On solutions of variable-order fractional differential equations
Numerical calculation of the fractional integrals and derivatives is the code to search fractional calculus and solve fractional differential equations. The exact solutions to fractional differential equations are compelling to get in real applications, due to the nonlocality and complexity of the fractional differential operators, especially for variable-order fractional differential equations. Therefore, it is significant to enhance numerical methods for fractional differential equations. In this work, we consider variable-order fractional differential equations by reproducing kernel method. There has been much attention in the use of reproducing kernels for the solutions to many problems in the recent years. We give an example to demonstrate how efficiently our theory can be implemented in practice.
[1] Coimbra, C. F. M. . Mechanics with variable-order differential operators. Ann. Phys., 12(11-12):692–703, 2003.
[2] Lin, R. , Liu, F. , Anh, V. , and Turner, I. . Stability and convergence of a new explicit finite-difference approximation for the variable-order nonlinear fractional diffusion equation. Appl. Math. Comput., 212(2):435– 445, 2009.
[3] Zhuang, P. , Liu, F. , Anh, V. , and Turner, I. . Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term. SIAM J. Numer. Anal., 47(3):1760–1781, 2009.
[4] Sun, H. , Chen, W. , and Chen, Y. Q. . Variable order fractional differential operators in anomalous diffusion modeling. Physica A: Statistical Mechanics and its Applications, 388(388):4586–4592, 2009.
[5] Sun, H. , Chen, W. , Li, C. , and Chen, Y. Q. . Finite difference schemes for variable-order time fractional diffusion equation. International Journal of Bifurcation and Chaos, 22(4)(22(4)):1250085 (16 pages), 2012.
[6] Sun, H. , Chen, W. , Wei, H. , and Chen, Y. . A comparative study of constant-order and variableorder fractional models in characterizing memory property of systems. Eur. Phys. J. Special Topics, 193(193):185–192, 2011.
[7] Aronszajn, N. . Theory of reproducing kernels. Trans. Amer. Math. Soc., 68:337–404, 1950.
[8] Geng, F. and Cui, M. . A reproducing kernel method for solving nonlocal fractional boundary value problems. Appl. Math. Lett., 25(5):818–823, 2012.
[9] Wang, Y. , Su, L. , Cao, X. , and Li, X. . Using reproducing kernel for solving a class of singularly perturbed problems. Comput. Math. Appl., 61(2):421– 430, 2011.
[10] Wu, B. Y. and Li, X. Y. . A new algorithm for a class of linear nonlocal boundary value problems based on the reproducing kernel method. Appl. Math. Lett., 24(2):156–159, 2011.
[11] Yao, H. and Lin, Y. . Solving singular boundaryvalue problems of higher even-order. J. Comput. Appl. Math., 223(2):703–713, 2009.
[12] Akg¨ul, A. . A new method for approximate solutions of fractional order boundary value problems. Neural Parallel Sci. Comput., 22(1-2):223–237, 2014.
[13] Akg¨ul, A. . New reproducing kernel functions. Math. Probl. Eng., pages Art. ID 158134, 10, 2015.
[14] Akg¨ul, A. , Inc, M. , Karatas, E. , and Baleanu, D. . Numerical solutions of fractional differential equations of Lane-Emden type by an accurate technique. Adv. Difference Equ., page 2015:220, 2015.
[15] Akg¨ul, A. and Kili¸cman, A. . Solving delay differential equations by an accurate method with interpolation. Abstr. Appl. Anal., pages Art. ID 676939, 7, 2015.
[16] Inc, M. and Akg¨ul, A. . Approximate solutions for MHD squeezing fluid flow by a novel method. Bound. Value Probl., pages 2014:18, 17, 2014
[17] Inc, M. and Akg¨ul, A. . Numerical solution of seventhorder boundary value problems by a novel method. Abstr. Appl. Anal., pages Art. ID 745287, 9, 2014.
[18] Inc, M. , Akg¨ul, A. , and Kili¸cman, A. . Explicit solution of telegraph equation based on reproducing kernel method. J. Funct. Spaces Appl., pages Art. ID 984682, 23, 2012.
[19] Inc, M. , Akg¨ul, A. , and Kılı¸cman, A. . A new application of the reproducing kernel Hilbert space method to solve MHD Jeffery-Hamel flows problem in nonparallel walls. Abstr. Appl. Anal., pages Art. ID 239454, 12, 2013.
[20] Inc, M. , Akg¨ul, A. , and Kili¸cman, A. . A novel method for solving KdV equation based on reproducing kernel Hilbert space method. Abstr. Appl. Anal., pages Art. ID 578942, 11, 2013.
[21] Inc, M. , Akg¨ul, A. , and Kılı¸cman, A. . Numerical solutions of the second-order one-dimensional telegraph equation based on reproducing kernel Hilbert space method. Abstr. Appl. Anal., pages Art. ID 768963, 13, 2013.
[22] Wang, J. , Liu, L. , Liu, L. , and Chen, Y. . Numerical solution for the variable order fractional partial differential equation with Bernstein polynomials. International Journal of Advancements in Computing Technology(IJACT), pages Volume 6, Number 3, May 2014.
[23] Cui, M. and Lin, Y. . Nonlinear numerical analysis in the reproducing kernel space. Nova Science Publishers Inc., New York, 2009.
[24] Cao, J. and Qiu, Y. . A high order numerical scheme for variable order fractional ordinary differential equation. Appl. Math. Lett., 61:88–94, 2016.