• <tr id="yyy80"></tr>
  • <sup id="yyy80"></sup>
  • <tfoot id="yyy80"><noscript id="yyy80"></noscript></tfoot>
  • 99热精品在线国产_美女午夜性视频免费_国产精品国产高清国产av_av欧美777_自拍偷自拍亚洲精品老妇_亚洲熟女精品中文字幕_www日本黄色视频网_国产精品野战在线观看 ?

    An Approximate Approach for Systems of Singular Volterra Integral Equations Based on Taylor Expansion

    2018-08-02 07:35:32MohsenDidgarandAlirezaVahidi
    Communications in Theoretical Physics 2018年8期

    Mohsen Didgarand Alireza Vahidi

    1Department of Mathematics,Guilan Science and Research Branch,Islamic Azad University,Rasht,Iran

    2Department of Mathematics,Rasht Branch,Islamic Azad University,Rasht,Iran

    3Department of Mathematics,College of Science,Yadegar-e-Emam Khomeyni(RAH)Shahr-e-Rey Branch,Islamic Azad University,Tehran,Iran

    AbstractIn this article,an extended Taylor expansion method is proposed to estimate the solution of linear singular Volterra integral equations systems.The method is based on combining the m-th order Taylor polynomial of unknown functions at an arbitrary point and integration method,such that the given system of singular integral equations is converted into a system of linear equations with respect to unknown functions and their derivatives.The required solutions are obtained by solving the resulting linear system.The proposed method gives a very satisfactory solution,which can be performed by any symbolic mathematical packages such as Maple,Mathematica,etc.Our proposed approach provides a significant advantage that the m-th order approximate solutions are equal to exact solutions if the exact solutions are polynomial functions of degree less than or equal to m.We present an error analysis for the proposed method to emphasize its reliability.Six numerical examples are provided to show the accuracy and the efficiency of the suggested scheme for which the exact solutions are known in advance.

    Key words:systems of singular Volterra integral equations(SSVIEs),systems of generalized Abel’s integral equations,error analysis,Taylor expansion

    1 Introduction

    Singular integral equations appear frequently in mathematical physics and have various applications in different fields including fluid mechanics,solid mechanics,quantum mechanics,bio-mechanics,astronomy,optics,electromagnetic theory,X-ray radiography,seismology,optical if ber evaluation,atomic scattering,radar ranging,electron emission,plasma diagnostics,and microscopy.[1]In recent years,approximate solution of integral equations has attracted great attention of many researchers[1?8]while numerical solution of weakly singular integral equations has been less considered.[9?18]Abel’s integral equation is one of the famous and important singular integral equations that arises from physical or mechanical models without passing through a differential equation.The general form of Abel’s integral equation is

    As mechanical description of Abel’s integral equation consider a point of mass moving in the gravity field on a smooth curve lying in a vertical plane.Let f(x)show the time in which the point mass reaches the lowest point while released from the height x.The problem is to find the equation of the curve.Abel’s integral equation is formulation of this problem.[1]

    Study and investigation of approximate solutions for systems of singular integral equations play a significant role in applied sciences,since they are not generally easy to solve analytically.Thus a variety of numerical and approximate methods have been developed to solve these,such as operational matrices,[19]homotopy perturbation method(HPM),[20]homotopy analysis method(HAM),[21]fractional differential transform method,[22]extrapolation method,[23]Legendre wavelets,[24]and Sinc approximation with the single exponential(SE).[25]

    Li[26]proposed a novel application of Taylor expansion method for approximate solution of linear ordinary differential equations with variable coefficients.The method expanded by Li and his co-authors to solve Abel’s integral equation,[18,27]Riccati equation,[28]an integral equation with fixed singularity for a cruciform crack,[29]a class of linear integro-differential equations,[30]and fractional integro-differential equations.[31]Vahidi and Didgar improved the Taylor expansion method proposed in Ref.[28]for determining the solution of Riccati equations.[32]Didgar and Ahmadi expanded the method proposed in Ref.[26]for solving systems of linear ordinary and fractional differential equations.[33]Moreover,Maleknejad and Damercheli[34]developed the method for solving linear second kind Volterra integral equations system.This investigation is an effort to propose a novel application of Taylor expansion[18,26?34]for solving systems of singular integral equations which possesses high accuracy.By expanding unknown functions as an m-th order Taylor poly-nomial and employing integration method,system of singular integral equations is converted into a new system of linear equations with respect to unknown functions and their derivatives.Then,intended approximate solutions can be obtained by solving the resulting linear system using a standard method.Besides simplicity and applicability,the considerable advantage of this method is that an m-th order approximation tends to the exact solution if it be a polynomial function of degree at most m.

