Numerical solution of linear time-fractional Kuramoto-Sivashinsky equation via quintic B-splines

References

• D. Anders and M. Dittmann, A higher-order finite element approach to the Kuramoto-Sivashinsky equation, ZAMM. Z. Angew Math. Mech. 92 (8) (2012), pp. 599–607.
   Web of Science ®Google Scholar
• C.M. Cazacu, L.I. Ignat, and A.F. Pazoto, Null-controllability of the linear Kuramoto-Sivashinsky equation on star-shaped trees, SIAM J. Control Optim. 56 (4) (2018), pp. 2921–2958.
   Web of Science ®Google Scholar
• E. Cerpa, P. Guzman, and A. Mercado, On the control of the linear Kuramoto-Sivashinsky equation, ESAIM: COCV 23 (1) (2017), pp. 165–194.
   Web of Science ®Google Scholar
• I. Dag, B. Saka, and D. Irk, Galerkin method for the numerical solution of the RLW equation using quintic B-splines, Comput. Appl. Math. 190 (1-2) (2006), pp. 532–547.
   Web of Science ®Google Scholar
• R. Ellahi, S.Z. Alamri, A. Basit, and A. Majeed, Effects of MHD and slip on heat transfer boundary layer flow over a moving plate based on specific entropy generation, J. Taibah Univ. Sci. 12 (4) (2018), pp. 476–482.
   Web of Science ®Google Scholar
• P. Gao, M. Chen, and Y. Li, Observability estimates and null controllability for forward and backward linear stochastic Kuramoto-Sivashinsky equation, SIAM J. Control Optim. 53 (1) (2015), pp. 475–500.
   Web of Science ®Google Scholar
• A.N. Golshan, Fractional generalized Kuramoto-Sivashinsky equation: Formulation and solution, Eur. Phys. J. Plus 134 (11) (2019), pp. Article ID 565.
   Web of Science ®Google Scholar
• C.A. Hall, On error bounds for spline interpolation, J. Approx. Theory 1 (2) (1968), pp. 209–218.
   Google Scholar
• Z. Hao, Z. Sun, and W. Cao, A fourth-order approximation of fractional derivatives with its applications, J. Comput. Phys. 281 (2015), pp. 787–805.
   Web of Science ®Google Scholar
• R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Company, Singapore, 2000, pp. 87–130.
   Google Scholar
• X. Hu and L. Zhang, A compact finite difference scheme for the fourth-order fractional diffusion-wave system, Comput. Phys. Commun. 182 (8) (2011), pp. 1645–1650.
   Web of Science ®Google Scholar
• A.A. Khan, S.R. Bukhari, and M. Marin, Effects of chemical reaction on third grade magnetohydrodynamics fluid flow under the influence of heat and mass transfer with variable reactive index, Heat Trans. Res. 50 (11) (2019), pp. 1061–1080.
   Web of Science ®Google Scholar
• A.A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Vol. 204, North-Holland Mathematics Studies, Elsevier, 2006.
   Google Scholar
• R. Klages, G. Radons, and I.M. Sokolov, Anomalous Transport: Foundations and Applications, Wiley, Weinheim, 2008.
   Google Scholar
• C. Li and G. Chen, Bifurcation analysis of the Kuramoto-Sivashinsky equation in one spatial dimension, Int. J. Bifurcat. Chaos 11 (09) (2001), pp. 2493–2499.
   Web of Science ®Google Scholar
• C. Li and G. Chen, Bifurcation from an equilibrium of the steady state Kuramoto-Sivashinsky equation in two spatial dimensions, Int. J. Bifurcat. Chaos 12 (1) (2002), pp. 103–114.
   Web of Science ®Google Scholar
• C. Li and Z. Yang, Symmetry-breaking bifurcation in O(2)-symmetric nonlinear large problems and its application to the Kuramoto-Sivashinsky equation in two spatial dimensions, Chaos Solit. Fractals 22 (2) (2004), pp. 451–468.
