THERMAL SCIENCE

International Scientific Journal

External Links

A NEW NUMERICAL TECHNIQUE FOR SOLVING FRACTIONAL SUB-DIFFUSION AND REACTION SUB-DIFFUSION EQUATIONS WITH A NON-LINEAR SOURCE TERM

ABSTRACT
In this paper, we are concerned with the fractional sub-diffusion equation with a non-linear source term. The Legendre spectral collocation method is introduced together with the operational matrix of fractional derivatives (described in the Caputo sense) to solve the fractional sub-diffusion equation with a non-linear source term. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. In addition, the Legendre spectral collocation methods applied also for a solution of the fractional reaction sub-diffusion equation with a non-linear source term. For confirming the validity and accuracy of the numerical scheme proposed, two numerical examples with their approximate solutions are presented with comparisons between our numerical results and those obtained by other methods.
KEYWORDS
PAPER SUBMITTED: 2014-11-26
PAPER REVISED: 2015-01-10
PAPER ACCEPTED: 2015-02-15
PUBLISHED ONLINE: 2015-08-02
DOI REFERENCE: https://doi.org/10.2298/TSCI15S1S25B
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2015, VOLUME 19, ISSUE Supplement 1, PAGES [S25 - S34]
REFERENCES
  1. Gutierrez, R. E., et al., Fractional Order Calculus: Basic Concepts and Engineering Applications, Math. Prob. Engin., 2010 (2010), ID 375858
  2. El-Saka, H. A. A., The Fractional-Order SIR and SIRS Epidemic Models with Variable Population Size, Math. Sci. Lett., 2 (2013), 3, pp. 195-200
  3. Bhrawy, A. H., et al., A Numerical Technique Based on the Shifted Legendre Polynomials for Solving the Time-Fractional Coupled Kdv Equations, Calcolo, 2015, DOI 10.1007/s10092-014-0132-x
  4. Oldham, K. B., Spanier, J., The Fractional Calculus, Academic Press, New York, USA, 1974
  5. Arafa, A. A. M. et al., Solutions of Fractional Order Model of Childhood Diseases with Constant Vaccination Strategy, Math. Sci. Lett., 1 (2012), 1, pp. 17-23
  6. Akinlar, M. A., et al., Numerical Solution of Fractional Benney Equation, Appl. Math. Info. Sci., 8 (2014), 4, pp. 1633-1637
  7. Atangana, A., Cloot, A. H., Stability and Convergence of the Space Fractional Variable-Order Schrodinger Equation, Advances in Difference Equations, 2013, (2013), 1, pp. 1-13
  8. Doha, E. H. et al., On Shifted Jacobi Spectral Approximations for Solving Fractional Differential Equations, Appl. Math. Comput., 219 (2013), 15, pp. 8042-8056
  9. Bhrawy, A. H., et al., New Numerical Approximations for Space-Time Fractional Burgers' Equations via a Legendre Spectral-Collocation Method, Rom. Rep. Phys, 2 (2015), 67, pp. 1-11
  10. Yi, M., Huang, J., Wavelet Operational Matrix Method for Solving Fractional Differential Equations with Variable Coefficients, Appl. Math. Comput., 230 (2014), pp. 383-394
  11. Neamaty, A., et al., Solving Fractional Partial Differential Equation by Using Wavelet Operational Method, J. Math. Comput. Sci., 7 (2013), pp. 230-240
  12. Wang, L. et al., Haar Wavelet Method for Solving Fractional Partial Differential Equations Numerically, Appl. Math. Comput., 227 (2014), pp. 66-76
  13. Wang Y., et al., Using Reproducing Kernel for Solving a Class of Fractional Partial Differential Equation with Non-Classical Conditions, Appl. Math. Comput., 219 (2013), pp. 5918-5925
  14. Pedas, A., Tamme, E., Numerical Solution of Non-linear Fractional Differential Equations by Spline Collocation Methods, J. Comput. Appl. Math., 255 (2014), pp. 216-230
  15. Yang,, X. J., Baleanu, D., Fractal Heat Conduction Problem Solved By Local Fractional Variation Iteration Method, Thermal Science, 17 (2013), 2, pp. 625-628
  16. Khalil, H., Khan, R. A., Operational Matrix of Integrations and Coupled System of Fredholm Integral Equations, Chinese Journal of Mathematics, 2014 (2014), ID 146013, doi.org/10.1155/2014/ 146013
  17. Baleanu, D., et al., Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems, Abstract and Applied Analysis, 2013 (2013), ID 546502
  18. Li, X., Wu, B., A Numerical Technique for Variable Fractional Functional Boundary Value Problems, Applied Mathematics Letters, 43 (2015), pp. 108-113
  19. Arqub, Abu, O., et al., Multiple Solutions of Non-linear Boundary Value Problems of Fractional Order: a New Analytic Iterative Technique, Entropy, 2014 (2014), 16, pp. 471-493
  20. Bhrawy, et al., A New Generalized Laguerre-Gauss Collocation Scheme For Numerical Solution Of Generalized Fractional Pantograph Equations, Romanian Journal of Physics, 59 (2014), 7-8, pp. 646-657
