第二类端点奇异Fredholm积分方程的分数阶退化核方法
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摘要
本文针对第二类端点奇异Fredholm积分方程构造基于分数阶Taylor展开的退化核方法,设计了两种计算格式,一是在全区间上使用分数阶Taylor展开式近似核函数,二是在包含奇点的小区间上采用分数阶插值,在剩余区间上采用分段二次多项式插值逼近核函数.讨论了两种退化核方法收敛的条件,并给出了混合插值法的收敛阶估计.数值算例表明对于非光滑核函数分数阶退化核方法有着良好的计算效果,且混合二次插值法比全区间上的分数阶退化核方法有着更广泛的适用范围.
This paper constructs fractional order degenerate kernel methods based on the fractional Taylor's expansion for Fredholm integral equations of the second kind with endpoint singularities. Two computational schemes are designed, one is approximating the kernel function by its fractional Taylor's expansion in the whole interval, and the other is constructing a fractional order interpolation in a small interval containing the singularity and using piecewise quadratic interpolation to approximate the kernel function in the remaining part of the interval. The conditions that the two degenerate kernel methods converge are discussed,and the convergence order is estimated for hybrid interpolation scheme. Numerical examples demonstrate that the fractional order degenerate kernel methods have good computational results for the kernel functions with endpoint singularities, and the hybrid quadratic interpolation scheme has broader scope of application than the fractional order degenerate kernel method in the whole interval.
This paper constructs fractional order degenerate kernel methods based on the fractional Taylor's expansion for Fredholm integral equations of the second kind with endpoint singularities. Two computational schemes are designed, one is approximating the kernel function by its fractional Taylor's expansion in the whole interval, and the other is constructing a fractional order interpolation in a small interval containing the singularity and using piecewise quadratic interpolation to approximate the kernel function in the remaining part of the interval. The conditions that the two degenerate kernel methods converge are discussed,and the convergence order is estimated for hybrid interpolation scheme. Numerical examples demonstrate that the fractional order degenerate kernel methods have good computational results for the kernel functions with endpoint singularities, and the hybrid quadratic interpolation scheme has broader scope of application than the fractional order degenerate kernel method in the whole interval.
引文
[1] Sloan I H. Error analysis for a class of degenerate-kernel methods[J]. Numer. Math., 1976, 25(3):231-238.
[2] Sloan I H. Convergence of degenerate kernel methods[J]. J. Aust. Math. Soc. Series B-Appl. Math.,1976, 19(4):422-431.
[3] Atkinson K E. The Numerical Solution of Integral Equations of the Second Kind[M]. Cambridge University Press, 1997.
[4] Karimi S, Jozi M. Numerical solution of the system of linear Fredholm integral equations based on degenerating kernels[J]. TWMS J. Pure Appl. Math., 2015, 6(1):111-119.
[5] Hammerlin G, Schumaker L L. Procedures for kernel approximation and solution of Fredholm integral equations of the second kind[J]. Numer. Math., 1980, 34:125-141.
[6] Wazwaz A M. Linear and Nonlinear Integral Equations:Methods and Applications[M]. Berlin:Springer-Verlag, 2011.
[7] Dellwo D R. Accelerated degenerate-kernel methods for linear integral equations[J]. J. Comput.Appl. Math., 1995, 58(2):135-149.
[8] Guebbai H, Grammont L. A new degenerate kernel method for a weakly singular integral equation[J]. Appl. Math. Comput., 2014, 230:414-427.
[9]沈以淡.积分方程(第3版)[M].北京:清华大学出版社,2012.
[10] Kaneko H, Xu Y S. Degenerate kernel method for Hammerstein equations[J]. Math. Comp., 1991,56(193):141-148.
[11] Ikebe Y. The Galerkin method for the numerical solution of Fredholm integral equations of the second kind[J]. SIAM Rev., 1972, 14(3):465-491.
[12] Okayama T, Matsuo T, Sugihara M. Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind[J]. J. Comput. Appl. Math., 2010, 234(4):1211-1227.
[13] Atkinson K E, Shampine L F. Algorithm876:Solving Fredholm integral equations of the second kind in Matlab[J]. ACM Trans. Math. Software, 2008, 34(4), Article 21:1-20.
[14]吕涛,黄晋.积分方程的高精度算法[M].北京:科学出版社,2013.
[15] Occorsio D, Russo M G. Numerical methods for Fredholm integral equations on the square[J].Appl. Math. Comput., 2011, 218(5):2318-2333.
[16] Orav-Puurand K, Pedas A, Vainikko G. Nystrom type methods for Fredholm integral equations with weak singularities[J]. J. Comput. Appl. Math., 2010, 234(9):2848-2858.
[17] Giuseppe M, Gradimir V. M. Well-conditioned matrices for numerical treatment of Fredholm integral equations of the second kind[J]. Numer. Linear Algebra Appl., 2009, 16(11-12):995-1011.
[18] Chen Q S, Lin F R. A modified Nystrom-Clenshaw-Curtis quadrature for integral equations with piecewise smooth kernels[J]. Appl. Numer. Math., 2014, 85:77-89.
[19]王同科,佘海艳,刘志方.分数阶光滑函数线性和二次插值公式余项估计[J].计算数学,2014, 36(4):393-406.
[20] Kolwankar K M. Recursive local fractional derivative[J]. arXiv preprint arXiv:1312.7675(2013).
