THERMAL SCIENCE

International Scientific Journal

Kamal Shah

,

Bahaaeldin Abdalla

,

Thabet Abdeljawad

,

Iyad Suwan

AN EFFICIENT MATRIX METHOD FOR COUPLED SYSTEMS OF VARIABLE FRACTIONAL ORDER DIFFERENTIAL EQUATIONS

ABSTRACT
We establish a powerful numerical algorithm to compute numerical solutions of coupled system of variable fractional order differential equations. Our numer­ical procedure is based on Bernstein polynomials. The mentioned polynomials are non-orthogonal and have the ability to produce good numerical results as compared to some other numerical method like wavelet. By variable fractional order differentiation and integration, some operational matrices are formed. On using the obtained matrices, the proposed coupled system is reduced to a system of algebraic equations. Using MATLAB, we solve the given equation for required results. Graphical presentations and maximum absolute errors are given to illustrate the results. Some useful features of our sachem are those that we need no discretization or collocation technique prior to develop operational matrices. Due to these features the computational complexity is much more reduced. Further, the efficacy of the procedure is enhanced by increasing the scale level. We also compare our results with that of Haar wavelet method to justify the useful­ness of our adopted method.
KEYWORDS
variable fractional order differential equations, coupled system, Bernstein polynomials
PAPER SUBMITTED: 2022-06-12
PAPER REVISED: 2022-06-24
PAPER ACCEPTED: 2022-07-04
PUBLISHED ONLINE: 2023-04-08
DOI REFERENCE: https://doi.org/10.2298/TSCI23S1195S
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2023, VOLUME 27, ISSUE Special issue 1, PAGES [195 - 210]
REFERENCES
  1. Machado, J. T., et al., Recent History of Fractional Calculas, Comun. Non-l. Sci. Num. Simul., 16 (2011), 3, pp. 1140-1153
  2. Kilbas, A. A., et al., Theory and Applications of Fractional Differential Equations, Elsavier, North-Holland Mathematics Studies, Amester Dam, Amsterdam, The Netherlands, 2006
  3. Hilfer, R., Threefold Introduction to Fractional Derivatives, in: Anomalous Transport: Foundations and Applications, Willy, New York, USA, 2008
  4. Miller, K. S., Ross, B., An Introduction the Fractional Calculas and Fractional Differential Equations, John Wiley and Sons, New York, USA, 1993
  5. Miller, K. S., Fractional Differential Equations, Journal Frac. Cal., 3 (1993), pp. 49-57
  6. Agarwal, R., et al., A Survey on Similar Differential Equations and Inclusions Involving Riemann-Liouville Fractional Derivative, Journal Adv. Diff. Equ., 10 (2009), 2, pp. 857-868
  7. Mainardi, F., An Historical Perspective on Fractional Calculus in Linear Viscoelasticity, Fractional Calculus and Applied Analysis, 15 (2012), 4, pp. 712-717
  8. Lakshmikantham, V., Vatsala, A. S., Basic Theory of Fractional Differential Equations, Non-Linear Analysis, Theory, Methods and Applications, 69 (2008), 8, pp. 2677-2682
  9. Zhou, Y., et al., Basic Theory of Fractional Differential Equations, World Scientific, Singapore, Singapore, 2016
  10. Zhou, Y., Existence and Uniqueness of Solutions for a System of Fractional Differential Equations, Fract. Calc. Appl. Anal., 12 (2009), 2, pp. 195-204
  11. Ali, Z., et al., On Ulam's Stability for a Coupled Systems of Non-Linear Implicit Fractional Differential Equations, Bulletin of the Malaysian Mathematical Sciences Society, 42 (2019), 5, pp. 2681-2699
  12. Sousa, E., Li, C., A Weighted Finite Difference Method for the Fractional Diffusion Equation Based on the Riemann-Liouville Derivative, Appl. Numer. Math., 90 (2015), Apr., pp. 22-37
