二阶时滞Volterra微积分方程数值研究（英文）
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摘要
本文研究二阶时滞Volterra微积分方程收敛问题.利用勒让德谱方法,获得方程的精确解与近似解及精确导数与近似导数误差在指定范数空间呈指数收敛结果,推广了二阶Volterra方程的结果.
In this paper, we use a Legendre-collocation spectral method to deal with the second order Volterra integro-differential equation, which contains vanishing delay.The convergence analysis for the proposed method is established in both L~2-norm and L~∞-norm. The goal is to provide a rigorous error analysis for the given equation. In the end of the paper, we give an example to confirm our deduce.
In this paper, we use a Legendre-collocation spectral method to deal with the second order Volterra integro-differential equation, which contains vanishing delay.The convergence analysis for the proposed method is established in both L~2-norm and L~∞-norm. The goal is to provide a rigorous error analysis for the given equation. In the end of the paper, we give an example to confirm our deduce.
引文
[1] BRUNNER H, LAMBERT J D. Stability of numerical methods for Volterra integro-differential equations[J]. Computing, 1974, 12:75-89.
[2] MAKROGLOU A. A block-by-block method for Volterra integro-differential equations with weaklysingular kernel[J]. Math. Comp., 1981, 37(155):95-99.
[3] ZHANG Y. The stability for integro-differential equations of Volterra type[J]. Mathematics Applicata,1989, 2:51-54.
[4] ENRIGHT W H,HU M. Continuous Runge-Kutta methods for neutral Volterra integro-differential equations with delay[J]. Appl. Numer. Math., 1997, 24(2):175-190.
[5] ZHANG C. The numerical stability of implicit Euler methods for Volterra integral-delay equations[J].Mathematics Applicata, 2000, 13:130-132.
[6] BRUNNER H. Collocation Methods for Volterra Integral and Related Functional Equations[M]. New York:Cambridge University Press, 2004.
[7] RAWASHDEH E, MCDOWELL D, RAKESH L. Polynomial spline collocation methods for secondorder Volterra integro-differential equations[J]. IJMMS, 2004, 56:3011-3022.
[8] TARANG M. Stability of the spline collocation method for second order Volterra integro-differential equations[J]. Math. Model. Anal., 2004, 9(1):79-90.
[9] WEI J, SHAN R, LIU W,et al. Differential transform method for solving the two-dimensional nonlinear Volterra integro-differential equation[J]. Mathematics Applicata, 2012, 25(3):691-696.
[10] ZHANG K, LI J. Collocation methods for a class of Volterra integral functional equations with multiple proportional delays[J]. Adv. Appl. Math. Mech., 2012,4(3):575-602.
[11] WEI Y, CHEN Y. Legendre spectral collocation methods for pantograph Volterra delay-integrodifferential equations[J]. J. Sci. Comput., 2013, 53(3):672-688.
[12] ZHENG W, CHEN Y. A spectral method for second order Volterra integro-differential equations with pantograph delay[J]. Advances in Applied Mathematics and Mechanics, 2013, 5(3):131-145.
[13] ZHENG W. Studies on the Volterra integral equation with linear delay[J]. Journal of Huanan University, 2017, 40:83-88.
[14] ZHENG W. Convergence analysis for fractional integral and differential equation with nonlinear delay[J]. Acta Scientiarum Naturaliun Universitatis Sunyatseni, 2018, 57(1):8-15.
[15] KATO T, MCLEOD J B. The functional-differential equation y'(x)=ay(λx)+by(x)[J]. Bull. Amer.Math. Soc., 1971, 77:891-937.
[16] ALI I, BRUNNER H, TANG T. A spectral method for pantograph-type delay differential equations and its convergence analysis[J]. J. Comput. Math., 2009,27:254-265.
[17] ALI I, BRUNNER H, TANG T. Spectral methods for pantograph-type differential and integral equations with multiple delays[J]. Front. Math. China, 2009, 4(1):49-61.
[18] CANUTO C, HUSSAINI M Y, QUARTERONI A, et al. Spectral Methods Fundamentals in Single Domains[M]. Belin:Springer-Verlag, 2006.