   Web of Science ®Google Scholar
• W.J. Liu and M. Krstic, Stability enhancement by boundary control in the Kuramoto-Sivashinsky equation, Nonlinear Anal. Ser. 43 (4) (2001), pp. 485–507.
   Web of Science ®Google Scholar
• K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
   Google Scholar
• R.C. Mittal and G. Arora, Quintic B-spline collocation method for numerical solution of the Kuramoto-Sivashinsky equation, Commun. Nonlinear Sci. Numer. Simulat. 15 (10) (2010), pp. 2798–2808.
   Web of Science ®Google Scholar
• K.B. Oldham and J. Spanier, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York, 1974.
   Google Scholar
• I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, Vol. 198, Elsevier, 1998.
   Google Scholar
• I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
   Google Scholar
• J.D. Rademacher and R.W. Wittenberg, Viscous shocks in the destabilized Kuramoto-Sivashinsky equation, J. Comput. Nonlinear Dyn. 1 (4) (2006), pp. 336–347.
   Web of Science ®Google Scholar
• S. Sahoo and S.S. Ray, New approach to find exact solutions of time-fractional Kuramoto-Sivashinsky equation, Physica A. 434 (2015), pp. 240–245.
   Web of Science ®Google Scholar
• B. Sepehrian and M. Lashani, A numerical Solution of the Burgers equation using quintic B-splines, in Proceedings of the World Congress on Engineering, Vol. 3, London, 2008.
   Google Scholar
• R. Shah, H. Khan, D. Baleanu, P. Kumam, and M. Arif, A semi-analytical method to solve family of Kuramoto-Sivashinsky equations, J. Taibah Univ. Sci. 14 (1) (2020), pp. 402–411.
   Web of Science ®Google Scholar
• S.S. Siddiqi and S.S. Arshed, Numerical swolution of time-fractional fourth-order partial differential equations, Int. J. Comput. Math. 92 (7) (2015), pp. 1496–1518.
   Web of Science ®Google Scholar
• T.H. Solomon, E.R. Weeks, and H.L. Swinney, Observations of anomalous diffusion and Lvy flights in a2-dimensional rotating flow, Phys. Rev. Lett. 71 (24) (1993), pp. 3975–3978.
   PubMed Web of Science ®Google Scholar
• B. Soltanalizadeh and M. Zarebnia, Numerical analysis of the linear and nonlinear Kuramoto-Sivashinskt equation by using differential transformation method, Int. J. Appl. Math. Mech. 7 (2011), pp. 63–72.
   Google Scholar
• M.A. Taneco-Hernández, V.F. Morales-Delgado, and J.F. Gómez-Aguilar, Fractional Kuramotoo-Sivashinsky equation with power law and stretched Mittag-Leffler kernel, Physica A. 527 (2019), pp. Article ID 121085.
   Web of Science ®Google Scholar
• H. Tariq and G. Akram, Quintic spline technique for time fractional fourth-order partial differential equation, Numer. Methods Partial Differ. Equ. 33 (2) (2017), pp. 445–466.
   Web of Science ®Google Scholar
• S.I. Zaki, A quintic B-spline finite elements scheme for the KdVB equation, Comput. Meth. Appl. Eng. 188 (1-3) (2000), pp. 121–134.
   Web of Science ®Google Scholar
• A. Zeeshan, F. Hussain, R. Ellahi, and K. Vafai, A study of gravitational and magnetic effects on coupled stress bi-phase liquid suspended with crystal and Hafnium particles down in steep channel, J. Mol. Liq. 286 (2019), pp. Article ID 110898).
   Web of Science ®Google Scholar
• Z. Zhao, X.Q. Jin, and M.M. Lin, Preconditioned iterative methods for space-time fractional advection diffusion equations, J. Comput. Phys. 319 (2016), pp. 266–279.
   Web of Science ®Google Scholar