一类微分方程的数值解法
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摘要
在记忆材料的热传导、多孔粘弹性介质的压缩、原子反应、动力学等问题中,常常碰到抛物型偏积分微分方程,对于该方程的数值求解,国外的V.thomee,W.Mclean,Ch.Lubich,L.Wahlbin,Sanz-Serna,E.G.Yanik,G.Fairweather,国内的陈传淼、黄元清、徐大等做了大量的研究,他们采用了有限元方法、谱配置方法及样条配置方法,但用Lubich的拉普拉斯变换数值逆离散却很少涉及.
本文考虑一类带弱奇异核抛物型偏积分微分方程时间、空间全离散格式,采用Lubich的拉普拉斯变换数值逆离散等方法进行数值计算,主要结果如下:
(1)给出偏积分微分方程空间x方向用有限差分法离散,时间t方向用Lubich的拉普拉斯变换数值逆离散的全离散格式,并进行数值计算。
(2)给出偏积分微分方程空间x方向用线性有限元离散,时间t方向用Lubich的拉普拉斯变换数值逆离散的全离散格式,并进行数值计算.
以上两种方法计算结果精度较高,并且计算也比较简便.
The integro-differential equation of parabolic type often occurs in application such as heat conduction in material with memory, comression of poro-viscoelastic media , nuclear reactor dynamics, ect. theres arelotsof documents of V. thomee, W. Mclean, Ch. Lubich, L. Wahlbin, Sanz-Serna, E. G. Yanik, G. Fairweather in overseas and Chuan-miao Chen、Yuan-qing Huang、Da-Xu in home. A lot of them use FEM ;Spectral collocation methods;Spline collocation methods.
but few of them use numerical inversion for the laplace transform of Lubich.
We study a partial integro-differential equations of parabolic type with a weakly singular kernel, which use numerical inversion for the laplace transform of Lubich for numerical calculation, main results follows:
(1) Give a kind of fully discrete scheme of a partial integro-differential equations which use finite difference method discrete in the direction of space x and numerical inversion for the laplace transform of Lubich in the direction of time t for numerical calculation.
(2) Give a kind of fully discrete scheme of a partial integro-differential equations which use linear finite element discrete in the direction of space x and numerical inversion for the laplace transform of Lubich in the direction of time t for numerical calculation.
Calculating result of two methods is accuratly higher, and calculate is simpler also.
本文考虑一类带弱奇异核抛物型偏积分微分方程时间、空间全离散格式,采用Lubich的拉普拉斯变换数值逆离散等方法进行数值计算,主要结果如下:
(1)给出偏积分微分方程空间x方向用有限差分法离散,时间t方向用Lubich的拉普拉斯变换数值逆离散的全离散格式,并进行数值计算。
(2)给出偏积分微分方程空间x方向用线性有限元离散,时间t方向用Lubich的拉普拉斯变换数值逆离散的全离散格式,并进行数值计算.
以上两种方法计算结果精度较高,并且计算也比较简便.
The integro-differential equation of parabolic type often occurs in application such as heat conduction in material with memory, comression of poro-viscoelastic media , nuclear reactor dynamics, ect. theres arelotsof documents of V. thomee, W. Mclean, Ch. Lubich, L. Wahlbin, Sanz-Serna, E. G. Yanik, G. Fairweather in overseas and Chuan-miao Chen、Yuan-qing Huang、Da-Xu in home. A lot of them use FEM ;Spectral collocation methods;Spline collocation methods.
but few of them use numerical inversion for the laplace transform of Lubich.
We study a partial integro-differential equations of parabolic type with a weakly singular kernel, which use numerical inversion for the laplace transform of Lubich for numerical calculation, main results follows:
(1) Give a kind of fully discrete scheme of a partial integro-differential equations which use finite difference method discrete in the direction of space x and numerical inversion for the laplace transform of Lubich in the direction of time t for numerical calculation.
(2) Give a kind of fully discrete scheme of a partial integro-differential equations which use linear finite element discrete in the direction of space x and numerical inversion for the laplace transform of Lubich in the direction of time t for numerical calculation.
