Hostname: page-component-77c89778f8-7drxs Total loading time: 0 Render date: 2024-07-20T07:17:30.478Z Has data issue: false hasContentIssue false
  • English
  • Français

A generalized scheme based on shifted Jacobi polynomials for numerical simulation of coupled systems of multi-term fractional-order partial differential equations

Part of: Miscellaneous topics - Partial differential equations Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  01 July 2017

Kamal Shah ,
Hammad Khalil  and
Rahmat Ali Khan
Kamal Shah
Affiliation:
Department of Mathematics, University of Malakand, Chakadara Dir(L), Khyber Pakhtunkhwa, Pakistan email kamalshah408@gmail.com
Hammad Khalil
Affiliation:
Department of Mathematics, University of Education (Attock Campus), Punjab, Pakistan email hammad.khalil@ue.edu.pk
Rahmat Ali Khan
Affiliation:
Department of Mathematics, University of Malakand, Chakadara Dir(L), Khyber Pakhtunkhwa, Pakistan email rahmat_alipk@yahoo.com

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Due to the increasing application of fractional calculus in engineering and biomedical processes, we analyze a new method for the numerical simulation of a large class of coupled systems of fractional-order partial differential equations. In this paper, we study shifted Jacobi polynomials in the case of two variables and develop some new operational matrices of fractional-order integrations as well as fractional-order differentiations. By the use of these operational matrices, we present a new and easy method for solving a generalized class of coupled systems of fractional-order partial differential equations subject to some initial conditions. We convert the system under consideration to a system of easily solvable algebraic equation without discretizing the system, and obtain a highly accurate solution. Also, the proposed method is compared with some other well-known differential transform methods. The proposed method is computer oriented. We use MatLab to perform the necessary calculation. The next two parts will appear soon.

MSC classification

Primary: 35R11: Fractional partial differential equations
Secondary: 65M06: Finite difference methods
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 20 , Issue 1 , 2017 , pp. 11 - 29
Copyright
© The Author(s) 2017 

