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An operational matrix based on the Independence polynomial of a complete bipartite graph for the Caputo fractional derivative


The vast applicability of fractional calculus to model physical phenomena in the form of fractional differential equations and their complexity has created a massive demand for efficient analytic and semi-analytic techniques to solve fractional differential equations. This paper has derived a new operational matrix using the independence polynomial of a complete bipartite graph to solve multi-order fractional differential equations. While deriving the operational matrix, the Caputo sense fractional derivatives have been considered. Series solutions are found by using the collocation matrix method. The main characteristic of this approach is that it reduces a complex fractional differential equation to a system of algebraic equations. The convergence analysis and the time complexity analysis of the proposed scheme are also presented in this paper. Six examples have been considered to illustrate the relevance and applicability of the method described. The results obtained are compared with the exact solutions. We have also compared our results with the ones obtained by other methods available in the literature.

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  1. 1.

    Oldham, K., Spanier, J.: The Fractional Calculus. Academic Press, New York (1974)

    MATH  Google Scholar 

  2. 2.

    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley-Blackwell, Hoboken (1993)

    MATH  Google Scholar 

  3. 3.

    Podlubny, I.: Fractional Differential Equations: An Introduction to Fractional Derivatives. Elsevier, Amsterdam (1998)

    MATH  Google Scholar 

  4. 4.

    Kilbas, A.A., Srivastava, H.M., Trujillo, J.: Theory and Applications of Fractional Differential Equations, vol. 204. Elsevier, North-Holland Mathematics Studies, (2006)

  5. 5.

    Hajipour, M., Jajarmi, A., Baleanu, D.: An efficient nonstandard finite difference scheme for a class of fractional chaotic systems. J. Comput. Nonlinear Dyn. 13(2), (2017)

  6. 6.

    Baleanu, D., Jajarmi, A., Hajipour, M.: A new formulation of the fractional optimal control problems involving Mittag-Leffler nonsingular kernel. J. Optim. Theory Appl. 175(3), 718–737 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Mirzaee, F., Samadyar, N.: Parameters estimation of HIV infection model of CD\({}_{4}^{+}\) T- cells by applying orthonormal Bernstein collocation method. Int. J. Biomath. 11(2), (2017)

  8. 8.

    Mirzaee, F., Samadyar, N.: On the numerical method for solving a system of nonlinear fractional ordinary differential equations arising in HIV infection of CD\({}_{4}^{+}\) T-cells. Iran. J. Sci. Technol. Trans. A Sci. 43(3), 1127–1138 (2019)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Baishya, C.: Dynamics of fractional stage structured predator prey model with prey refuge. Indian J. Ecol. 47, 1118–1124 (2020)

    Google Scholar 

  10. 10.

    Abdo, M.S., Panchal, S.K., Shah, K., Abdeljawad, T.: Existence theory and numerical analysis of three species prey-predator model under Mittag-Leffler power law. Adv. Differ. Equ. (2020)

  11. 11.

    Baishya, C., Achar, S.J., Veeresha, P., Prakasha, D.G.: Dynamics of a fractional epidemiological model with disease infection in both the populations, Chaos: An Interdisciplinary. J. Nonlinear Sci. 31(4), (2021)

  12. 12.

    Baishya, C.: Dynamics of fractional Holling type-II predator-prey model with prey refuge and additional food to predator. J. Appl. Nonlinear Dyn. 10(2), 315–328 (2020)

    MathSciNet  Article  Google Scholar 

  13. 13.

    He, J.: Nonlinear oscillation with fractional derivative and its applications. In: Conference on Vibrating Engineering’98, pp. 288–291. Dalian, China (1998)

  14. 14.

    He, J.: Some applications of nonlinear fractional differential equations and their approximations. Bull. Sci. Technol. Soc. 15, 86–90 (1999)

    Google Scholar 

  15. 15.

    Mainardi, F.: Fractional calculus: Some basic problems in continuum and statistical mechanics, Fractals and Fractional Calculus in Continuum Mechanics, pp. 291–348, (2012)

  16. 16.

    Baillie, R.: Long memory processes and fractional integration in econometrics. J. Econom. 73(1), 5–59 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Prakasha, D.G., Veeresha, P.: Analysis of Lakes pollution model with Mittag-Leffler kernel. J. Ocean Eng. Sci. 5(4), 310–322 (2020)

    Article  Google Scholar 

  18. 18.

    Veeresha, P., Prakasha, D.G.: A reliable analytical technique for fractional Caudrey-Dodd-Gibbon equation with Mittag-Leffler kernel. Nonlinear Eng. 9, 319–328 (2020)

    Article  Google Scholar 

  19. 19.

    Pundikala, V., Gowda, P.D., Dumitru, B., Dumitru, B.: Analysis of fractional Swift-Hohenberg equation using a novel computational technique. Math. Methods Appl. Sci. 43(4), 1970–1987 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Veeresha, P., Prakasha, D.G.: Solution for fractional generalized Zakharov equations with Mittag-Leffler function, Results Eng. vol. 5, (2020)

  21. 21.

