基于块脉冲函数求解第一类Volterra积分方程的数值分析（英文）
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摘要
本文主要基于块脉冲函数求解第一类Volterra积分方程。介绍了块脉冲函数的定义和性质,基于块脉冲函数的性质及其积分算子矩阵数值求解第一类Volterra积分方程,给出了相应的数值格式,证明数值解的存在唯一性,以及相应数值方法的1阶收敛性。数值算例验证了理论结果的正确性。
It mainly focuses on how to solve Volterra integral equations of the first kind based on blockpulse functions. The definition and properties of block pulse functions are introduced briefly. Based on the properties of block pulse functions and their operational matrix of integration,Volterra integral equation of the first kind is solved numerically,and the corresponding numerical scheme is given. The existence and uniqueness of the approximate solution are proved. The convergence with order 1 of the corresponding numerical method is proved in detail. Some numerical experiments are given to illustrate the theoretical results.
It mainly focuses on how to solve Volterra integral equations of the first kind based on blockpulse functions. The definition and properties of block pulse functions are introduced briefly. Based on the properties of block pulse functions and their operational matrix of integration,Volterra integral equation of the first kind is solved numerically,and the corresponding numerical scheme is given. The existence and uniqueness of the approximate solution are proved. The convergence with order 1 of the corresponding numerical method is proved in detail. Some numerical experiments are given to illustrate the theoretical results.
引文
[1] BRUNNER H.Discretization of Volterra integral equations of the first kind II [J].Numerische Mathematik,1978,30:117-136.
[2] BRUNNER H.Collocation methods for Volterra integral and related functional differential equations [M].Cambridge:Cambridge University Press,2004.
[3] KAUTHEN J P,BRUNNER H.Continuous collocation approximations to solutions of first kind Volterra equations [J].Mathematics of Computation,1997,66:1441-1459.
[4] BRUNNER H.Superconvergence of collocation methods for Volterra integral equations of the first kind [J].Computing,1979,21:151-157.
[5] BRUNNER H.A note on collocation methods for Volterra integral equations of the first kind [J].Computing,1979,23:179-187.
[6] EGGERMONT P P B.Collocation as a projection method and superconvergence for Volterra integral equations of the first kind [M].Treatment of Integral Equations by Numerical Methods,London:Academic Press,1982.
[7] EGGERMONT P P B.Collocation for Volterra integral equations of the first kind with iterated kernel [J].SIAM Journal on Numerical Analysis,1983,20:1032-1048.
[8] EGGERMONT P P B.Improving the accuracy of collocation solutions of Volterra integral equations of the first kind by local interpolation [J].Numerische Mathematik,1986,48:263-279.
[9] LIANG H,BRUNNER H.Discrete superconvergence of collocation solutions for first kind volterra integral equations [J].Journal of Integral Equations and Applications,2012,24:359-391.
[10] BRUNNER H,DAVIES P J,DUNCAN D B.Discontinuous Galerkin approximations for Volterra integral equations of the first kind [J].IMA Journal of Numerical Analysis,2009,29:856-881.
[11] BRUNNER H,DAVIES P J,DUNCAN D B.Global convergence and local superconvergence of first kind Volterra integral equation approximations[J].IMA Journal of Numerical Analysis,2012,32:1117-1146.
[12] BABOLIAN E,MASOURI Z.Direct method to solve Volterra integral eqution of the first kind using operational matrix with block-pulse functions[J].Journal of Computational and Applied Mathematics,2008,220:51-57.
[13] MASOURI Z,BABOLIAN E,HATAMZADEH-VARMAZYAR S.An expansion-iterative method for numerically solving Volterra integral equation of the first kind [J].Computers and Mathematics with Applications,2010,59:1491-1499.
[14] MALEKNEJAD K,RAHIMI B.Modification of block pulse functions and their application to solve numerically Volterra integral equation of the first kind [J].Communications in Nonlinear Science and Numerical Simulation,2011,16:2469-2477.
[15] MALEKNEJAD K,SAFDARI H,NOURI M.Numerical solution of an integral equations system of the first kind by using an operational matrix with block pulse functions [J].International Journal of Systems Science,2011,42:195-199.
[2] BRUNNER H.Collocation methods for Volterra integral and related functional differential equations [M].Cambridge:Cambridge University Press,2004.
[3] KAUTHEN J P,BRUNNER H.Continuous collocation approximations to solutions of first kind Volterra equations [J].Mathematics of Computation,1997,66:1441-1459.
[4] BRUNNER H.Superconvergence of collocation methods for Volterra integral equations of the first kind [J].Computing,1979,21:151-157.
[5] BRUNNER H.A note on collocation methods for Volterra integral equations of the first kind [J].Computing,1979,23:179-187.
[6] EGGERMONT P P B.Collocation as a projection method and superconvergence for Volterra integral equations of the first kind [M].Treatment of Integral Equations by Numerical Methods,London:Academic Press,1982.
[7] EGGERMONT P P B.Collocation for Volterra integral equations of the first kind with iterated kernel [J].SIAM Journal on Numerical Analysis,1983,20:1032-1048.
[8] EGGERMONT P P B.Improving the accuracy of collocation solutions of Volterra integral equations of the first kind by local interpolation [J].Numerische Mathematik,1986,48:263-279.
[9] LIANG H,BRUNNER H.Discrete superconvergence of collocation solutions for first kind volterra integral equations [J].Journal of Integral Equations and Applications,2012,24:359-391.
[10] BRUNNER H,DAVIES P J,DUNCAN D B.Discontinuous Galerkin approximations for Volterra integral equations of the first kind [J].IMA Journal of Numerical Analysis,2009,29:856-881.
[11] BRUNNER H,DAVIES P J,DUNCAN D B.Global convergence and local superconvergence of first kind Volterra integral equation approximations[J].IMA Journal of Numerical Analysis,2012,32:1117-1146.
[12] BABOLIAN E,MASOURI Z.Direct method to solve Volterra integral eqution of the first kind using operational matrix with block-pulse functions[J].Journal of Computational and Applied Mathematics,2008,220:51-57.
[13] MASOURI Z,BABOLIAN E,HATAMZADEH-VARMAZYAR S.An expansion-iterative method for numerically solving Volterra integral equation of the first kind [J].Computers and Mathematics with Applications,2010,59:1491-1499.
[14] MALEKNEJAD K,RAHIMI B.Modification of block pulse functions and their application to solve numerically Volterra integral equation of the first kind [J].Communications in Nonlinear Science and Numerical Simulation,2011,16:2469-2477.
[15] MALEKNEJAD K,SAFDARI H,NOURI M.Numerical solution of an integral equations system of the first kind by using an operational matrix with block pulse functions [J].International Journal of Systems Science,2011,42:195-199.