THERMAL SCIENCE

International Scientific Journal

NUMERICAL COMPUTATION OF THE TIME NON-LINEAR FRACTIONAL GENERALIZED EQUAL WIDTH MODEL ARISING IN SHALLOW WATER CHANNEL

ABSTRACT
The generalized equal width model is an important non-linear dispersive wave model which is naturally used to describe physical situations in a water channel. In this work, we implement the idea of the interpolation by radial basis function to obtain numerical solution of the non-linear time fractional generalized equal width model defined by Caputo sense. In this technique, firstly, a time discretization is accomplished via the finite difference approach and the non-linear term is linearized by a linearization method. Afterwards, with the help of the radial basis function approximation method is used to discretize the spatial derivative terms. The stability of the method is theoretically discussed using the von Neumann (Fourier series) method. Numerical results and comparisons are presented which illustrate the validity and accuracy of our proposed concepts.
KEYWORDS
PAPER SUBMITTED: 2020-04-14
PAPER REVISED: 2020-05-20
PAPER ACCEPTED: 2020-05-26
PUBLISHED ONLINE: 2020-10-25
DOI REFERENCE: https://doi.org/10.2298/TSCI20S1049C
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2020, VOLUME 24, ISSUE Supplement 1, PAGES [S49 - S58]
REFERENCES
  1. Oldham, K. B., Spanier, J., The Fractional Calculus, Mathematics in Science and Engineering, Elsevier, Amsterdam, The Netherlands, Vol. 111, 1974
  2. Kaya, D., A Numerical Simulation of Solitary-Wave Solutions of the Generalized Regularized Long-Wave Equation, Applied Mathematics and Computation, 149 (2004), 3, pp. 833-841
  3. Kaya, D., El-Sayed, S. M., An Application of the Decomposition Method for the Generalized KdV and RLW Equations, Chaos, Solitons and Fractals, 17 (2003), 5, pp. 869-877
  4. Peregrine, D., Calculations of the Development of an Undular Bore, Journal of Fluid Mechanics, 25 (1966), 2, pp. 321-330
  5. Peregrine, D. H., Long Waves on a Beach, Journal of Fluid Mechanics, 27 (1967), 4, pp. 815-827
  6. Benjamin,T. B., et al., Model Equations for Long Waves in Non-Linear Dispersive Systems, Philosophical Transactions of the Royal Society of London A, 272 (1972), 1220, pp. 47-78
  7. Courtenay Lewis, J., Tjon, J., Resonant Production of Solitons in the RLW Equation, Physics Letters A, 73 (1979), 4, pp. 275-279
  8. Zaki, S., A Least-Squares Finite Element Scheme for the EW Equation, Computer Methods in Applied Mechanics and Engineering, 189 (2000), 2, pp. 587-594
  9. Gardner, L., Gardner, G., Solitary Waves of the Equal Width Wave Equation, Journal of Computational Physics, 101 (1992), 1, pp. 218-223
  10. Inc, M., et al., Modified Variational Iteration Method for Straight Fins with Temperature Dependent Thermal Conductivity, Thermal Science, 22 (2018), Suppl. 1, pp. S229-S236
  11. Akgul, A., et al., New Method for Investigating the Density-Dependent Diffusion Nagumo Equation, Thermal Science, 22 (2018), Suppl. 1, pp. S143-S152
  12. Moallem, G. R., et al., A Numerical Scheme to Solve Variable Order Diffusion-Wave Equations, Thermal Science, 23 (2019), Suppl. 6, pp. S2063-S2071
  13. Ganji, R., Jafari, H., Numerical Solution of Variable Order Integro-Differential Equations, Advanced Mathematical Models and Applications, 4 (2019), 1, pp. 64-69
  14. Aziz, R., Kumawat, Y., Marichev-Saigo-Maeda Fractional Calculus Operators with Extended Mittag-Leffler Function and Generalized Galue Type Struve Function, Advanced Mathematical Models and Applications, 4 (2019), 3, pp. 210-223
  15. Jafari, H., Tajadodi, H., New Method for Solving a Class of Fractional Partial Differential Equations with Applications, Thermal Science, 22 (2018), Suppl. 1, pp. S277-S286
  16. Khan, H., et al., Existence and Data Dependence Theorems for Solutions of an ABC-Fractional Order Impulsive System, Chaos, Solitons and Fractals, 131 (2020), 109477
  17. Khan, A., et al., Analytical Solutions of Time-Fractional Wave Equation by Double Laplace Transform Method, The European Physical Journal Plus, 134 (2019), 4, 163
  18. Khan, H., et al., A Singular ABC-Fractional Differential Equation with p-Laplacian Operator, Chaos, Solitons and Fractals, 129 (2019), Dec., pp. 56-61
  19. Khan, H., et al., A Fractional Order HIV-TB Coinfection Model with Non-Singular Mittag-Leffler Law, Mathematical Methods in the Applied Sciences, 43 (2020), 6, pp. 3786-3806
  20. Golbabai, A., et al., Numerical Investigation of the Time Fractional Mobile-Immobile Advection-Dispersion Model Arising from Solute Transport in Porous Media, International Journal of Applied and Computational Mathematics, 5 (2019), 3, 50
  21. Rashidinia, J., Rasoulizadeh, M. N., Numerical Methods Based on Radial Basis Function-Generated Finite Difference (RBF-FD) for Solution of GKdVB Equation, Wave Motion, 90 (2019), Aug., pp. 152-167
  22. Fasshauer, G. E., Meshfree Approximation Methods with MATLAB: (With CD-ROM), World Scientific Publishing Company, Singapore, Vol. 6, 2007
  23. Tolstykh, A., Shirobokov, D., On Using Radial Basis Functions in A Finite Difference Mode with Applications to Elasticity Problems, Computational Mechanics, 33 (2003), 1, pp. 68-79
  24. Merdan, M., et al., Numerical Solution of Time-Fraction Modified Equal Width Wave Equation, Engineering Computations, 29 (2012), 7, pp. 766-777

© 2022 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