International Scientific Journal


In this work, radial basis function collocation method (RBFCM) is implemented for generalized time fractional Gardner equation (GTFGE). The RBFCM is meshless and easy-to-implement in complex geometries and higher dimensions, therefore, it is highly demanding. In this work, the Caputo derivative of fractional order ξ ∈ (0, 1] is used to approximate the first order time derivative whereas, Crank-Nicolson scheme is hired to approximate space derivatives. The numerical solutions are presented and discussed, which demonstrate that the method is effective and accurate.
PAPER REVISED: 2022-03-04
PAPER ACCEPTED: 2022-03-14
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2023, VOLUME 27, ISSUE Special issue 1, PAGES [121 - 128]
  1. El-Ajou, A., et al., New Results on Fractional Power Series: Theories and Applications, Entropy, 15 (2013), 12, pp. 5305-5323
  2. Abu Arqub, O., et al., Multiple Solutions of Non-Linear Boundary Value Problems of Fractional Order: A New Analytic Iterative Technique, Entropy, 16 (2014), 1, pp. 471-493
  3. Beyer, H., Kempfle, S., Definition of Physical Consistent Damping Laws with Fractional Derivatives, Z. Angew. Math. Mech., 75 (1995), 8, pp. 623-635
  4. He, J., Approximate Analytic Solution for Seepage Flow with Fractional Derivatives in Porous Media, Comput. Methods Appl. Mech. Eng., 167 (1998), 1-2, pp. 57-68
  5. He, J., Some Applications of Non-Linear Fractional Differential Equations and Their Approximations, Bull. Sci. Technol., 15 (1999), pp. 86-90
  6. Caputo, M., Linear Models of Dissipation whose Q is Almost Frequency Independent - Part II: Geophys. J. Int., 13 (1967), 5, pp. 529-539
  7. Machado, J., Entropy Analysis of Integer and Fractional Dynamical Systems, Non-Linear Dyn., 62 (2010), May, pp. 371-378
  8. Cifani, S., Jakobsen, E. R., Entropy Solution Theory for Fractional Degenerate Convection-Diffusion Equations, Ann. Inst. Henri Poincare, Anal. Non-Linear, 28 (2011), 3, pp. 413-441
  9. Mathieu, B., et al., Fractional Differentiation for Edge Detection, Signal Process, 83 (2003), 11, pp. 2421-2432
  10. Al-Smadi, M., et al., Numerical Multistep Approach for Solving Fractional Partial Differential Equations, Int. J. Comput. Meth., 14 (2017), 3, 1750029
  11. Miller, K. S., Ross, B., An Introduction the Fractional Calculus and Fractional Differential Equations, John Willey and Sons, New York, USA, 1993
  12. Podlubny, I., Fractional Differential Equations, Academic Press, New York, USA, 1999
  13. Guo, S., et al., Time-Fractional Gardner Equation for Ion-Acoustic Waves in Negative-Ion-Beam Plasma with Negative Ions and Non-Thermal Non-Extensive Electrons, Phys. Plasmas, 22 (2015), 052306
  14. Kansa, E., Multiquadrics-A Scattered Data Approximation Scheme with Applications to Computational Fluid Dynamics-I, Surface Approximations and Partial Derivative Estimates, Comput. Math. Appl., 19 (1990), 8-9, pp. 127-145
  15. Zerroukat, M., A Numerical Method for Heat Transfer Problems Using Collocation and Radial Basis Functions, Int. J. Numer. Meth. Eng., 42 (1998), 7, pp. 1263-1278
  16. Zerroukat, M., Explicit and Implicit Meshless Methods for Linear Advection-Diffusion-Type Partial Differential Equations, Int. J. Numer. Meth. Eng., 48 (2000), 1, pp. 19-35
  17. Shakeel, M., et al., Meshless Technique for the Solution of Time-Fractional Partial Differential Equations Having Real-World Applications, Journal Funct. Spaces., 2020 (2020), ID8898309
  18. Ahmad, I., et al., Local Meshless Differential Quadrature Collocation Method for Time-Fractional PDE, Discrete and Continuous Dynamical Systems-S, 13 (2020), 2641
  19. Shakeel, M., et al., Numerical Solution and Characteristic Study of Time-Fractional Shocks Collision, Phys. Scr., 96 (2021), 045214
  20. Liu, F., et al., Time Fractional Advection Dispersion Equation, Appl. Math. Comput., 13 (2003), Sept., pp. 233-245
  21. Micchelli, C., Interpolation of Scatterted Data: Distance Matrix and Conditionally Positive Definite Functions, Construct. Approx., 2 (1986), Dec., pp. 11-22

© 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