两边空间分数阶对流-扩散方程的一种加权显式有限差分方法
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摘要
考虑两边空间分数阶对流-扩散方程的初边值问题,基于Grünwald公式和移位Grünwald-Letnikov公式,提出一种加权显式有限差分解法.利用傅里叶变换和特征值法,得到差分格式的稳定性.然后使用最大模估计法证明在相同的条件下,所提出的差分格式是收敛的.最后通过数值例子说明所提出的差分格式是可靠和有效的,并对方程的数值解与精确解进行比较,验证了文中的理论结果.
Based on Grünwald formula and the shifted Grünwald-Letnikov formula,a weighted explicit finite difference method is proposed to solve initial boundary value problems of two-sided space fractional advection diffusion equation. Their stability is analyzed by means of Fourier transform and eigenvalue analysis. Using the technique of maximum norm analysis,it is proved that the scheme convergent under the same condition. Illustrative example is included to demonstrate the validity and applicability of the scheme,and a comparison between the exact analytical and the numerical prediction is made to demonstrate the theoretical results.
Based on Grünwald formula and the shifted Grünwald-Letnikov formula,a weighted explicit finite difference method is proposed to solve initial boundary value problems of two-sided space fractional advection diffusion equation. Their stability is analyzed by means of Fourier transform and eigenvalue analysis. Using the technique of maximum norm analysis,it is proved that the scheme convergent under the same condition. Illustrative example is included to demonstrate the validity and applicability of the scheme,and a comparison between the exact analytical and the numerical prediction is made to demonstrate the theoretical results.
引文
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[19]马亮亮,田富鹏.变系数空间分数阶对流-扩散方程的隐式差分近似[J].中北大学学报(自然科学版),2014,35(1):11-14.
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[21]马亮亮.变系数阶空间分数阶对流-扩散方程的有限差分解法[J].沈阳大学学报(自然科学版),2013,25(4):341-344.
[22]马亮亮.变时间分数阶非定常对流扩散方程的数值分析[J].辽东学院学报(自然科学版),2013,20(3):220-223.
[23]马亮亮,刘冬兵.一类变时间分数阶含源项非定常奇异摄动对流扩散方程的数值分析[J].沈阳大学学报(自然科学版),2013,25(5):424-427.
[24]苏丽娟,王文洽.双边空间分数阶对流-扩散方程的一种有限差分解法[J].山东大学学报(理学版),2009,44(10):26-29.
[25]MEERSCHAERT M,TADJERAN C.Finite difference approximations for fractional advection-dispersion flow equations[J].J Comput Appl Math,2004,172(1):65-77.
[26]MEERSCHAET M,TADJERAN C.Finite difference approximations for two-sided space fractional partial differential equations[J].Appl Num Math,2006,56(1):80-90.
[27]夏源,吴吉春.分数阶对流-弥散方程的数值求解[J].南京大学学报(自然科学版),2007,43(4):441-446.
[28]LIU F,ANH V,TURNER I.Numerical solution of the space fractional Fokker-Plank equation[J].J Comput Appl Math,2004,166(1):209-219.
[29]马维元,刘华.两边空间-时间分数阶扩散方程的加权有限差分格式[J].华东师范大学学报(自然科学版),2012(3):41-48.
[30]张红玉,崔明荣.两类分数阶对流-扩散方程的有限差分方法[J].山东大学学报(理学版),2012,47(6):40-48.
[31]汪向艳,朱琳,芮洪兴.空间分数阶扩散方程的一种加权显式差分方法[J].宁夏大学学报(自然科学版),2014,35(1):1-5.
[32]徐仲,张凯院,陆全,等.矩阵简明教程[M].北京:科学出版社,2001.
[2]HILFER R.Applications of Fractional Calculus in Physics[M].Singapore:Word Scientific,2000.
[3]MORTON K W,MARYERS D F.Numerical Solution of Partial Differential Equation[M].Cambridge:Cambridge University Press,2005.
[4]LUCHKO Y.Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation[J].J Math Anal Appl,2011,374(2):538-548.
[5]马亮亮,刘冬兵.二维变系数空间分数阶电报方程数值解[J].辽宁工程技术大学学报(自然科学版),2014,33(3):429-432.
[6]马亮亮,刘冬兵.一类维空间Riesz分数阶扩散方程的解析解[J].合肥工业大学学报(自然科学版),2014,37(4):506-509.
