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Based on the recently developed local radial basis function method, we devise an implicit local radial basis function scheme, which is intrinsic mesh-free, for solving time fractional diffusion equations. In this paper the L1 scheme and the local radial basis function method are applied for temporal and spatial discretization, respectively, in which the time-marching iteration is performed implicitly. The robustness and accuracy of this proposed implicit local radial basis function method are demonstrated by the numerical example. Furthermore, the sensitivities of the shape parameter c and the number of nodes in the local sub-domain to the accuracy of numerical solutions are also investigated.
PAPER REVISED: 2015-01-21
PAPER ACCEPTED: 2015-02-02
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THERMAL SCIENCE YEAR 2015, VOLUME 19, ISSUE Supplement 1, PAGES [S59 - S67]
  1. Povstenko, Y. Z., Fractional Heat Conduction Equation and Associated Thermal Stress, Journal of Thermal Stresses, 28 (2004), 1, pp. 83-102
  2. Gabano, J. D., Poinot, T., Estimation of Thermal Parameters Using Fractional Modelling, Signal Processing, 91 (2011), 4, pp. 938-948
  3. Caputo, M., Diffusion of Fluids in Porous Media with Memory, Geothermics, 28 (1999), 1, pp. 113-130
  4. Chechkin, A., et al., Fractional Diffusion in Inhomogeneous Media, Journal of Physics A: Mathematical and General, 38 (2005), 42, L679
  5. Mainardi, F., Fractional Relaxation-Oscillation and Fractional Diffusion-Wave Phenomena, Chaos, Solitons & Fractals, 7 (1996), 9, pp. 1461-1477
  6. Yin, C., et al., Fractional-Order Sliding Mode Based Extremum Seeking Control of a Class of Nonlinear Systems, Automatica, 50 (2014), 12, pp. 3173-3181
  7. Li, C., Peng, G., Chaos in Chen's System with a Fractional Order, Chaos, Solitons & Fractals, 22 (2004), 2, pp. 443-450
  8. Baleanu, D., et al., Local Fractional Variational Iteration Algorithms for the Parabolic Fokker-Planck Equation Defined on Cantor Sets, Progr. Fract. Differ. Appl., 1 (2014), 1, pp. 1-10
  9. Yang, X.-J., Baleanu, D., Fractal Heat Conduction Problem Solved by Local Fractional Variation Iteration Method, Thermal Science, 17 (2013), 2, pp. 625-628
  10. Sun, H. G., et al., A Comparative Study of Constant-Order and Variable-Order Fractional Models in Characterizing Memory Property of Systems, The European Physical Journal Special Topics, 193 (2011), 1, pp. 185-192
  11. Bagley, R. L., Torvik, P. J., A Theoretical Basis for the Application of Fractional Calculus to Viscoelasticity, Journal of Rheology, 27 (1983), 3, pp. 201-210
  12. Coimbra, C. F. M., Mechanics with Variable-Order Differential Operators, Annalen der Physik, 12 (2003), 11-12, pp. 692-703
  13. Chen, W., et al., A Variable-Order Time-Fractional Derivative Model for Chloride Ions Sub-Diffusion in Concrete Structures, Fractional Calculus and Applied Analysis, 16 (2013), 1, pp. 76-92
  14. Zhang, Y., et al., Particle Tracking for Time-Fractional Diffusion, Physical Review E, 78 (2008), 3, 036705
  15. Tadjeran, C., Meerschaert, M. M., A Second-Order Accurate Numerical Method for the Two- Dimensional Fractional Diffusion Equation, J. Comput. Phys., 220 (2007), 2, pp. 813-823
  16. Murio, D., Implicit Finite Difference Approximation for Time Fractional Diffusion Equations, Computers & Mathematics with Applications, 56 (2008), 4, pp. 1138-1145
  17. Sun, H., et al., Finite Difference Schemes for Variable-Order Time Fractional Diffusion Equation, International Journal of Bifurcation and Chaos, 22 (2012), 4, pp. 1-16.