[19] CHEN Y, TANG T. Spectral methods for weakly singular Volterra integral equations with smooth solutions[J]. J. Comput. Appl. Math., 2009, 233(4):938-950.
[20] GUO B Y, WANG L L. Jacobi interpolation approximations and their applications to singular differential equations[J]. Adv. Comput. Math., 2001, 14:227-276.
[21] SHEN J, TANG T. Spectral and High-Order Methods with Applications[M]. Beijing:Science Press,2006.
[22] QU C K, WONG R. Szego's conjecture on Lebesgue constants for Legendre series[J]. Pacific J. Math.,1988, 135:157-188.
[2] MAKROGLOU A. A block-by-block method for Volterra integro-differential equations with weaklysingular kernel[J]. Math. Comp., 1981, 37(155):95-99.
[3] ZHANG Y. The stability for integro-differential equations of Volterra type[J]. Mathematics Applicata,1989, 2:51-54.
[4] ENRIGHT W H,HU M. Continuous Runge-Kutta methods for neutral Volterra integro-differential equations with delay[J]. Appl. Numer. Math., 1997, 24(2):175-190.
[5] ZHANG C. The numerical stability of implicit Euler methods for Volterra integral-delay equations[J].Mathematics Applicata, 2000, 13:130-132.
[6] BRUNNER H. Collocation Methods for Volterra Integral and Related Functional Equations[M]. New York:Cambridge University Press, 2004.
[7] RAWASHDEH E, MCDOWELL D, RAKESH L. Polynomial spline collocation methods for secondorder Volterra integro-differential equations[J]. IJMMS, 2004, 56:3011-3022.
[8] TARANG M. Stability of the spline collocation method for second order Volterra integro-differential equations[J]. Math. Model. Anal., 2004, 9(1):79-90.
[9] WEI J, SHAN R, LIU W,et al. Differential transform method for solving the two-dimensional nonlinear Volterra integro-differential equation[J]. Mathematics Applicata, 2012, 25(3):691-696.
[10] ZHANG K, LI J. Collocation methods for a class of Volterra integral functional equations with multiple proportional delays[J]. Adv. Appl. Math. Mech., 2012,4(3):575-602.
[11] WEI Y, CHEN Y. Legendre spectral collocation methods for pantograph Volterra delay-integrodifferential equations[J]. J. Sci. Comput., 2013, 53(3):672-688.
[12] ZHENG W, CHEN Y. A spectral method for second order Volterra integro-differential equations with pantograph delay[J]. Advances in Applied Mathematics and Mechanics, 2013, 5(3):131-145.
[13] ZHENG W. Studies on the Volterra integral equation with linear delay[J]. Journal of Huanan University, 2017, 40:83-88.
[14] ZHENG W. Convergence analysis for fractional integral and differential equation with nonlinear delay[J]. Acta Scientiarum Naturaliun Universitatis Sunyatseni, 2018, 57(1):8-15.
[15] KATO T, MCLEOD J B. The functional-differential equation y'(x)=ay(λx)+by(x)[J]. Bull. Amer.Math. Soc., 1971, 77:891-937.
[16] ALI I, BRUNNER H, TANG T. A spectral method for pantograph-type delay differential equations and its convergence analysis[J]. J. Comput. Math., 2009,27:254-265.
[17] ALI I, BRUNNER H, TANG T. Spectral methods for pantograph-type differential and integral equations with multiple delays[J]. Front. Math. China, 2009, 4(1):49-61.
[18] CANUTO C, HUSSAINI M Y, QUARTERONI A, et al. Spectral Methods Fundamentals in Single Domains[M]. Belin:Springer-Verlag, 2006.
[19] CHEN Y, TANG T. Spectral methods for weakly singular Volterra integral equations with smooth solutions[J]. J. Comput. Appl. Math., 2009, 233(4):938-950.
[20] GUO B Y, WANG L L. Jacobi interpolation approximations and their applications to singular differential equations[J]. Adv. Comput. Math., 2001, 14:227-276.
[21] SHEN J, TANG T. Spectral and High-Order Methods with Applications[M]. Beijing:Science Press,2006.
[22] QU C K, WONG R. Szego's conjecture on Lebesgue constants for Legendre series[J]. Pacific J. Math.,1988, 135:157-188.