  13. Abdulaziz, O., et al., Solving Systems of Fractional Differential Equations by Homotopy-Perturbation Method, Phys. Lett. A, 372 (2008), 4, pp. 451-459
  14. Suarez, L., Shokooh, A., An Eigenvector Expansion Method for the Solution of Motion Containing Fractional Derivatives, Journal Appl. Mech., 64 (1997), 3, pp. 629-635
  15. Erturk, V. S., Momani, S., Solving Systems of Fractional Differential Equations Using Differential Transform Method, Journal Comput. Appl. Math., 215 (2008), 1, pp. 142-151
  16. Daftardar-Gejji, V., Jafari, H., Solving a Multi-Order Fractional Differentail Equation Using Adomian Decomposition, Appl. Math., 189 (2007), 1, pp. 541-548
  17. Odibat, Z., Momani, S., Application of Variational Iteration Method to Non-Linear Differential Equations of Fractional Order, Int. J. Non-linear Sci. Numer. Simul., 7 (2006), 1, pp. 15-27
  18. Amer, Y. A., et al., Solving Fractional Integro-Differential Equations by Using Sumudu Transform Method and Hermite Spectral Collocation Method, Computers, Materials and Continua, 54 (2018), 22, pp. 161-180
  19. Saadatmandi, A., Mohabbati, M., Numerical Solution of Fractional Telegraph Equation Via the Tau Method, Math. Rep., 17 (2011), 2, pp. 155-166
  20. Meyers, S. D., et al., An Introduction Wavelet Analysis in Oceanography and Meteorology: With Application the Dispersion of Yanai Waves, Monthly Weather Review, 121 (1993), 10, pp. 2858-2866
  21. Saadatmandi, S., Dehghan, M., A New Operational Matrix for Solving Fractional-Order Differential Equations, Comput. Math. Appl., 59 (2010), 3, pp.1326-1336
  22. Samko, S. G., Ross, B., Integration and Differentiation a Variable Fractional Order, Integral Transform. Spec. Funct. 1, (1993), 4, pp. 277-300
  23. Lorenzo, C. F., Hartley, T. T., Variable Order and Distributed Order Fractional Operators, Non-Linear Dyn., 29 (2002), Feb., pp. 57-98
  24. Coimbra, C., Mechanics with Variable-Order Differential Operators, Ann. Phys., 515 (2003), 11-12, pp. 692-703
  25. Sun, H., et al., Variable-Order Fractional Differential Operators in Anomalous Diffusion Modelling, Physica A, 388 (2009), 45864592
  26. Xu, Y., He, Z., Existence and Uniqueness Results for Cauchy Problem of Variable-Order Fractional Differential Equations, Journal Appl. Math. Computing, 43 (2013), 1, pp. 295-306
  27. Valerio, D., Sa da Costa, J., Variable-Order Fractional Derivatives and Their Numerical Approximations, Signal Process, 91 (2011), 3, pp. 470-483
  28. Saadatmandi, A., Bernstein Operational Matrix of Fractional Derivatives and Its Applications, Appl.Math. Model., 38 (2014), 4, pp. 1365-1372
  29. Bushnaq, S., et al., Computation of Numerical Solutions to Variable Order Fractional Differential Equations by Using Non-Orthogonal Basis, AIMS Mathematics, 7 (2022), 6, pp. 10917-10938
  30. Nikolaevich, T. A., Systems of Differential Equations Containing Small Parameters in the Derivatives, Matematicheskii Sbornik, 73 (1952), 3, pp. 575-586
  31. Kimeu, J. M., Fractional Calculus: Definitions and Applications, M. Sc. theses and Specialist Projects, Western Kentucky University, Bowling Green, Kentucky, 2009
  32. Abdeljawad, T., et al., Efficient Sustainable Algorithm for Numerical Solutions of Systems of Fractional Order Differential Equations by Haar Wavelet Collocation Method, Alexandria Engineering Journal, 59 (2020), 4, pp. 2391-2400
  33. Gottlieb, S., Gottlieb, D., Spectral methods, Scholarpedia, 4 (2009), 9, 7504
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An efficient matrix method for coupled systems of variable fractional order differential equations

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