Skip to main content Accessibility help
×
Home

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

  • Kamal Shah (a1), Hammad Khalil (a2) and Rahmat Ali Khan (a3)

Abstract

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.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

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

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

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

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

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

Copyright

References

Hide All
1. 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, 15421548.
2. Arikoglu, A. and Ozkol, I., ‘Solution of fractional differential equations by using differential transform method’, Chaos Solitons Fractals 34 (2007) no. 5, 14731481.
3. Ayaz, F., ‘Solutions of the system of differential equations by differential transform method’, Appl. Math. Comput. 147 (2004) 547567.
4. Chen, C. and Hsiao, C., ‘Haar wavelet method for solving lumped and distributed parameter systems’, IEE Press Contr. Theor. Appl. 144 (1997) 8794.
5. 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, 448479.
6. 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, 935949.
7. 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) 174195.
8. Erturk, V. S. and Momani, S., ‘Solving systems of fractional differential equations using differential transform method’, J. Comput. Appl. Math. 215 (2008) 142151.
9. Eslahchi, M. R. and Dehghan, M., ‘Application of Taylor series in obtaining the orthogonal operational matrix’, Comput. Math. Appl. 61 (2011) no. 6, 25962604.
10. 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, 15171535.
11. Folland, G. B., Introduction to partial differential equations , 2nd edn (Princeton University Press, Princeton, NJ, 1995).
12. Gasea, M. and Sauer, T., ‘On the history of multivariate polynomial interpolation’, J. Comput. Appl. Math. 122 (2000) 2335.
13. Gorenflo, R., Mainardi, F., Scalas, E. and Raberto, M., ‘Fractional calculus and continuous time finance III’, Math. Finance (2000) 171180.
14. 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) 584595.
15. Hilfer, R., Applications of fractional calculus in physics (World Scientific Publishing Company, Singapore, 2000).
16. 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) 220229.
17. Ibrahim, R. W., ‘Solutions to systems of arbitrary-order differential equations in complex domains’, Electron. J. Differential Equations 46 (2014) 113.
18. 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) 19621969.
19. Katica, R. and Hedrih, S., ‘Dynamics of multi-pendulum systems with fractional order creep elements’, J. Theoret. Appl. Mech. 46 (2008) no. 3, 483509.
20. Katica, R. and Hedrih, S., ‘Fractional order hybrid system dynamics’, Proc. Appl. Math. Mech. 13 (2013) 2526.
21. 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, 736748.
22. 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) 19381953.
23. 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, 25542567.
24. 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, 165185.
25. 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) 2535.
26. Kilbas, A. A., Srivastava, H. M. and Trujillo, J., Theory and applications of fractional differential equations (Elsevier Science, Amsterdam, 2006).
27. 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, 11491162.
28. Li, Y. L., ‘Solving a nonlinear fractional differential equation using Chebyshev wavelets’, Nonlinear Sci. Numer. Simul. 15 (2010) 22842292.
29. 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) 304315.
30. 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) 19311950.
31. Liu, F., Anh, V. and Turner, I., ‘Numerical solution of the space fractional Fokker–Planck equation’, J. Comput. Appl. Math. 66 (2005) 209219.
32. 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) 1524.
33. Metzler, R. and Klafter, J., ‘The random walk’s guide to anomalous diffusion: a fractional dynamics approach’, Phy. Rep. 339 (2000) no. 1, 177.
34. Metzler, R. and Klafter, J., ‘Boundary value problems for fractional diffusion equations’, Phys. A: Stat. Mech. Appl. 278 (2005) 107125.
35. 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).
36. 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.
37. 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) 609621.
38. Oldham, K. B., ‘Fractional differential equations in electrochemistry’, Adv. Eng. Soft. 41 (2010) 912.
39. Paraskevopolus, P. N, Saparis, P. D. and Mouroutsos, S. G., ‘The Fourier series operational matrix of integration’, Int. J. Syst. Sci. 16 (1985) 171176.
40. Parthiban, V. and Balachandran, K., ‘Solutions of system of fractional partial differential equations’, Appl. Appl. Math. 8 (2013) no. 1, 289304.
41. Podlubny, I., Fractional differential equations (Academic Press, San Diego, CA, 1999).
42. Razzaghi, M. and Yousefi, S., ‘The Legendre wavelets operational matrix of integration’, Internat. J. Systems Sci. 32 (2001) 495502.
43. 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) 805810.
44. Rehman, M. and Khan, R. A., ‘The Legendre wavelet method for solving fractional differential equation’, Commun. Nonlinear Sci. Numer. Simul. 16 (2011) 41634173.
45. 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) 26302637.
46. 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, 673678.
47. Rosikin, Y. and Shitikova, M., ‘Application of fractional calculus for dynamic problems of solid mechanics’, Amer. Soc. Mech. Eng. 63 (2010) 010801 152.
48. 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) 14671474.
49. Saadatmandi, A. and Deghan, M., ‘A new operational matrix for solving fractional-order differential equation’, Comput. Math. Appl. 59 (2010) 13261336.
50. 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) 240246.
51. 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, 171181.
52. 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, 112.
53. 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, 397409.
54. 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, 233248.
55. Torvik, P. J. and Bagley, R. L., ‘On the appearance of fractional derivatives in the behaviour of real materials’, J. Appl. Mech. 51 (1984) 294298.
56. 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, 203209.
57. Wang, Y. and Fan, Q., ‘The second kind Chebyshev wavelet method for solving fractional differential equation’, Appl. Math. Comput. 218 (2012) 8592.
58. Wazwaz, M., ‘The decomposition method applied to systems of partial differential equations and to the reaction diffusion Brusselator model’, Appl. Math. Comput. 110 (2000) 251264.
59. Wensheng, S., ‘Computer simulation and modeling of physical and biological processes using partial differential equations’, University of Kentucky Doctoral Dissertations, Lexington, KY, 2007.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed