinear fuzzy Hammerstein–Volterra delay integral equations, Fuzzy Set. Syst. 309 (2017), pp. 131–144. doi: 10.1016/j.fss.2016.04.004
   Web of Science ®Google Scholar
• M. Yi, L. Wang and J. Huang, Legendre wavelets method for the numerical solution of fractional integro-differential equations with weakly singular kernel, Appl. Math. Model. 40 (2016), pp. 3422–3437. doi: 10.1016/j.apm.2015.10.009
   Web of Science ®Google Scholar
• L. Zhu and Q. Fan, Solving fractional nonlinear Fredholm integro-differential equations by the second kind Chebyshev wavelets, Commun. Nonlinear Sci. Numer. Simul. 17 (2012), pp. 2333–2341. doi: 10.1016/j.cnsns.2011.10.014
   Web of Science ®Google Scholar
• J.-S. Gu and W.-S. Jiang, The Haar wavelets operational matrix of integration, Int. J. Syst. Sci. 27 (1996), pp. 623–628. doi: 10.1080/00207729608929258
   Web of Science ®Google Scholar
• L. Wang, Y. Ma and Z. Meng, Haar wavelet method for solving fractional partial differential equations numerically, Appl. Math. Comput. 227 (2014), pp. 66–76. doi: 10.1016/j.amc.2013.11.004
   Web of Science ®Google Scholar
• Y.M. Chen, X.N. Han and L.C. Liu, Numerical solution for a class of linear system of fractional differential equations by the Haar wavelet method and the convergence analysis, Comput. Model. Eng. Sci. 97 (2014), pp. 391–405.
   Web of Science ®Google Scholar