    The paper is arranged as follows.In Sec.2,a method for systems of singular integral equations is described.An error analysis is given in Sec.3.In Sec.4,the accuracy and efficiency of the method is illustrated by considering six numerical examples.Section 5 is devoted to conclusions.

    2 Description of the Method

    Consider the following system of singular Volterra integral equations

    where λij(i,j= 1,...,n)are real constants,gij(x)(i,j=1,...,n)and fi(x)(i=1,...,n)are given functions in C(I)where I is the interval of interest.The ψj(x)(j=1,...,n)are unknown functions to be determined and kijare singular kernels of the form

    In this section we aim to show how the Taylor expansion method can be applied to the approximate solutions of singular system(2).Toward this end,we convert the SSVIEs into a system of linear equations with respect to unknown functions and their derivatives.This needs the desired solutions ψj(t)to be m+1 times continuously differentiable on the interval I,in other words ψj∈ Cm+1(I).Therefore,for ψj∈ Cm+1(I),the unknown functions ψj(t)can be expressed in terms of the m-th order Taylor series at an arbitrary point x∈I as

    where Ej,m(t,x)indicates the Lagrange error bound

    for some point ξjbetween x and t.Generally,the Lagrange error bound Ej,m(t,x)becomes sufficiently small as m gets great enough.Especially,if the solutions ψj(t)are polynomial functions of degree up to m,then the Lagrange error bound becomes zero,namely,the obtained approximate solutions of system(2)yield the exact solutions.Based on the aforementioned assumption,by omitting the last Lagrange error bound,we consider the truncated Taylor expansion ψj(t)as

    Inserting the approximate relation(6),for unknown function ψj(t),into Eq.(2)and in view of Eq.(3),we obtain

    In fact,Eq.(2)was converted into a linear system of ordinary differential equations with respect to ψj(x)and its derivatives up to order m.In other word,we have obtained n linear equations in Eq.(7)with respect to n×(m+1)unknown functionsfor k=0,...,m,j=1,...,n.In the following,we want to determineby solving a system of linear equations.In order to achieve this goal,other n×m independent linear equations with respect toare needed,which can be obtained by integrating both sides of Eq.(2)m times with respect to x from 0 to s and with the help of changing the order of the integrations.Thus,we have

    where

    in which the variable s has replaced by x,for simplicity.Similarly,we apply the Taylor expansion again and substituting(6)for ψj(t)into Eq.(8)results in

    for l=1,...,m.

    In this way,Eqs.(7)and(10)construct a system of linear equations with respect to the unknown functions ψj(x)and its derivatives up to order m.In the following,we indicate this system as

    where

    and in coefficient matrix(12),the first n rows refer to coefficients ofin Eq.(7)for k=0,...,m,j=1,...,n and the other rows refer to coefficients ofinfor k=1,...,m are determined by solving the resulting linear system but in point of fact,it is ψj(x)that we need.

    3 Error Analysis

    This section belongs to the stability analysis of the scheme and the error analysis proposed in Ref.[18]will be expanded for derived m-th order approximate solution of singular integral equations system(2)in order to get theoretical features about the convergence of the suggested method.We assume that the exact solutions ψj(t)are infinitely differentiable on the interval I;so ψj(t)can be expanded as an uniformly convergent Taylor series in I as follows Eq.(10)for l=1,...,m.Ultimately,the resulting system(11)can be solved by any appropriate method to obtain unknown functions.We note that not only ψj(x)but also

    Using the proposed method given in Sec.2,SSVIEs can be converted into an equivalent system of linear equations with respect to unknown functionsk=0,1,...as

    where

    Hence,under the solvability conditions of system(16)and letting B=V?1,the unique solution of system(16)is represented as

    We rewrite relation(19)in an alternative matrix form as

    Accordingly,we can find out that the vector Ψnconsists of the first n(m+1)elements of the exact solution vector Ψ must satisfy the following relation

    According to the proposed process,the unique solution of SSVIEs(2)can be denoted as

    where Ψnis replaced bynas its approximate solution.