Calculating result of two methods is accuratly higher, and calculate is simpler also.
引文
[1]C.Chen,V.Thomee,and L.B.Wahlbin.Finite element approximation of a parabolic integro-differential equations with a weakly singular kernel.Math.Comp.,1992,Vol.58:pp.587-602.
[2]W.Mclean,V.Thomee.Numerical solution of an evolution equation with a positive type memory term.J.Austral.Math.Soc.Ser.,1993,B35:pp.23-70.
[3]J.M.Sane-Serna.A Numerical method for a partial integro-differential equations.SIAM.J.Numer.Anal.,1998,Vol.25:pp.319-327.
[4]Xu Da.The global behaviour of time discretization for an abstract Volterra equation In Hilbert space.CALCOLO,1997,Vol.34:pp.71-104.
[5]E.G.Yanik,G.Fairweather.Finite element methods for parabolic and hyperbolic partial integro-differential equations,Nonlinear Analysis.Theory.Methods & Appl.,1998 Vol.12(8):pp.785-809.
[6]YiYan,G.Fairweather.Orthogonal spline collocation methods for some partial integro-differential equations.SIAM.J.Numer.Anal,1992,Vol.29(8):pp.755-768.
[7]K.Adolfsson,M.Enelund.Stig Larsson Adaptive discretization of an integro-differential equation with a weakly singular convolution kernel.Comput.Methods Appl.Mech.Engrg.,2003,Vol.192:pp.5285-5304.
[8]W.Mclean,V.Thomee.Time disretization of an evolution equation via Laplace transforms.IMA.J.Numer.Anal.,2004,Vol.24:pp.439-463.
[9]Tao Tang.Superconvergence of numerical solution to weakly singular Volerra integro-differential equations.Numer.Math.,1992,Vol.61:pp.373-382.
[10]HuangYun-Qing.Time discretization scheme for an integro-differential equation of paraholic type.J.Comp.Math.,1994,Vol.12(3):pp.259-263.
[11]D.L..Jagerman.An Inversion Technique for the Laplace Transform with Application to Approximation.B.S.T.J.,1978,57,No.3:pp.669-710.
[12]D.L.Jagerman.An Inversion Technique for the Laplace Transform,B.S.T.J., 1982,61,No.8:pp.1995-2002.
[13]Zhi-Zhong Sun.An Unconditionally Stable and 0(τ~2+h~4)Order L~∞ Convergent Difference Scheme for Linear Parabolic Equations with Variable Coefficients.Numer Methods Partial Differential Eq.,2001,Vol.17:pp.619-631.
[14]V.Thomee and W.Mclean.Time Discretization of an Evolution Equation via Laplace Transforms.AMR,2003,No.7:pp.1-27.
[15]C.Lubich.Convolution Quadrature and Discretized Operational Calculus Ⅰ.Numer.Math.,1998,Vol.52:pp.129-145.
[16]C.Lubich.Convolution Quadrature and Discretized Operational Calculus Ⅱ.Numer.Math.,1998,Vol.52:pp.412-425.
[17]Xu-Da.On the Discretization in Time for a Parabolic Integrodifferential Equation with a Weakly Singular Kernel Ⅰ:Smooth Initial Data.Appl.Math.Comp.,1993,Vol.58:pp.1-27.
[18]Xu-Da.On the Discretization in Time for a Parabolic Integrodifferential Equation with a Weakly Singular Kernel Ⅱ:Non Smooth Initial Data.Appl.Math.Comp.,1993,Vol.58:pp.29-61.
[19]杨晓霖,徐大.拉普拉斯变换的数字逆在微分方程中的应用[J].湖南师范大学自然科学学报,2004,Vol.27(2):pp.21-25.
[20]杨晓淼,徐大.拉普拉斯变换的数字逆在偏微分方程中的应用[J].南华大学学报,2005,Vol.19(2):pp.10-13.
[21]陆金甫,关治.偏微分方程数值解法[M].第二版.北京:清华大学出版社,2003.
