拟逆正则化方法结合离散随机扰动反演初值问题
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摘要
利用离散随机扰动探讨时间分数阶扩散方程的反演初值问题,这类问题是不适定的,即问题的解(如果存在)不连续依赖于测量数据.利用拟逆正则化方法,得到问题的一个正则近似解,并且给出在先验正则化参数选取规则下的收敛性估计.数值结果表明拟逆正则化方法解决此类问题是有效和稳定的.
The inversion of initial value problem of fractional diffusion equation is explored with discrete random noise. This problem is ill-posed, i.e., the solution(if it exists) does not depend continuously on the measured data. The quasi-reversibility regularization method is used to obtain a regularized approximate solution and the convergence estimate is given under a priori parameter choice rule. Numerical results show that this method will be effective and stable.
The inversion of initial value problem of fractional diffusion equation is explored with discrete random noise. This problem is ill-posed, i.e., the solution(if it exists) does not depend continuously on the measured data. The quasi-reversibility regularization method is used to obtain a regularized approximate solution and the convergence estimate is given under a priori parameter choice rule. Numerical results show that this method will be effective and stable.
引文
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[7] WEI Z L,LI Q D,CHE J L.Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative [J].Mathematical Analysis and Applications,2010,367:260-272.
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[9] WEI Y,WANG J G.A modified quasi-boundary value method for the backward time-fractional diffusion problem [J].Applied Numerical Mathematics,2014,48:603-621.
[10] TUAN N H,NANE E R.Inverse source problem for time fractional diffusion with discrete random noise [J].Statistical Problem Letter,2017,120:126-134.
[11] PODLUBNY I.Fractional differential equations:an introduction to fractional derivatives fractional differential equations to methods of their solution an some of their applications [M].San Diego:Academic Press Inc,1999.
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[13] ELDEN L.Approximations for a Cauchy problem for the heat equation [J].Inverse Problem,1987,3:263-273.
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[15] YANG F,FU C L.Two regularization methods for identification of the heat source depending only on spatial variable for the heat equation [J].Inverse Ill-Posed Problems,2009,17:1-16.
[16] YANG F,FU C L.Two regularization methods to identify time-dependent heat source through an internal measurement of temperature [J].Mathematics and Computer Modelling,2011,53:793-804.
[17] 杨帆,任玉鹏,张盼,等.Poisson方程未知源识别的拟边界正则化方法 [J].兰州理工大学学报,2017,43(2):152-155.
[18] PODLUBNY I.Fractional differential equation [M].San Diego,Calif USA:Academy Press,1999.
[19] LIU J J,YAMAMOTO M.A backward problem for the time-fractional diffusion equation [J].Applicable Analysis,2010,89(11):1769-1788.
[20] EUBANK R L.Nonparametric regression and spline smoothing [M].2th ed.New York:Marcel Dekker Inc,1999.
[2] YUSTE S B,ACEDO L,LINDENBERG K.Reaction front in an A+B→C reaction subdiffusion process [J].Physical Review E,2004,69(3):03612601-03612610.
[3] HENEY B I,LANGLANDS T A M,WEARNE S L.Fractional cable models for spiny neuronal dendrites [J].Physical Review Letter,2008,100(12):103-128.
[4] SCALAS E,GORENFLO R,MAINARDI F.Fractional calculus and continuous-time finance [J].Physics Applied,2000,284(1):376-384.
[5] YANG F,ZHANG P,LI X X.The truncation method for the Cauchy problem of the inhomogeneous Helmholtz equation [J].Applicable Analysis,2019,98(5):991-1004.
[6] YANG F,SUN Y R,LI X X,et al.The quasi-boundary value method for identifying the initial value of heat equation on a columnar symmetric domain [J].Numerical Algorithms,2018,224:1-17.
[7] WEI Z L,LI Q D,CHE J L.Initial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative [J].Mathematical Analysis and Applications,2010,367:260-272.
[8] YANG M,LIU J J.Solving a final value fractional diffusion problem by boundary condition regularization [J].Applied Numerical Mathematics,2013,66:45-58.
[9] WEI Y,WANG J G.A modified quasi-boundary value method for the backward time-fractional diffusion problem [J].Applied Numerical Mathematics,2014,48:603-621.
[10] TUAN N H,NANE E R.Inverse source problem for time fractional diffusion with discrete random noise [J].Statistical Problem Letter,2017,120:126-134.
[11] PODLUBNY I.Fractional differential equations:an introduction to fractional derivatives fractional differential equations to methods of their solution an some of their applications [M].San Diego:Academic Press Inc,1999.
[12] WEBER C F.Analysis and solution of the ill-posed inverse conduction problem [J].Heat Mass Transfer,1981,24:1783-1792.
[13] ELDEN L.Approximations for a Cauchy problem for the heat equation [J].Inverse Problem,1987,3:263-273.
[14] QIAN Z,FU C L.Fourth-order modified method for the Cauchy problem for the Laplace equation [J].Computational Applied Mathematics,2006,192:205-218.
[15] YANG F,FU C L.Two regularization methods for identification of the heat source depending only on spatial variable for the heat equation [J].Inverse Ill-Posed Problems,2009,17:1-16.
[16] YANG F,FU C L.Two regularization methods to identify time-dependent heat source through an internal measurement of temperature [J].Mathematics and Computer Modelling,2011,53:793-804.
[17] 杨帆,任玉鹏,张盼,等.Poisson方程未知源识别的拟边界正则化方法 [J].兰州理工大学学报,2017,43(2):152-155.
[18] PODLUBNY I.Fractional differential equation [M].San Diego,Calif USA:Academy Press,1999.
[19] LIU J J,YAMAMOTO M.A backward problem for the time-fractional diffusion equation [J].Applicable Analysis,2010,89(11):1769-1788.
[20] EUBANK R L.Nonparametric regression and spline smoothing [M].2th ed.New York:Marcel Dekker Inc,1999.