    Subtracting Eq.(22)from Eq.(21)leads to

    Now,we expand the right-hand side of Eq.(23),the first n elements of the vector at the left-hand side of Eq.(23)can be expressed as

    where

    4 Numerical Examples

    In this section,six numerical examples are considered in order to establish the applicability and the accuracy of the proposed method.The results are compared with previous reports results to illustrate that the suggested method is not only accurate but also quite stable.In the following examples,absolute errors of the m-th order approximate values ψi,m(x)and the corresponding exact values ψi(x)asare determined.All computations were performed using Mathematica 8.

    Example 1Consider the following system of singular Volterra integral equations[19]

    with the exact solutions ψ1(x)=x and ψ2(x)=1.

    Using the proposed method in Sec.2,we obtain the approximate solutions of the problem(27)and it is important to note that after converting system(27)into a system of linear equations the Mathematica command“LinearSolve” is used for the new system.We can find by setting m=1,the first-order approximate solution yields the exact solution as expected,since the m-th order approximate solution yields the exact solution if the exact solution is a polynomial function of degree up to m.

    This example has been solved by operational matrices of piecewise constant orthogonal functions[19]on the interval[0,1).We present the maximum of the absolute errors obtained from Ref.[19]in Table 1.

    Table 1 The maximum of the absolute errors in Ref.[19].

    Table 2 Absolute errors of Example 2 for ψ1(x).

    Example 2Consider the following integral equations system[22,25]

    Example 3The following system of integral equations

    is considered in Ref.[25]with the exact solutions ψ1(x)=x and ψ2(x)=1.Employing the process described in Sec.2,by setting m=1,the first-order approximate solution of Eq.(29)results in the exact solution,as expected.This example was used in Ref.[25]and has been solved by SE-Sinc method.Tables 6 and 7 are related to the numerical results obtained from Ref.[25].

    Table 3 Absolute errors of Example 2 for ψ2(x).

    Table 4 Absolute errors of Example 2 by SE-Sinc method in Ref.[25]for ψ1(x).

    Table 6 Absolute errors of Example 3 by SE-Sinc method in Ref.[25]for ψ1(x).

    Example 4Consider the following system of Abel’s integral equations of the second kind[21]

    Table 7 Absolute errors of Example 3 by SE-Sinc method in Ref.[25]for ψ2(x).

    From Tables 8 and 9,we can find that the accuracy of our results is quite satisfactory and more accurate results can be obtained by taking higher-order m.This example was used in Ref.[21]and has been solved by homotopy analysis method.Figures 1 and 2 are related to the absolute errors of ψ1(x)and ψ2(x),respectively,in Ref.[21].From Tables 8 and 9 and Figs.1 and 2,we observe that the results obtained by Taylor expansion method are much better than those obtained in Ref.[21].

    Example 5Consider the following system of Abel’s integral equations of the second kind[21]with the exact solutions ψ(x)=x and

    1We obtain the approximate solutions by setting m=1,5,10,15.In the following,absolute errors are shown in Tables 10 and 11.From Tables 10 and 11,we observe that the accuracy of our results is quite satisfactory and more accurate results can be obtained by taking higherorder m.This example was used in Ref.[21]and has been solved by homotopy analysis method.Figures 3 and 4 are related to the absolute errors of ψ1(x)and ψ2(x),respectively,in Ref.[21].From Tables 10 and 11 and Figs.3 and 4,we can find that the results obtained by Taylor expansion method are much better than those obtained in Ref.[21].

    Table 8 Absolute errors of Example 4 for ψ1(x).

    Table 9 Absolute errors of Example 4 for ψ2(x).

    Table 10 Absolute errors of Example 5 for ψ1(x).