References

Aksikas, I., Fuxman, A., Forbes, J. F. and Winkin, J., ‘LQ control design of a class of hyperbolic PDE systems: application to fixed-bed reactor’, Automatica 45 (2009) no. 6, 1542–1548.CrossRefGoogle Scholar
Arikoglu, A. and Ozkol, I., ‘Solution of fractional differential equations by using differential transform method’, Chaos Solitons Fractals 34 (2007) no. 5, 1473–1481.CrossRefGoogle Scholar
Ayaz, F., ‘Solutions of the system of differential equations by differential transform method’, Appl. Math. Comput. 147 (2004) 547–567.Google Scholar
Chen, C. and Hsiao, C., ‘Haar wavelet method for solving lumped and distributed parameter systems’, IEE Press Contr. Theor. Appl. 144 (1997) 87–94.CrossRefGoogle Scholar
Dehghan, M., Manafian, J. and Saadatmandi, A., ‘Solving nonlinear fractional partial differential equations using the homotopy analysis method’, Numer. Methods Partial Differential Equations 26 (2010) no. 2, 448–479.CrossRefGoogle Scholar
Dehghan, M., Manafian, J. and Saadatmandi, A., ‘The solution of the linear fractional partial differential equations using the homotopy analysis method’, Z. Naturforsch. A 65a (2010) no. 11, 935–949.CrossRefGoogle Scholar
Dehghan, M., Safarpoorand, M. and Abbaszadeh, M., ‘Two high-order numerical algorithms for solving the multi-term time fractional diffusion-wave equations’, J. Comput. Appl. Math. 290 (2015) 174–195.CrossRefGoogle Scholar
Erturk, V. S. and Momani, S., ‘Solving systems of fractional differential equations using differential transform method’, J. Comput. Appl. Math. 215 (2008) 142–151.CrossRefGoogle Scholar
Eslahchi, M. R. and Dehghan, M., ‘Application of Taylor series in obtaining the orthogonal operational matrix’, Comput. Math. Appl. 61 (2011) no. 6, 2596–2604.CrossRefGoogle Scholar
Fackeldey, K. and Krause, R., ‘Multiscale coupling in function space weak coupling between molecular dynamics and continuum mechanics’, Int. J. Numer. Methods Eng. 79 (2012) no. 12, 1517–1535.CrossRefGoogle Scholar
Folland, G. B., Introduction to partial differential equations , 2nd edn (Princeton University Press, Princeton, NJ, 1995).Google Scholar
Gasea, M. and Sauer, T., ‘On the history of multivariate polynomial interpolation’, J. Comput. Appl. Math. 122 (2000) 23–35.Google Scholar
Gorenflo, R., Mainardi, F., Scalas, E. and Raberto, M., ‘Fractional calculus and continuous time finance III’, Math. Finance (2000) 171–180.Google Scholar
Hedrih, K. S., ‘Transversal creep vibrations of a beam with fractional derivative constitutive relation order. I-Partial fractional differential equation. II-Stochastic stability of the beam dynamic shape, under axial bounded noise excitation’, Proceedings of Fourth International Conference on Nonlinear Mechanics (ICNM-IV), Shanghai, P.R. China (eds Chien, W. Z. et al. ; 2002) 584–595.Google Scholar
Hilfer, R., Applications of fractional calculus in physics (World Scientific Publishing Company, Singapore, 2000).CrossRefGoogle Scholar
Hu, Y., Luo, Y. and Lu, Z., ‘Analytical solution of the linear fractional differential equation by Adomian decomposition method’, J. Comput. Appl. Math. 215 (2008) 220–229.CrossRefGoogle Scholar
Ibrahim, R. W., ‘Solutions to systems of arbitrary-order differential equations in complex domains’, Electron. J. Differential Equations 46 (2014) 1–13.Google Scholar
Jafari, H. and Seifi, S., ‘Solving a system of nonlinear fractional partial differential equations using homotopy analysis method’, Commun. Nonlinear Sci. Numer. Simul. 14 (2009) 1962–1969.CrossRefGoogle Scholar
Katica, R. and Hedrih, S., ‘Dynamics of multi-pendulum systems with fractional order creep elements’, J. Theoret. Appl. Mech. 46 (2008) no. 3, 483–509.Google Scholar
Katica, R. and Hedrih, S., ‘Fractional order hybrid system dynamics’, Proc. Appl. Math. Mech. 13 (2013) 25–26.Google Scholar
Kayedi-Bardeh, A., Eslahchi, M. R. and Dehghan, M., ‘A method for obtaining the operational matrix of the fractional Jacobi functions and applications’, J. Vib. Control 20 ( 2014) no. 5, 736–748.CrossRefGoogle Scholar
Khalil, H. and Khan, R. A., ‘A new method based on Legendre polynomials for solutions of the fractional two-dimensional heat conduction equation’, Comput. Math. Appl. 67 (2014) 1938–1953.CrossRefGoogle Scholar
Khalil, H. and Khan, R. A., ‘A new method based on Legendre polynomials for solution of system of fractional order partial differential equations’, Int. J. Comput. Math. 91 (2014) no. 12, 2554–2567.Google Scholar
Khalil, H. and Khan, R. A., ‘Extended spectral method for fractional order three-dimensional heat conduction problem’, Prog. Fract. Differ. Appl. 1 (2015) no. 3, 165–185.Google Scholar
Khan, R. A. and Rehman, M., ‘Existence of multiple positive solutions for a general system of fractional differential equations’, Commun. Appl. Nonlinear Anal. 18 (2011) 25–35.Google Scholar
Kilbas, A. A., Srivastava, H. M. and Trujillo, J., Theory and applications of fractional differential equations (Elsevier Science, Amsterdam, 2006).Google Scholar
Lakestani, M., Dehghan, M. and Pakchin, S. I., ‘The construction of operational matrix of fractional derivatives using B-spline functions’, Commun. Nonlinear Sci. Numer. Simul. 17 (2012) no. 3, 1149–1162.CrossRefGoogle Scholar
Li, Y. L., ‘Solving a nonlinear fractional differential equation using Chebyshev wavelets’, Nonlinear Sci. Numer. Simul. 15 (2010) 2284–2292.CrossRefGoogle Scholar
Lin, L.-L., Li, Z.-Y. and Lin, B., ‘Engineering waveguide-cavity resonant side coupling in a dynamically tunable ultracompact photonic crystal filter’, Phys. Rev. B 72 (2005) 304–315.CrossRefGoogle Scholar
