THERMAL SCIENCE

International Scientific Journal

Jumana H. S. Alkhalissi

,

Ibrahim Emiroglu

,

Aydin Secer

,

Mustafa Bayram

THE GENERALIZED GEGENBAUER-HUMBERTS WAVELET FOR SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS

ABSTRACT
In this paper we present a new method of wavelets, based on generalized Gegen­bauer-Humberts polynomials, named generalized Gegenbauer-Humberts wave­lets. The operational matrix of integration are derived. By using the proposed method converted linear and non-linear fractional differential equation a system of algebraic equations. In addition, discussed some examples to explain the efficiency and accuracy of the presented method.
KEYWORDS
block-pulse functions, operational matrix or integration, the generalized Gegenbauer-Humberts polynomial, fractional calculus, orthogonal polynomials
PAPER SUBMITTED: 2020-04-25
PAPER REVISED: 2020-05-29
PAPER ACCEPTED: 2020-06-01
PUBLISHED ONLINE: 2020-10-25
DOI REFERENCE: https://doi.org/10.2298/TSCI20S1107A
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2020, VOLUME 24, ISSUE Supplement 1, PAGES [S107 - S118]
REFERENCES
  1. Oldham, K. B., Spanier, J., The Fractional Calculus, Academic Press, New York, USA, 1974, pp. 417-422
  2. Meerschaert, M., Tadjeran, C., Finite Difference Approximations for Two-Sided Space-Fractional Partial Differential Equations, Applied Numerical Mathematics, 56 (2006), 1, pp. 80-90
  3. Arikoglu, A.,Ozkol, I., Solution of Fractional Differential Equations by Using Differential Transform Method, Chaos, Solitons and Fractals, 34 (2007), 5, pp. 1473-1481
  4. Gaul, L., et al., Damping Description Involving Fractional Operators, Mechanical Systems and Signal Processing, 5 (1991), 2, pp. 81-88
  5. Choudhary, A., et al., Numerical Simulation of a Fractional Model of Temperature Distribution and Heat Flux in the Semi Infinite Solid, Alexandria Engineering Journal, 55 (2016), 1, pp. 87-91
  6. El-Wakil, S. A., et al., Adomian Decomposition Method for Solving Fractional Non-Linear Differential Equations, Applied Mathematics and Computation, 182 (2006), 1, pp.313-324
  7. Odibat, Z., Momani, S., Application of Variational Iteration Method to Non-Linear Differential Equations of Fractional Order, International Journal of Non-linear Sciences and Numerical Simulation, 7 (2006), 1, pp. 27-34
  8. Saadatmandi, A., Bernstein Operational Matrix of Fractional Derivatives and Its Applications, Applied Mathematical Modelling, 38 (2014), 4, pp. 1365-1372
  9. Jajarmi, A., et al., A New Fractional Modelling and Control Strategy for the Outbreak of Dengue Fever, Physica A: Statistical Mechanics and its Applications, 535 (2019), 122524
  10. Bhatter, S., et al., Fractional Modified Kawahara Equation with Mittag-Leffler Law, Chaos, Solitons and Fractals, 131 (2020), 109508
  11. Goswami, A., et al., An Efficient Analytical Technique for Fractional Partial Differential Equations Occurring in Ion Acoustic Waves in Plasma, Journal of Ocean Engineering and Science, 4 (2019), 2, pp. 85-99
  12. Baleanu, D., et al., New Features of the Fractional Euler-Lagrange Equations for a Physical System Within Non-Singular Derivative Operator, The European Physical Journal Plus, 134 (2019), 181
  13. Jajarmi, A., et al., A New Feature of the Fractional Euler-Lagrange Equations for a Coupled Oscillator Using a Non-Singular Operator Approach, Frontiers in Physics, 7 (2019), 196
  14. Kilicman, A., et al., Kronecker Operational Matrices for Fractional Calculus and Some Applications, Applied Mathematics and Computation, 187 (2007), 1, pp. 250-265
  15. Saadatmandi, A., Dehghan, M., A New Operational Matrix for Solving Fractional-Order Differential Equations, Computers and Mathematics with Application, 59 (2010), 3, pp. 1326-1336
  16. Chen, C., Hsiao, C., Haar Wavelet Method for Solving Lumped and Distributed-Parameter Systems, IEE Proceedings - Control Theory and Applications, 144 (1997), 1, pp. 87-94
  17. Elgindy, K. T., Smith-Miles, K. A., Solving Boundary Value Problems, Integral, and Integro-Differential Equations Using Gegenbauer Integration Matrices, Journal of Computational and Applied Mathematics, 237 (2013), 1, pp. 307-325
  18. Rehman, M. U., Saeeds, U., Gegenbauer Wavelets Operational Matrix Method for Fractional Differential Equations, Journal of the Korean Mathematical Society, 52 (2015), 5, pp. 1069-1096
  19. Srivastava, H. M., et al., An Application of the Gegenbauer Wavelet Method for the Numerical Solution of the Fractional Bagley-Torvik Equation, Russian Journal of Mathematical Physic, 26 (2019), 1, pp. 77-93
  20. Saeed, U., Wavelet Quasilinearization Methods for Fractional Differential Equations, Ph. D. thesis, School of Natural Sciences National University of Sciences and Technology Pakistan, Islamabad, Pakistan, 2015
  21. Tian-Xiao, H., Characterizations of Orthogonal Generalized Gegenbauer-Humbert Polynomials and Orthogonal Sheffer-Type Polynomial, Journal of Computational Analysis and Applications, 13 (2011), 4, pp. 1-23
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The generalized Gegenbauer-Humberts wavelet for solving fractional differential equations

© 2024 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, Belgrade, Serbia. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International licence