    Example 6Consider the following singular integral equations system of the first kind[1]

    with the exact solutions ψ1(x)=1+x+x3and ψ2(x)=1?x?x3.We evaluate the approximate solutions by setting m=1,2,3 and the obtained absolute errors are shown in Tables 12 and 13.We observe that the accuracy of our results is quite satisfactory and the third-order approximate solution yields the exact value,as expected.

    Table 11 Absolute errors of Example 5 for ψ2(x).

    Table 12 Absolute errors of Example 6 for ψ1(x).

    5 Conclusion

    The main objective of this investigation was to present a new application of Taylor expansion to conveniently solve linear singular integral equations systems.By employing the Taylor expansion of unknown functions at an arbitrary point and integration method,the SSVIEs has been converted into a system of linear equations with respect to unknown functions and their derivatives.The stability analysis of the method was also carried out and we have demonstrated the practicality and efficiency of our proposed method by several numerical examples.In particular for such cases when the exact solutions are polynomial functions of degree at most m,the derived m-th order approximations are equal to exact solutions.

    Table 13 Absolute errors of Example 6 for ψ2(x).

    Fig.1 The absolute error of ψ1(x)in Ref.[21].

    Fig.2 The absolute error of ψ2(x)in Ref.[21].

    Fig.3 The absolute error of ψ1(x)in Ref.[21].

    Fig.4 The absolute error of ψ2(x)in Ref.[21].