Linge, S., Sundnes, J., Hanslien, M., Lines, G. T. and Tveito, A., ‘Numerical solution of the bidomain equations’, Phil. Trans. Ser. A. Math. Phys. Eng. Sci. 367 (2009) 1931–1950.Google ScholarPubMed
Liu, F., Anh, V. and Turner, I., ‘Numerical solution of the space fractional Fokker–Planck equation’, J. Comput. Appl. Math. 66 (2005) 209–219.Google Scholar
Maleknedjad, K., Shahrezaee, M. and Khatami, H., ‘Numerical solution of integral equation system of the second kind by block-pulse function’, Appl. Math. Comput. 166 (2005) 15–24.Google Scholar
Metzler, R. and Klafter, J., ‘The random walk’s guide to anomalous diffusion: a fractional dynamics approach’, Phy. Rep. 339 (2000) no. 1, 1–77.CrossRefGoogle Scholar
Metzler, R. and Klafter, J., ‘Boundary value problems for fractional diffusion equations’, Phys. A: Stat. Mech. Appl. 278 (2005) 107–125.CrossRefGoogle Scholar
Moghadam, A. A., Aksikas, I., Dubljevic, S. and Forbes, J. F., ‘LQ control of coupled hyperbolic PDEs and ODEs: application to a CSTR-PFR system’, Proceedings of the 9th International Symposium on Dynamics and Control of Process Systems (DYCOPS 2010), Leuven, Belgium (eds Kothare, M. et al. ; 2010).Google Scholar
Mohamed, M. A. and Torky, M. Sh., ‘Solution of linear system of partial differential equations by Legendre multiwavelet and Chebyshev multiwavelet’, Intl J. Sci. Innov. Math. Res. 2 (2014) no. 12, 966–976.Google Scholar
Nemati, S. and Ordokhani, Y., ‘Legendre expansion methods for the numerical solution of nonlinear 2D Fredholm integral equations of the second kind’, J. Appl. Math. Inform. 31 (2013) 609–621.CrossRefGoogle Scholar
Oldham, K. B., ‘Fractional differential equations in electrochemistry’, Adv. Eng. Soft. 41 (2010) 9–12.CrossRefGoogle Scholar
Paraskevopolus, P. N, Saparis, P. D. and Mouroutsos, S. G., ‘The Fourier series operational matrix of integration’, Int. J. Syst. Sci. 16 (1985) 171–176.CrossRefGoogle Scholar
Parthiban, V. and Balachandran, K., ‘Solutions of system of fractional partial differential equations’, Appl. Appl. Math. 8 (2013) no. 1, 289–304.Google Scholar
Podlubny, I., Fractional differential equations (Academic Press, San Diego, CA, 1999).Google Scholar
Razzaghi, M. and Yousefi, S., ‘The Legendre wavelets operational matrix of integration’, Internat. J. Systems Sci. 32 (2001) 495–502.CrossRefGoogle Scholar
Razzaghi, M. and Yousefi, S., ‘Sine-cosine wavelets operational matrix of integration and its application in the calculus of variation’, Int. J. Syst. Sci. 33 (2002) 805–810.CrossRefGoogle Scholar
Rehman, M. and Khan, R. A., ‘The Legendre wavelet method for solving fractional differential equation’, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 4163–4173.CrossRefGoogle Scholar
Rehman, M. and Khan, R. A., ‘A note on boundary value problems for a coupled system of fractional differential equations’, Comput. Math. Appl. 61 (2011) 2630–2637.CrossRefGoogle Scholar
Richard, G. and Sarma, P. R. R., ‘Reduced order model for induction motors with two rotor circuits’, IEEE Trans. Energy Conv. 9 (1994) no. 4, 673–678.CrossRefGoogle Scholar
Rosikin, Y. and Shitikova, M., ‘Application of fractional calculus for dynamic problems of solid mechanics’, Amer. Soc. Mech. Eng. 63 (2010) 010801 1–52.Google Scholar
Saadatmandi, A. and Deghan, M., ‘Numerical solution of a mathematical model for capillary formation in tumor angiogenesis via the tau method’, Commun. Numer. Method. Eng. 24 (2008) 1467–1474.CrossRefGoogle Scholar
Saadatmandi, A. and Deghan, M., ‘A new operational matrix for solving fractional-order differential equation’, Comput. Math. Appl. 59 (2010) 1326–1336.CrossRefGoogle Scholar
Shah, K., Khalil, H. and Khan, R. A., ‘Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations’, Chaos Solitons Fractals 77 (2015) 240–246.CrossRefGoogle Scholar
Shah, K., Zeb, S. and Khan, R. A., ‘Existence and uniqueness of solutions for fractional order m-point boundary value problems’, Frac. Diff. Calc. 5 (2015) no. 2, 171–181.Google Scholar
Shah, K., Ali, A. and Khan, R. A., ‘Degree theory and existence of positive solutions to coupled systems of multi-point boundary value problems’, Bound. Value Probl. 2016 (2016) no. 43, 1–12.CrossRefGoogle Scholar
Sundnes, J., Lines, G. T., Mardal, K. A. and Tveito, A., ‘Multigrid block preconditioning for a coupled system of partial differential equations modeling the electrical activity in the heart’, Comp. Method. Biomech. Biomed. Eng. 5 (2002) no. 6, 397–409.CrossRefGoogle ScholarPubMed
Sundnes, J., Lines, G. T. and Tveito, A., ‘An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso’, Math. Biosci. 194 (2005) no. 2, 233–248.CrossRefGoogle ScholarPubMed
Torvik, P. J. and Bagley, R. L., ‘On the appearance of fractional derivatives in the behaviour of real materials’, J. Appl. Mech. 51 (1984) 294–298.CrossRefGoogle Scholar
Wald, R. M., ‘Construction of solutions of gravitational, electromagnetic or other perturbation equations from solutions of decoupled equations’, Phy. Rev. Lett. 41 (1978) no. 4, 203–209.CrossRefGoogle Scholar
Wang, Y. and Fan, Q., ‘The second kind Chebyshev wavelet method for solving fractional differential equation’, Appl. Math. Comput. 218 (2012) 85–92.Google Scholar
Wazwaz, M., ‘The decomposition method applied to systems of partial differential equations and to the reaction diffusion Brusselator model’, Appl. Math. Comput. 110 (2000) 251–264.Google Scholar
Wensheng, S., ‘Computer simulation and modeling of physical and biological processes using partial differential equations’, University of Kentucky Doctoral Dissertations, Lexington, KY, 2007.Google Scholar