[21] Liu Z F, Wang T K, Gao G H. A local fractional Taylor expansion and its computation for insufficiently smooth functions[J]. East Asian Journal on Applied Mathematics, 2015, 5(2):176-191.
[22] Wang T K, Li N, Gao G H. The asymptotic expansion and extrapolation of trapezoidal rule for integrals with fractional order singularities[J]. Int. J. Comput. Math., 2015, 92(3):579-590.
[23] Wang T K, Liu Z F, Zhang Z Y. The modified composite Gauss type rules for singular integrals using Puiseux expansions[J]. Math. Comp, 2017, 86(303):345-373.
[24] Wang T K, Zhang Z Y, Liu Z F. The practical Gauss type rules for Hadamard finite-part integrals using Puiseux expansions[J]. Adv. Comput. Math., 2017, 43(2):319-350.
[25] Wang T K, Gu Y S, Zhang Z Y. An algorithm for the inversion of Laplace transforms using Puiseux expansions[J]. Numer. Algorithms, 2018, 78(1):107-132.
[26]樊梦,王同科,常慧宾.非光滑函数的分数阶插值公式[J].计算数学,2016, 38(2):212-224.
[2] Sloan I H. Convergence of degenerate kernel methods[J]. J. Aust. Math. Soc. Series B-Appl. Math.,1976, 19(4):422-431.
[3] Atkinson K E. The Numerical Solution of Integral Equations of the Second Kind[M]. Cambridge University Press, 1997.
[4] Karimi S, Jozi M. Numerical solution of the system of linear Fredholm integral equations based on degenerating kernels[J]. TWMS J. Pure Appl. Math., 2015, 6(1):111-119.
[5] Hammerlin G, Schumaker L L. Procedures for kernel approximation and solution of Fredholm integral equations of the second kind[J]. Numer. Math., 1980, 34:125-141.
[6] Wazwaz A M. Linear and Nonlinear Integral Equations:Methods and Applications[M]. Berlin:Springer-Verlag, 2011.
[7] Dellwo D R. Accelerated degenerate-kernel methods for linear integral equations[J]. J. Comput.Appl. Math., 1995, 58(2):135-149.
[8] Guebbai H, Grammont L. A new degenerate kernel method for a weakly singular integral equation[J]. Appl. Math. Comput., 2014, 230:414-427.
[9]沈以淡.积分方程(第3版)[M].北京:清华大学出版社,2012.
[10] Kaneko H, Xu Y S. Degenerate kernel method for Hammerstein equations[J]. Math. Comp., 1991,56(193):141-148.
[11] Ikebe Y. The Galerkin method for the numerical solution of Fredholm integral equations of the second kind[J]. SIAM Rev., 1972, 14(3):465-491.
[12] Okayama T, Matsuo T, Sugihara M. Sinc-collocation methods for weakly singular Fredholm integral equations of the second kind[J]. J. Comput. Appl. Math., 2010, 234(4):1211-1227.
[13] Atkinson K E, Shampine L F. Algorithm876:Solving Fredholm integral equations of the second kind in Matlab[J]. ACM Trans. Math. Software, 2008, 34(4), Article 21:1-20.
[14]吕涛,黄晋.积分方程的高精度算法[M].北京:科学出版社,2013.
[15] Occorsio D, Russo M G. Numerical methods for Fredholm integral equations on the square[J].Appl. Math. Comput., 2011, 218(5):2318-2333.
[16] Orav-Puurand K, Pedas A, Vainikko G. Nystrom type methods for Fredholm integral equations with weak singularities[J]. J. Comput. Appl. Math., 2010, 234(9):2848-2858.
[17] Giuseppe M, Gradimir V. M. Well-conditioned matrices for numerical treatment of Fredholm integral equations of the second kind[J]. Numer. Linear Algebra Appl., 2009, 16(11-12):995-1011.
[18] Chen Q S, Lin F R. A modified Nystrom-Clenshaw-Curtis quadrature for integral equations with piecewise smooth kernels[J]. Appl. Numer. Math., 2014, 85:77-89.
[19]王同科,佘海艳,刘志方.分数阶光滑函数线性和二次插值公式余项估计[J].计算数学,2014, 36(4):393-406.
[20] Kolwankar K M. Recursive local fractional derivative[J]. arXiv preprint arXiv:1312.7675(2013).
[21] Liu Z F, Wang T K, Gao G H. A local fractional Taylor expansion and its computation for insufficiently smooth functions[J]. East Asian Journal on Applied Mathematics, 2015, 5(2):176-191.
[22] Wang T K, Li N, Gao G H. The asymptotic expansion and extrapolation of trapezoidal rule for integrals with fractional order singularities[J]. Int. J. Comput. Math., 2015, 92(3):579-590.
[23] Wang T K, Liu Z F, Zhang Z Y. The modified composite Gauss type rules for singular integrals using Puiseux expansions[J]. Math. Comp, 2017, 86(303):345-373.
[24] Wang T K, Zhang Z Y, Liu Z F. The practical Gauss type rules for Hadamard finite-part integrals using Puiseux expansions[J]. Adv. Comput. Math., 2017, 43(2):319-350.
[25] Wang T K, Gu Y S, Zhang Z Y. An algorithm for the inversion of Laplace transforms using Puiseux expansions[J]. Numer. Algorithms, 2018, 78(1):107-132.
[26]樊梦,王同科,常慧宾.非光滑函数的分数阶插值公式[J].计算数学,2016, 38(2):212-224.