# 变空间分数阶扩散方程微分阶数的数值反演Numerical Inversion for the Fractional Order in the Variable-Order Space-Fractional Diffusion Equation

• 全文下载: PDF(1091KB)    PP.589-598   DOI: 10.12677/AAM.2017.64069
• 下载量: 434  浏览量: 535   科研立项经费支持

We consider a variable-order fractional advection-diffusion equation. Explicit approximations for the equation are proposed. Stability and convergence of the methods are discussed. Moreover, the homotopy regularization algorithm is applied to solve the inverse problem, and numerical examples are presented.

 [1] Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley, New York. [2] Podlubny, I. (1999) Fractional Differential Equations. Academic, San Diego. [3] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam. [4] 陈文, 孙洪广, 李西成, 等. 力学与工程问题的分数阶导数建模[M]. 北京: 科学出版社, 2010. [5] 郭柏灵, 蒲学科, 黄凤辉. 分数阶偏微分方程及其数值解[M]. 北京: 科学出版社, 2011. [6] Coimbra, C.F.M. (2003) Mechanica with Variable-Order Differential Operators. Annalen der Physik, 12, 692-703. [7] Copper, G.R.J. and Rown, D.R.C. (2004) Filtering Using Variable Order Vertical Derivatives. Computers and Geosciences, 30, 455-459. https://doi.org/10.1016/j.cageo.2004.03.001 [8] Tseng, C.C. (2006) Design of Variable and Adaptive Fractional Order FIR Differentiators. Signal Processing, 86, 2554-2566. https://doi.org/10.1016/j.sigpro.2006.02.004 [9] Chen, C.M., Liu, F. and Anh, V. (2010) Numerical Schemes with High Spatial Accuracy for a Variable-Order Anomalous Subdiffusion Equation. SIAM Journal on Scientific Computing, 32, 1740-1760. https://doi.org/10.1137/090771715 [10] Chen, C.M., Liu, F. and Anh, V. (2011) Turner I Numerical Simulation for the Variable-Order Galilei Invariant Advection Diffusion Equation with a Nonlinear Source Term. Applied Mathematics and Computation, 217, 5729-5742. https://doi.org/10.1016/j.amc.2010.12.049 [11] Chen, C.M., Liu, F., Turner, I., Anh, V. and Chen, Y. (2013) Numerical Approximation for a Variable-Order Nonlinear Reaction-Subdiffusion Equation. Numerical Algorithms, 63, 265-290. https://doi.org/10.1007/s11075-012-9622-6 [12] Zhuang, P., Liu, F., Anh, V. and Turner, I. (2009) Numerical Methods for the Variable-Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term. SIAM Journal on Numerical Analysis, 47, 1760-1781. [13] 马维元, 张海东, 邵亚斌. 非线性变阶分数阶扩散方程的全隐差分格式[J]. 山东大学学报(自然科学版), 2013, 48(2): 93-97. [14] 马亮亮, 刘冬兵. Coimbra 变时间分数阶扩散方程–波动方程的新隐式差分法[J]. 西南师范大学学报(自然科学版), 2015, 40(3): 25-31. [15] 刘迪, 孙春龙, 李功胜, 贾现正. 变分数阶扩散方程微分阶数的数值反演[J]. 应用数学进展, 2015, 4(4): 326-335. [16] 贾现正, 张大利, 李功胜, 等. 等空间-时间分数阶变系数对流扩散方程微分阶数的数值反演[J]. 计算数学, 2014, 36(2): 113-132.