## THERMAL SCIENCE

International Scientific Journal

### 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**

PAPER SUBMITTED: 2022-06-12

PAPER REVISED: 2022-06-24

PAPER ACCEPTED: 2022-07-04

PUBLISHED ONLINE: 2023-04-08

- Machado, J. T., et al., Recent History of Fractional Calculas, Comun. Non-l. Sci. Num. Simul., 16 (2011), 3, pp. 1140-1153
- Kilbas, A. A., et al., Theory and Applications of Fractional Differential Equations, Elsavier, North-Holland Mathematics Studies, Amester Dam, Amsterdam, The Netherlands, 2006
- Hilfer, R., Threefold Introduction to Fractional Derivatives, in: Anomalous Transport: Foundations and Applications, Willy, New York, USA, 2008
- Miller, K. S., Ross, B., An Introduction the Fractional Calculas and Fractional Differential Equations, John Wiley and Sons, New York, USA, 1993
- Miller, K. S., Fractional Differential Equations, Journal Frac. Cal., 3 (1993), pp. 49-57
- 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
- Mainardi, F., An Historical Perspective on Fractional Calculus in Linear Viscoelasticity, Fractional Calculus and Applied Analysis, 15 (2012), 4, pp. 712-717
- Lakshmikantham, V., Vatsala, A. S., Basic Theory of Fractional Differential Equations, Non-Linear Analysis, Theory, Methods and Applications, 69 (2008), 8, pp. 2677-2682
- Zhou, Y., et al., Basic Theory of Fractional Differential Equations, World Scientific, Singapore, Singapore, 2016
- Zhou, Y., Existence and Uniqueness of Solutions for a System of Fractional Differential Equations, Fract. Calc. Appl. Anal., 12 (2009), 2, pp. 195-204
- 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
- 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
- Abdulaziz, O., et al., Solving Systems of Fractional Differential Equations by Homotopy-Perturbation Method, Phys. Lett. A, 372 (2008), 4, pp. 451-459
- 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
- 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
- Daftardar-Gejji, V., Jafari, H., Solving a Multi-Order Fractional Differentail Equation Using Adomian Decomposition, Appl. Math., 189 (2007), 1, pp. 541-548
- 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
- 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
- Saadatmandi, A., Mohabbati, M., Numerical Solution of Fractional Telegraph Equation Via the Tau Method, Math. Rep., 17 (2011), 2, pp. 155-166
- 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
- Saadatmandi, S., Dehghan, M., A New Operational Matrix for Solving Fractional-Order Differential Equations, Comput. Math. Appl., 59 (2010), 3, pp.1326-1336
- Samko, S. G., Ross, B., Integration and Differentiation a Variable Fractional Order, Integral Transform. Spec. Funct. 1, (1993), 4, pp. 277-300
- Lorenzo, C. F., Hartley, T. T., Variable Order and Distributed Order Fractional Operators, Non-Linear Dyn., 29 (2002), Feb., pp. 57-98
- Coimbra, C., Mechanics with Variable-Order Differential Operators, Ann. Phys., 515 (2003), 11-12, pp. 692-703
- Sun, H., et al., Variable-Order Fractional Differential Operators in Anomalous Diffusion Modelling, Physica A, 388 (2009), 45864592
- 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
- Valerio, D., Sa da Costa, J., Variable-Order Fractional Derivatives and Their Numerical Approximations, Signal Process, 91 (2011), 3, pp. 470-483
- Saadatmandi, A., Bernstein Operational Matrix of Fractional Derivatives and Its Applications, Appl.Math. Model., 38 (2014), 4, pp. 1365-1372
- 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
- Nikolaevich, T. A., Systems of Differential Equations Containing Small Parameters in the Derivatives, Matematicheskii Sbornik, 73 (1952), 3, pp. 575-586
- Kimeu, J. M., Fractional Calculus: Definitions and Applications, M. Sc. theses and Specialist Projects, Western Kentucky University, Bowling Green, Kentucky, 2009
- 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
- Gottlieb, S., Gottlieb, D., Spectral methods, Scholarpedia, 4 (2009), 9, 7504