    国产一区二区在线观看日韩| 一本久久中文字幕| av在线观看视频网站免费| 又爽又黄a免费视频| 亚洲高清免费不卡视频| 久久精品影院6| 国产高清不卡午夜福利| 国产在视频线在精品| 丝袜美腿在线中文| 国产成年人精品一区二区| 1000部很黄的大片| 97超碰精品成人国产| 91精品国产九色| 国产成人a∨麻豆精品| 精品人妻偷拍中文字幕| 国产成人a区在线观看| 18禁在线播放成人免费| 久久99精品国语久久久| 国产在线男女| 久久久国产成人精品二区| 观看美女的网站| 精品一区二区免费观看| 精品人妻偷拍中文字幕| 国产毛片a区久久久久| 午夜精品一区二区三区免费看| 婷婷色av中文字幕| 亚洲国产精品成人久久小说 | 国产乱人偷精品视频| 美女cb高潮喷水在线观看| 国产一区亚洲一区在线观看| 亚洲激情五月婷婷啪啪| 国产伦一二天堂av在线观看| 乱码一卡2卡4卡精品| 99热这里只有是精品在线观看| 亚洲最大成人av| 久久99热这里只有精品18| 两个人的视频大全免费| 亚洲欧美成人精品一区二区| 好男人视频免费观看在线| 久久亚洲精品不卡| 男的添女的下面高潮视频| 日本一本二区三区精品| 美女cb高潮喷水在线观看| 美女xxoo啪啪120秒动态图| 国产私拍福利视频在线观看| 国产精品.久久久| 亚洲av电影不卡..在线观看| 日本一二三区视频观看| 有码 亚洲区| 成人国产麻豆网| 夜夜看夜夜爽夜夜摸| 91精品一卡2卡3卡4卡| 亚洲人成网站高清观看| 国产亚洲av嫩草精品影院| 国产成人91sexporn| 欧美bdsm另类| 国产av一区在线观看免费| 六月丁香七月| 九九久久精品国产亚洲av麻豆| 国产乱人视频| 在线国产一区二区在线| 秋霞在线观看毛片| 在线观看午夜福利视频| 亚洲精品影视一区二区三区av| 男人和女人高潮做爰伦理| 国产黄色视频一区二区在线观看 | 久久久久网色| 一个人免费在线观看电影| 亚洲精品乱码久久久v下载方式| 极品教师在线视频| 日本黄色片子视频| 久久久国产成人免费| 久久九九热精品免费| 在现免费观看毛片| 精品人妻熟女av久视频| 人妻少妇偷人精品九色| 久久久久网色| 听说在线观看完整版免费高清| 国产精品一及| 一区二区三区高清视频在线| 一个人看视频在线观看www免费| 国产大屁股一区二区在线视频| 国产人妻一区二区三区在| 国产精品女同一区二区软件| 色噜噜av男人的天堂激情| 欧美潮喷喷水| 日日干狠狠操夜夜爽| 久久这里有精品视频免费| 久久精品国产99精品国产亚洲性色| 听说在线观看完整版免费高清| 欧美日韩乱码在线| 嫩草影院精品99| 乱人视频在线观看| 亚洲内射少妇av| 色哟哟·www| 91久久精品国产一区二区三区| a级毛片免费高清观看在线播放| 久久久久久久午夜电影| 九九热线精品视视频播放| 日韩成人av中文字幕在线观看| 日韩av在线大香蕉| 毛片一级片免费看久久久久| 亚洲无线在线观看| 国产视频首页在线观看| 国产精品久久久久久精品电影| 亚洲欧美精品专区久久| 欧美+亚洲+日韩+国产| 久久精品人妻少妇| 欧美+亚洲+日韩+国产| 91aial.com中文字幕在线观看| 日韩av不卡免费在线播放| 亚洲一区二区三区色噜噜| 12—13女人毛片做爰片一| 亚洲国产精品国产精品| 亚洲国产精品国产精品| 高清毛片免费看| 成人鲁丝片一二三区免费| 嘟嘟电影网在线观看| 国内久久婷婷六月综合欲色啪| 亚洲美女搞黄在线观看| 人妻少妇偷人精品九色| 国产高清有码在线观看视频| 3wmmmm亚洲av在线观看| 少妇的逼水好多| 中文字幕久久专区| 人妻少妇偷人精品九色| 欧美xxxx黑人xx丫x性爽| 亚洲自偷自拍三级| 国产免费一级a男人的天堂| 一个人看视频在线观看www免费| 日本一本二区三区精品| 直男gayav资源| 亚洲国产精品国产精品| 中文字幕制服av| 久久精品久久久久久久性| 色视频www国产| 最后的刺客免费高清国语| 亚洲av免费高清在线观看| 中国美女看黄片| 99久久精品国产国产毛片| 亚洲国产精品成人久久小说 | 欧美人与善性xxx| 欧美不卡视频在线免费观看| 岛国毛片在线播放| 人妻制服诱惑在线中文字幕| 国产毛片a区久久久久| 日韩欧美 国产精品| 精品久久久久久久久亚洲| 日日撸夜夜添| 尾随美女入室| 精品久久久久久久人妻蜜臀av| 免费人成在线观看视频色| 中文字幕免费在线视频6| 日韩高清综合在线| 在线观看美女被高潮喷水网站| 色吧在线观看| 黄片wwwwww| 久久人人爽人人爽人人片va| 观看美女的网站| 久久久久久久午夜电影| 国产私拍福利视频在线观看| 99国产极品粉嫩在线观看| 