[22]曾金平.数值计算方法[M].湖南:湖南大学出版社,2004.
[23]钟玉泉.复变函数论[M].第二版.北京:高等教育出版社,1987.
[2]W.Mclean,V.Thomee.Numerical solution of an evolution equation with a positive type memory term.J.Austral.Math.Soc.Ser.,1993,B35:pp.23-70.
[3]J.M.Sane-Serna.A Numerical method for a partial integro-differential equations.SIAM.J.Numer.Anal.,1998,Vol.25:pp.319-327.
[4]Xu Da.The global behaviour of time discretization for an abstract Volterra equation In Hilbert space.CALCOLO,1997,Vol.34:pp.71-104.
[5]E.G.Yanik,G.Fairweather.Finite element methods for parabolic and hyperbolic partial integro-differential equations,Nonlinear Analysis.Theory.Methods & Appl.,1998 Vol.12(8):pp.785-809.
[6]YiYan,G.Fairweather.Orthogonal spline collocation methods for some partial integro-differential equations.SIAM.J.Numer.Anal,1992,Vol.29(8):pp.755-768.
[7]K.Adolfsson,M.Enelund.Stig Larsson Adaptive discretization of an integro-differential equation with a weakly singular convolution kernel.Comput.Methods Appl.Mech.Engrg.,2003,Vol.192:pp.5285-5304.
[8]W.Mclean,V.Thomee.Time disretization of an evolution equation via Laplace transforms.IMA.J.Numer.Anal.,2004,Vol.24:pp.439-463.
[9]Tao Tang.Superconvergence of numerical solution to weakly singular Volerra integro-differential equations.Numer.Math.,1992,Vol.61:pp.373-382.
[10]HuangYun-Qing.Time discretization scheme for an integro-differential equation of paraholic type.J.Comp.Math.,1994,Vol.12(3):pp.259-263.
[11]D.L..Jagerman.An Inversion Technique for the Laplace Transform with Application to Approximation.B.S.T.J.,1978,57,No.3:pp.669-710.
[12]D.L.Jagerman.An Inversion Technique for the Laplace Transform,B.S.T.J., 1982,61,No.8:pp.1995-2002.
[13]Zhi-Zhong Sun.An Unconditionally Stable and 0(τ~2+h~4)Order L~∞ Convergent Difference Scheme for Linear Parabolic Equations with Variable Coefficients.Numer Methods Partial Differential Eq.,2001,Vol.17:pp.619-631.
[14]V.Thomee and W.Mclean.Time Discretization of an Evolution Equation via Laplace Transforms.AMR,2003,No.7:pp.1-27.
[15]C.Lubich.Convolution Quadrature and Discretized Operational Calculus Ⅰ.Numer.Math.,1998,Vol.52:pp.129-145.
[16]C.Lubich.Convolution Quadrature and Discretized Operational Calculus Ⅱ.Numer.Math.,1998,Vol.52:pp.412-425.
[17]Xu-Da.On the Discretization in Time for a Parabolic Integrodifferential Equation with a Weakly Singular Kernel Ⅰ:Smooth Initial Data.Appl.Math.Comp.,1993,Vol.58:pp.1-27.
[18]Xu-Da.On the Discretization in Time for a Parabolic Integrodifferential Equation with a Weakly Singular Kernel Ⅱ:Non Smooth Initial Data.Appl.Math.Comp.,1993,Vol.58:pp.29-61.
[19]杨晓霖,徐大.拉普拉斯变换的数字逆在微分方程中的应用[J].湖南师范大学自然科学学报,2004,Vol.27(2):pp.21-25.
[20]杨晓淼,徐大.拉普拉斯变换的数字逆在偏微分方程中的应用[J].南华大学学报,2005,Vol.19(2):pp.10-13.
[21]陆金甫,关治.偏微分方程数值解法[M].第二版.北京:清华大学出版社,2003.
[22]曾金平.数值计算方法[M].湖南:湖南大学出版社,2004.
[23]钟玉泉.复变函数论[M].第二版.北京:高等教育出版社,1987.