  21. Mokhtary, P., Reconstruction of Exponentially Rate of Convergence to Legendre Collocation Solution of a Class of Fractional Integro-Differential Equations, J. Comput. Appl. Math., 279 (2015), pp. 145-158
  22. Parvizi, M., et al., Numerical Solution of Fractional Advection-Diffusion Equation with a Non-linear Source Term, Numerical Algorithms, (2014), doi: 10.1007/s11075-014-9863-7
  23. Canuto, C., et al., Spectral Methods: Fundamentals in Single Domains, Springer-Verlag, New York (2006)
  24. Doha, E. H. et al., Jacobi Spectral Galerkin Method for Elliptic Neumann Problems, Numerical Algorithms, 50 (2009), 1, pp. 67-91
  25. Aminikhah, H., Hosseini, S., Numerical Solution of Linear System of Integro-Differential Equations by Using Chebyshev Wavelet Method, Math. Sci. Lett., 4 (2015), 1, pp. 45-50
  26. Doha, E. H., et al., A Chebyshev-Gauss-Radau Scheme For Non-linear Hyperbolic System of First Order, Appl. Math. Info. Sci., 8 (2014), 2, 535-544
  27. Doha, E. H., et al., Jacobi-Gauss-Lobatto Collocation Method For The Numerical Solution Of 1+ 1 nonlinear Schrödinger Equations, Journal of Computational Physics, 261 (2014), pp. 244-255
  28. Xiao-Yong, Z., Junlin, L., Convergence Analysis of Jacobi Pseudo-Spectral Method for the Volterra Delay Integro-Differential Equations, Appl. Math. Info. Sci., 9 (2015), 1, pp. 135-145
  29. Abdelkawy, M. A., et al., A Method Based on Legendre Pseudo-Spectral Approximations for Solving Inverse Problems of Parabolic Types Equations, Math. Sci. Lett., 4 (2015), 1, pp. 81-90
  30. Bhrawy, A. H., Zaky, M. A., Numerical Simulation for Two-Dimensional Variable-Order Fractional Non-linear Cable Equation, Non-linear Dynamics, (2015) doi:10.1007/s11071-014-1854-7
  31. Bhrawy, A. H., An Efficient Jacobi Pseudospectral Approximation for Non-linear Complex Generalized Zakharov System, Applied Mathematics and Computations, 247 (2014), pp. 30-46
  32. Doha, E. H., et al., Numerical Treatment of Coupled Non-linear Hyperbolic Klein-Gordon Equations, Romanian Journal of Physics, 59 (2014), 3-4, pp. 247-264
  33. Doha, E. H., et al., A New Jacobi Operational Matrix: An Application for Solving Fractional Differential Equations, Appl. Math. Model., 36 (2012), pp. 4931-4943
  34. Saadatmandi, A., Dehghan, M., A New Operational Matrix for Solving Fractional-Order Differential Equations, Comput. Math. Appl., 59 (2010), pp. 1326-1336
  35. Ma, X., Huang, C., Spectral Collocation Method for Linear Fractional Integro-Differential Equations, Appl. Math. Model., 38 (2014), pp. 1434-1448
  36. Eslahchi, M. R., et al. Application of the Collocation Method for Solving Non-linear Fractional Integro- Differential Equations, J. Comput. Appl. Math., 257 (2014), pp. 105-128
  37. Saadatmandi, A., Dehghan, M., A Tau Approach for Solution of the Space Fractional Diffusion Equation, Comput. Math. Appl., 62 (2011), pp. 1135-1142
  38. Bhrawy, A. H., Baleanu, D., A Spectral Legendre-Gauss-Lobatto Collocation Method for a Space- Fractional Advection Diffusion Equations with Variable Coefficients, Reports in Mathematical Physics, 72 (2013), 2, pp. 219-233
  39. Liu, F. et al., Numerical Method and Analytical Technique of the Modified Anomalous Subdiffusion Equation with a Non-linear Source Term, J. Comput. Appl. Math., 231 (2009), pp. 160-176
  40. Mohebbi, A. et al., A High-Order and Unconditionally Stable Scheme for the Modified Anomalous Fractional Sub-Diffusion Equation with a Non-linear Source Term, J. Comput. Phys., 240 (2013), pp. 36-48
  41. Bhrawy, A. H., et al., A Spectral Tau Algorithm Based on Jacobi Operational Matrix for Numerical Solution of time Fractional Diffusion-Wave Equations, Journal of Computational Physics, (2014) doi10.1016/j.jcp.2014. 03.039
  42. Bhrawy, A. H., Zaky, M. A., A Method Based on the Jacobi Tau Approximation for Solving multi-Term Time-Space Fractional Partial Differential Equations, Journal of Computational Physics, 281 (2015), Jan., pp. 876-895
  43. Abbaszade, M., Mohebbi, A., Fourth-Order Numerical Solution of a Fractional PDE with the Non-linear Source Term in the Electroanalytical Chemistry, Iranian Journal of Mathematical Chemistry, 3 (2012), 2, pp. 195-220
  44. Ding, H., Li, C., Mixed Spline Function Method for Reaction-Subdiffusion Equations, J. Comput. Phys., 242 (2013), pp. 103-123

© 2019 Society of Thermal Engineers of Serbia. Published by the Vinča Institute of Nuclear Sciences, 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