Iterative reproducing kernel method for nonlinear variable-order space fractional diffusion equations

References

• S. Abbasbandy, B. Azarnavid, and M.S. Alhuthali, A shooting reproducing kernel Hilbert space method for multiple solutions of nonlinear boundary value problems, J. Comput. Appl. Math. 279 (2015), pp. 293–305. doi: 10.1016/j.cam.2014.11.014
   Web of Science ®Google Scholar
• O. Abu Arqub, M. AL-Smadi, S. Momani, and T. Hayat, Numerical solutions of fuzzy differential equations using reproducing kernel Hilbert space method, Soft Comput. (2015). DOI:10.1007/s00500-0151707-4.
   PubMed Web of Science ®Google Scholar
• N. Aronszajn, Theory of reproducing kernel, Trans. Amer. Math. Soc. 168 (1950), pp. 1–50.
   Google Scholar
• C. Coimbra, Mechanics with variable-order differential operators, Ann. Phys. 12 (2003), pp. 692–703. doi: 10.1002/andp.200310032
   Web of Science ®Google Scholar
• M.G. Cui and Y.Z. Lin, Nonlinear Numerical Analysis in Reproducing Kernel Space, Nova Science Pub Inc, New York, 2009.
   Google Scholar
• M. Dehghan and M. Abbaszadeh, Analysis of the element free Galerkin (EFG) method for solving fractional cable equation with Dirichlet boundary condition, Appl. Numer. Math. 109 (2016), pp. 208–234. doi: 10.1016/j.apnum.2016.07.002
   Web of Science ®Google Scholar
• M. Dehghan and M. Abbaszadeh, Element free Galerkin approach based on the reproducing kernel particle method for solving 2D fractional Tricomi-type equation with Robin boundary condition, Comput. Math. Appl. 73 (2017), pp. 1270–1285. doi: 10.1016/j.camwa.2016.11.020
   Web of Science ®Google Scholar
• F.Z. Geng and M.G. Cui, Solving a nonlinear system of second order boundary value problems, J. Math. Anal. Appl. 327 (2007), pp. 1167–1181. doi: 10.1016/j.jmaa.2006.05.011
   Web of Science ®Google Scholar
• F.Z. Geng and Z.Q. Tang, Piecewise shooting reproducing kernel method for linear singularly perturbed boundary value problems, Appl. Math. Lett. 62 (2016), pp. 1–8. doi: 10.1016/j.aml.2016.06.009
   Web of Science ®Google Scholar
• F.Z. Geng and Z.Q. Tang, A new numerical method for singularly perturbed turning point problems with two boundary layers based on reproducing kernel method, Calcolo 54 (2017), pp. 515–526. doi: 10.1007/s10092-016-0196-x
   Web of Science ®Google Scholar
• F.Z. Geng, S.P. Qian, and S. Li, A numerical method for singularly perturbed turning point problems with an interior layer, J. Comput. Appl. Math. 255 (2013), pp. 97–105. doi: 10.1016/j.cam.2013.04.040
   Web of Science ®Google Scholar
• R. Ketabchi, R. Mokhtari, and E. Babolian, Some error estimates for solving Volterra integral equations by using the reproducing kernel method, J. Comput. Appl. Math. 273 (2015), pp. 245–250. doi: 10.1016/j.cam.2014.06.016
   Web of Science ®Google Scholar
• C.P. Li and H.F. Ding, Higher order finite difference method for the reaction and anomalous-diffusion equation, Appl. Math. Model. 38 (2014), pp. 3802–3821. doi: 10.1016/j.apm.2013.12.002
   Web of Science ®Google Scholar
• X.Y. Li and B.Y. Wu, Error estimation for the reproducing kernel method to solve linear boundary value problems, J. Comput. Appl. Math. 243 (2013), pp. 10–15. doi: 10.1016/j.cam.2012.11.002
   Web of Science ®Google Scholar
• X.Y. Li and B.Y. Wu, A numerical technique for variable fractional functional boundary value problems, Appl. Math. Lett. 43 (2015), pp. 108–113. doi: 10.1016/j.aml.2014.12.012
   Web of Science ®Google Scholar
• X.Y. Li and B.Y. Wu, A new numerical method for variable order fractional functional differential equations, Appl. Math. Lett. 68 (2017), pp. 80–86. doi: 10.1016/j.aml.2017.01.001
   Web of Science ®Google Scholar
• X.Y. Li and B.Y. Wu, A new reproducing kernel method for variable order fractional boundary value problems for functional differential equations, J. Comput. Appl. Math. 311 (2017), pp. 387–393. doi: 10.1016/j.cam.2016.08.010
   Web of Science ®Google Scholar
• R. Lin, F. Liu, V. Anh, and I. Turner, Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractonal diffusion equation, Appl. Math. Comput. 212 (2009), pp. 435–445.
   Web of Science ®Google Scholar
• C.W. Lv and C.J. Xu, Error analysis of a high order method for time-fractional diffusion equations, SIAM J. Sci. Comput. 38 (2016), pp. A2699–A2724. doi: 10.1137/15M102664X
   Web of Science ®Google Scholar
• R.F. Ren, H.B. Li, W. Jiang, and M.Y. Song, An efficient Chebyshev–Tau method for solving the space fractional diffusion equations, Appl. Math. Comput. 224 (2013), pp. 259–267.
   Web of Science ®Google Scholar
• R. Salehi and M. Dehghan, A generalized moving least square reproducing kernel method, J. Comput. Appl. Math. 249 (2013), pp. 120–132. doi: 10.1016/j.cam.2013.02.005
   Web of Science ®Google Scholar
• S. Samko, Fractional integration and differentiation of variable order, Anal. Math. 21 (1995), pp. 213–236. doi: 10.1007/BF01911126
   Google Scholar
• S. Samko, Fractional integration, differentiation of variable order: An overview, Nonlinear Dyn. 71 (2013), pp. 653–662. doi: 10.1007/s11071-012-0485-0
   Web of Science ®Google Scholar
• S. Shen, F. Liu, J. Chen, I. Turner, and V. Anh, Numerical techniques for the variable order time fractional diffusion equation, Appl. Math. Comput. 218 (2012), pp. 10861–10870.
   Web of Science ®Google Scholar
• F.Y. Song and C.J. Xu, Spectral direction splitting methods for two-dimensional space fractional diffusion equations, J. Comput. Phys. 299 (2015), pp. 196–214. doi: 10.1016/j.jcp.2015.07.011
   Web of Science ®Google Scholar
• H.G. Sun, W. Chen, and C.P. Li, Finite difference schemes for variable-order time fractional diffusion equation, Int. J. Bifurcat. Chaos 22 (2012), pp. 1–16.
   Web of Science ®Google Scholar
• C. Yu, K.L. Teo, and H.H. Dam, Design of allpass variable fractional delay filter with signed powers-of-two coefficients, Signal Process. 95 (2014), pp. 32–42. doi: 10.1016/j.sigpro.2013.08.005
   Web of Science ®Google Scholar