[7]马亮亮.时间分数阶扩散方程的数值解法[J].数学的实践与认识,2013,43(10):248-253.
[8]马亮亮.一种Caputo分数阶反应-扩散方程初边值问题的隐式差分格式[J].贵州师范大学学报(自然科学版),2013,31(1):58-61.
[9]马亮亮,田富鹏.空间分数阶Edwards-Wilkinson方程的显式差分近似[J].沈阳大学学报(自然科学版),2013,25(3):250-252.
[10]马亮亮,刘冬兵.一类反常次扩散方程Neumann问题的有限差分格式收敛性分析[J].五邑大学学报(自然科学版),2014,28(1):1-4.
[11]马亮亮,刘冬兵.变系数分数阶反应-扩散方程的数值解法[J].沈阳大学学报(自然科学版),2014,26(1):76-80.
[12]马亮亮.时间分数阶扩散方程的隐式差分近似[J].贵州师范大学学报(自然科学版),2014,32(2):79-82.
[13]刘冬兵,马亮亮.变时间分数阶反应扩散方程的数值分析[J].江南大学学报(自然科学版),2014,13(1):109-112.
[14]马亮亮,刘冬兵.高维非齐次时间分数阶电报方程的基本解[J].四川师范大学学报(自然科学版),2015,38(1):77-83.
[15]LIU F,ZHUANG P,ANH V,et al.Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation[J].Appl Math Comput,2007,191(1):2-20.
[16]JIANG H,LIU F,TURNER I,et al.Analytical solutions for the multi-term time-space Caputo-Riesz fractional advectiondiffusion equations on a finite domain[J].J Math Anal Appl,2012,389(2):1117-1127.
[17]CARELLA A R,DORAO C A.Least-squares spectral method for the solution of a fractional advection-dispersion equation[J].J Comput Phys,2013,232(1):33-45.
[18]马亮亮,刘冬兵.高维分数阶cable方程隐式差分逼近[J].辽宁工程技术大学学报(自然科学版),2014,33(4):544-547.
[19]马亮亮,田富鹏.变系数空间分数阶对流-扩散方程的隐式差分近似[J].中北大学学报(自然科学版),2014,35(1):11-14.
[20]马亮亮.一种时间分数阶对流扩散方程的隐式差分近似[J].西北民族大学学报(自然科学版),2013,34(1):7-12.
[21]马亮亮.变系数阶空间分数阶对流-扩散方程的有限差分解法[J].沈阳大学学报(自然科学版),2013,25(4):341-344.
[22]马亮亮.变时间分数阶非定常对流扩散方程的数值分析[J].辽东学院学报(自然科学版),2013,20(3):220-223.
[23]马亮亮,刘冬兵.一类变时间分数阶含源项非定常奇异摄动对流扩散方程的数值分析[J].沈阳大学学报(自然科学版),2013,25(5):424-427.
[24]苏丽娟,王文洽.双边空间分数阶对流-扩散方程的一种有限差分解法[J].山东大学学报(理学版),2009,44(10):26-29.
[25]MEERSCHAERT M,TADJERAN C.Finite difference approximations for fractional advection-dispersion flow equations[J].J Comput Appl Math,2004,172(1):65-77.
[26]MEERSCHAET M,TADJERAN C.Finite difference approximations for two-sided space fractional partial differential equations[J].Appl Num Math,2006,56(1):80-90.
[27]夏源,吴吉春.分数阶对流-弥散方程的数值求解[J].南京大学学报(自然科学版),2007,43(4):441-446.
[28]LIU F,ANH V,TURNER I.Numerical solution of the space fractional Fokker-Plank equation[J].J Comput Appl Math,2004,166(1):209-219.
[29]马维元,刘华.两边空间-时间分数阶扩散方程的加权有限差分格式[J].华东师范大学学报(自然科学版),2012(3):41-48.
[30]张红玉,崔明荣.两类分数阶对流-扩散方程的有限差分方法[J].山东大学学报(理学版),2012,47(6):40-48.
[31]汪向艳,朱琳,芮洪兴.空间分数阶扩散方程的一种加权显式差分方法[J].宁夏大学学报(自然科学版),2014,35(1):1-5.
[32]徐仲,张凯院,陆全,等.矩阵简明教程[M].北京:科学出版社,2001.