    Veeresha, P., Prakasha, D.G., Jagdev, S.: Solution for fractional forced KdV equation using fractional natural decomposition method, Results Eng. vol. 5, (2020)

  22. 22.

    Momani, S., Shawagfeh, N.: Decomposition method for solving fractional Riccati differential equations. Appl. Math. Comput. 182, 1083–1092 (2006)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Wang, Q.: Numerical solutions for fractional KdV-Burgers equation by Adomian decomposition method. Appl. Math. Comput. 182, 1048–1055 (2006)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Odibat, Z., Momani, S.: The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics. Comput. Math. Appl. 58, 2199–2208 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25.

    Khan, Y., Faraz, N., Yildirim, A., Wu, Q.: Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science. Comput. Math. Appl. 62(5), 2273–2278 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Sweilam, N., Khader, M., Al-Bar, R.: Numerical studies for a multi-order fractional differential equation. Phys. Lett. A 371, 26–33 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Hashim, I., Abdulaziz, O., Momani, S.: Homotopy analysis method for fractional ivps. Commun. Nonlinear Sci. Numer. Simul. 14, 674–684 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28.

    Saratha, S.R., Bagyalakshmi, M., G. Sai Sundara Krishnan, : Fractional generalised homotopy analysis method for solving nonlinear fractional differential equations. Comput. Appl. Math. 39(2), (2020)

  29. 29.

    Wei, G., Veeresha, P., Prakasha, D., Baskonus, G., H.M., Yel.: New numerical results for the time-fractional phi-four equation using a novel analytical approach, Appl. Math. Comput. vol. 478, (2020)

  30. 30.

    Veeresha, P., Prakasha, D.G., Singh, J., Khan, I., Kumar, D.: Analytical approach for fractional extended Fisher-Kolmogorov equation with Mittag-Leffler kernel. Adv. Differ. Equ. (2020)

  31. 31.

    Veeresha, P., Prakasha, D., Singh, J.: A novel approach for nonlinear equations occurs in ion acoustic waves in plasma with Mittag-Leffler law. Eng. Comput. Int. J. Comput.-Aided Eng. 37(6), 1865–1897 (2020)

    Google Scholar 

  32. 32.

    Rawashdeh, E.: Numerical solution of fractional integro-differential equations by collocation method. Appl. Math. Comput. 176, 1–6 (2006)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Diethelm, K., Ford, N.J., Freed, A.D.: A predictor corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1), 3–22 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Jafarian, A., Mokhtarpour, M., Baleanu, D.: Artificial neural network approach for a class of fractional ordinary differential equation. Neural Comput. Appl. 28(4), 765–773 (2017)

    Article  Google Scholar 

  35. 35.

    Ervin, V., Roop, J.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Part. Differ. Equ. 22, 558–576 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Abuasad, S., Hashim, I., Abdul Karim, S.A.: Modified fractional reduced differential transform method for the solution of multiterm time fractional diffusion equations. Adv. Math. Phys. (2019)

  37. 37.

    Samadyar, N., Mirzaee, F.: Orthonormal Bernoulli polynomials collocation approach for solving stochastic Itô-Volterra integral equations of Abel type. Int. J. Numer. Model. Electron. Netw. Dev. Fields 33(1), (2020)

  38. 38.

    Mirzaee, F., Samadyar, N.: Explicit representation of orthonormal Bernoulli polynomials and its application for solving Volterra-Fredholm-Hammerstein integral equations. SeMA J. 77(1), 81–96 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39.

    Kazem, S., Abbasbandy, S., Kumar, S.: Fractional-order legendre functions for solving fractional-order differential equations. Appl. Math. Model. 7(37), 5498–5510 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  40. 40.

    Kazem, S.: An integral operational matrix based on Jacobi polynomials for solving fractional-order differential equations. Appl. Math. Model. 37, 1126–1136 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  41. 41.

    Doha, E., Bhrawy, A., Ezz-Eldien, S.: Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations. Appl. Math. Model. 35(4), 5662–5672 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  42. 42.

    Han, W., Chen, Y.-M., Liu, D.-Y., Li, X.-L., Boutat, D.: Numerical solution for a class of multi-order fractional differential equations with error correction and convergence analysis. Adv. Differ. Equ. 2018,(2018)

  43. 43.

    Mirzaee, F., Alipour, S.: Cubic B-approximation for linear stochastic integro-differential equation of fractional order. J. Comput. Appl. Math. 366,(2020)

  44. 44.

    Isah, A., Phang, C.: New operational matrix of derivative for solving non-linear fractional differential equations via Genocchi polynomials. J.King Saud Univ. 31, 1–7 (2019)

    Article  Google Scholar 

  45. 45.

    Mirzaee, F., Bimesl, S.: A new approach to numerical solution of second-order linear hyperbolic partial differential equations arising from physics and engineering. Results Phys. 3, 241–247 (2013)

    Article  Google Scholar 

  46. 46.