  18. Lin, R., et al., Stability and Convergence of a New Explicit Finite-Difference Approximation for the Variable-Order Nonlinear Fractional Diffusion Equation. Applied Mathematics and Computation, 212 (2009), 2, pp. 435-445
  19. Chen, C. S., et al., The Method of Approximate Particular Solutions for Solving Certain Partial Differential Equations, Numerical Methods for Partial Differential Equations, 28 (2012), 2, pp. 506-522
  20. Liu, Q., et al., An Implicit Rbf Meshless Approach for Time Fractional Diffusion Equations, Computational Mechanics, 48 (2011), 1, pp. 1-12
  21. Gu, Y. T., Meshfree Methods and Their Comparisons, International Journal of Computational Methods, 2 (2005), 4, pp. 477-515
  22. Chinchapatnam, P. P., et al., Unsymmetric and Symmetric Meshless Schemes for the Unsteady Convection- Diffusion Equation, Computer Methods in Applied Mechanics and Engineering, 195 (2006), 19-22, pp. 2432-2453
  23. Fedoseyev, A. I., et al., Improved Multiquadric Method for Elliptic Partial Differential Equations Via Pde Collocation on the Boundary, Computers & Mathematics with Applications, 43 (2002), 3-5, pp. 439- 455
  24. Uddin, M., On the Selection of a Good Value of Shape Parameter in Solving Time-Dependent Partial Differential Equations Using Rbf Approximation Method, Applied Mathematical Modelling, 38 (2014), 1, pp. 135-144
  25. Rippa, S., An Algorithm for Selecting a Good Value for the Parameter C in Radial Basis Function Interpolation, Advances in Computational Mathematics, 11 (1999), 2-3, pp. 193-210
  26. Huang, C. S., et al., On the Increasingly Flat Radial Basis Function and Optimal Shape Parameter for the Solution of Elliptic Pdes, Engineering Analysis with Boundary Elements, 34 (2010), 9, pp. 802-809
  27. Kansa, E. J., Hon, Y. C., Circumventing the Ill-Conditioning Problem with Multiquadric Radial Basis Functions: Applications to Elliptic Partial Differential Equations, Computers & Mathematics with Applications, 39 (2000), 7-8, pp. 123-137
  28. Luh, L.-T., The Mystery of the Shape Parameter IV, Engineering Analysis with Boundary Elements, 48 (2014), Nov., pp. 24-31
  29. Yao, G., et al., Assessment of Global and Local Meshless Methods Based on Collocation with Radial Basis Functions for Parabolic Partial Differential Equations in Three Dimensions, Engineering Analysis with Boundary Elements, 36 (2012), 11, pp. 1640-1648
  30. Sarler, B., Vertnik, R., Meshfree Explicit Local Radial Basis Function Collocation Method for Diffusion Problems, Computers & Mathematics with Applications, 51 (2006), 8, pp. 1269-1282
  31. Hon, Y. C., et al., Local Radial Basis Function Collocation Method for Solving Thermo-Driven Fluid- -Flow Problems with Free Surface, Engineering Analysis with Boundary Elements, (2015), in press
  32. Divo, E., Kassab, A. J., An Efficient Localized Radial Basis Function Meshless Method for Fluid Flow and Conjugate Heat Transfer, Journal of Heat Transfer, 129 (2007), 2, pp. 124-136
  33. Kovacevic, I., Sarler, B., Solution of a Phase-Field Model for Dissolution of Primary Particles in Binary Aluminum Alloys by an R-Adaptive Mesh-Free Method, Materials Science and Engineering: A, 413 (2005), Dec., pp. 423-428
  34. Lee, C. K., et al., Local Multiquadric Approximation for Solving Boundary Value Problems, Computational Mechanics, 30 (2003), 5-6, pp. 396-409
  35. Yao, G., et al., A Localized Approach for the Method of Approximate Particular Solutions, Computers & Mathematics with Applications, 61 (2011), 9, pp. 2376-2387
  36. Hon, Y. C., et al., An Adaptive Greedy Algorithm for Solving Large Rbf Collocation Problems, Numerical Algorithms, 32 (2003), 1, pp. 13-25
  37. Sarler, B., From Global to Local Radial Basis Function Collocation Method for Transport Phenomena, in Advances in Meshfree Techniques, (Eds.: V. M. A. Leitao, C. J. S. Alves, C. Armando Duarte), 2007, Springer, The Netherlands, pp. 257-282
  38. Gao, G.-H., et al., A New Fractional Numerical Differentiation Formula to Approximate the Caputo Fractional Derivative and its Applications, Journal of Computational Physics, 259 (2014), Feb., pp. 33-50
  39. Vertnik, R., Sarler, B., Meshless Local Radial Basis Function Collocation Method for Convective- Diffusive Solid-Liquid Phase Change Problems, International Journal of Numerical Methods for Heat and Fluid Flow, 16 (2006), 5, pp. 617-640
  40. Sarler, B., A Radial Basis Function Collocation Approach in Computational Fluid Dynamics, Computer Modelling in Engineering & Sciences, 7 (2005), 2, pp. 185-193

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