    Mirzaee, F., Bimesl, S.: Numerical solutions of systems of high-order Fredholm integro-differential equations using Euler polynomials. Appl. Math. Model. 39(22), 6767–6779 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  47. 47.

    Abd-Elhameed, W.: On solving linear and nonlinear sixth-order two point boundary value problems via an elegant harmonic numbers operational matrix of derivativesd. Comput. Model. Eng. Sci. 101, 159–185 (2014)

    MathSciNet  MATH  Google Scholar 

  48. 48.

    Mirzaee, F., Alipour, S.: Fractional-order orthogonal Bernstein polynomials for numerical solution of nonlinear fractional partial Volterra integro-differential equations. Math. Methods Appl. Sci. 42(6), 1870–1893 (2019)

    MathSciNet  MATH  Article  Google Scholar 

  49. 49.

    Mirzaee, F., Samadyar, N.: Numerical solution based on two-dimensional orthonormal Bernstein polynomials for solving some classes of two-dimensional nonlinear integral equations of fractional order. Appl. Math. Comput. 344, 191–203 (2019)

    MathSciNet  MATH  Google Scholar 

  50. 50.

    Mirzaee, F., Samadyar, N.: Convergence of 2D-orthonormal Bernstein collocation method for solving 2D-mixed Volterra–Fredholm integral equations, Transactions of A. Razmadze Mathematical Institute, vol. 172, no. 3, Part B, pp. 631–641, (2018)

  51. 51.

    Ramane, H.S., Shiralashetti, S.C., Mundewadi, R.A., Jummannaver, R.B.: Numerical solution of Fredholm integral equations using Hosoya polynomial of path graphs. Am. J. Numer. Anal 5(1), 11–15 (2018)

    Google Scholar 

  52. 52.

    Agheli, B.: Approximate solution for solving fractional Riccati differential equations via trigonometric basic functions, Transactions of A. Razmadze Mathematical Institute, vol. 172, no. 3, Part A, pp. 299–308, (2018)

  53. 53.

    Kumbinarasaiah, S.: A new approach for the numerical solution for nonlinear Klein-Gordon equation. SeMA J. 77, 435–456 (2020)

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Koc, M., nd Cakmak, A.B., Kurnaz, A., Uslu, K.: A new Fibonacci type collocation procedure for boundary value problems, Adv. Differ. Equ., vol. 262, (2013)

  55. 55.

    Mirzaee, F., Hoseini, S.F.: Solving singularly perturbed differential-difference equations arising in science and engineering with Fibonacci polynomials. Results Phys. 3, 134–141 (2013)

    Article  Google Scholar 

  56. 56.

    Mirzaee, F., Hoseini, S.: Application of Fibonacci collocation method for solving Volterra-Fredholm integral equations. Appl. Math. Comput. 273, 637–644 (2016)

    MathSciNet  MATH  Google Scholar 

  57. 57.

    Abd-Elhameed, W.M., Youssri, Y.H.: A novel operational matrix of Caputo fractional derivatives of Fibonacci polynomials: Spectral solutions of fractional differential equations, Entropy, vol. 18, no. 10, (2016)

  58. 58.

    Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  59. 59.

    Aphithana, A., Ntouyas, S.K., Tariboon, J.: Existence and uniqueness of symmetric solutions for fractional differential equations with multi-order fractional integral conditions, Boundary Value Problems, vol. 2015, (2015)

  60. 60.

    Hesameddini, E., Rahimi, A., Asadollahifard, E.: On the convergence of a new reliable algorithm for solving multi-order fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 34, 154–164 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  61. 61.

    Torvik, R., J, P., Bagley : On the appearance of the fractional derivative in the behavior of real materials. ASME J. Appl. Mech. 51, 294–298 (1984)

  62. 62.

    Matthew Ferrin, G.: Independence Polynomials. University of South Carolina, Columbia (2014)

    Google Scholar 

  63. 63.

    Ford, N.J., Connolly, J.: Systems-based decomposition schemes for the approximate solution of multi-term factional differential equations. J. Comput. Appl. Math. 229:282–391

  64. 64.

    Jafari, H., Khalique, C., Ramezani, M., Tajadodi, H.: Numerical solution of fractional differential equations by using fractional B-spline. Open Phys. J. 11, 1372–1376 (2013)

    Google Scholar 

  65. 65.

    Youssri, Y.H.: A new operational matrix of Caputo fractional derivatives of fermat polynomials: an application for solving the Bagley-Torvik equation. Adv. Differ. Equ. 2017, 73 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  66. 66.

    Saadatmandi, A., Dehghan, M.: A new operational matrix for solving fractional-order differential equations. Comput. Math. Appl. 59(3), 1326–1336 (2010)

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to Chandrali Baishya.

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Baishya, C. An operational matrix based on the Independence polynomial of a complete bipartite graph for the Caputo fractional derivative. SeMA (2021).

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  • Caputo fractional derivative
  • Independence polynomial
  • Complete bipartite graph
  • Fractional multi-order equation
  • Operational matrix

Mathematics Subject Classification

  • 26A33
  • 65M70