国产极品天堂在线| 日韩精品青青久久久久久| 精品人妻视频免费看| 久久婷婷人人爽人人干人人爱| 自拍偷自拍亚洲精品老妇| 此物有八面人人有两片| 大型黄色视频在线免费观看| 亚洲电影在线观看av| 变态另类成人亚洲欧美熟女| www.av在线官网国产| 成人高潮视频无遮挡免费网站| 亚洲精华国产精华液的使用体验 | 久久人人精品亚洲av| 村上凉子中文字幕在线| 亚洲精品成人久久久久久| 亚洲成人中文字幕在线播放| 久久精品国产自在天天线| 日韩精品有码人妻一区| 国产一区二区激情短视频| 青青草视频在线视频观看| 久久6这里有精品| 欧美色视频一区免费| 啦啦啦啦在线视频资源| 级片在线观看| 别揉我奶头 嗯啊视频| 久久精品国产鲁丝片午夜精品| 久久久a久久爽久久v久久| 国产伦在线观看视频一区| 国产免费男女视频| 亚洲第一区二区三区不卡| 精品人妻偷拍中文字幕| 国产不卡一卡二| 九九热线精品视视频播放| 欧美一区二区亚洲| 欧美性感艳星| 日日干狠狠操夜夜爽| 国产精华一区二区三区| 少妇人妻精品综合一区二区 | 亚洲三级黄色毛片| 国产激情偷乱视频一区二区| 最近最新中文字幕大全电影3| 又黄又爽又刺激的免费视频.| 97超碰精品成人国产| 人体艺术视频欧美日本| 又粗又爽又猛毛片免费看| 美女大奶头视频| av又黄又爽大尺度在线免费看 | 又粗又爽又猛毛片免费看| 亚洲精品成人久久久久久| 有码 亚洲区| 性插视频无遮挡在线免费观看| 欧美一区二区国产精品久久精品| 97热精品久久久久久| 97在线视频观看| 亚洲欧洲日产国产| 亚洲国产欧洲综合997久久,| av国产免费在线观看| 久久人人精品亚洲av| 亚洲欧美成人综合另类久久久 | 久久99精品国语久久久| 久久精品国产清高在天天线| 狂野欧美激情性xxxx在线观看| av卡一久久| 免费看av在线观看网站| 最近最新中文字幕大全电影3| 黄色一级大片看看| 男女啪啪激烈高潮av片| 久久婷婷人人爽人人干人人爱| 久久精品综合一区二区三区| 在线观看美女被高潮喷水网站| 中国美女看黄片| av女优亚洲男人天堂| 如何舔出高潮| 一级毛片aaaaaa免费看小| 看片在线看免费视频| 国产午夜精品论理片| www.av在线官网国产| 夜夜看夜夜爽夜夜摸| 男女啪啪激烈高潮av片| 日韩中字成人| 日韩欧美三级三区| 亚洲av一区综合| 亚洲天堂国产精品一区在线| 夜夜夜夜夜久久久久| 成人av在线播放网站| 国产乱人偷精品视频| 国产精品麻豆人妻色哟哟久久 | 日韩欧美精品v在线| 村上凉子中文字幕在线| 国产精品久久久久久精品电影| 爱豆传媒免费全集在线观看| 中文字幕熟女人妻在线| 欧美性猛交╳xxx乱大交人| 卡戴珊不雅视频在线播放| 在线免费观看不下载黄p国产| 亚洲在线自拍视频| 熟妇人妻久久中文字幕3abv| 日韩国内少妇激情av| 亚洲第一区二区三区不卡| 亚洲av中文av极速乱| 26uuu在线亚洲综合色| 国产精品人妻久久久影院| 九九热线精品视视频播放| 夜夜爽天天搞| 99在线视频只有这里精品首页| 如何舔出高潮| 国产色婷婷99| h日本视频在线播放| 久久久午夜欧美精品| 国产精品一及| 超碰av人人做人人爽久久| 国内精品宾馆在线| 午夜爱爱视频在线播放| 亚洲美女搞黄在线观看| 亚洲成av人片在线播放无| 成人毛片60女人毛片免费| 久久精品国产自在天天线| 一级二级三级毛片免费看| 1024手机看黄色片| 久久人妻av系列| 久久精品国产亚洲av涩爱 | 一夜夜www| 免费观看的影片在线观看| 日韩 亚洲 欧美在线| 国产91av在线免费观看| 91精品一卡2卡3卡4卡| 成人毛片60女人毛片免费| 91麻豆精品激情在线观看国产| 观看美女的网站| 久久精品国产亚洲网站| 精品国内亚洲2022精品成人| 欧美三级亚洲精品| 国产蜜桃级精品一区二区三区| 亚洲av熟女| 欧美3d第一页| 国产亚洲精品av在线| 成人毛片a级毛片在线播放| 亚洲精品色激情综合| av视频在线观看入口| 亚洲一区高清亚洲精品| 麻豆一二三区av精品| av在线蜜桃| 欧美精品国产亚洲| 可以在线观看的亚洲视频| 国产麻豆成人av免费视频| 天堂影院成人在线观看| 精品人妻一区二区三区麻豆| 男女那种视频在线观看| 国产色婷婷99| 午夜免费男女啪啪视频观看| 内射极品少妇av片p| 欧美一区二区精品小视频在线| 精品国产三级普通话版| 精品久久久久久久久av| 一区二区三区免费毛片| 免费黄网站久久成人精品| 国产精品99久久久久久久久| 亚洲精品日韩在线中文字幕 | 中文字幕免费在线视频6| 久久精品国产亚洲av涩爱 | 免费在线观看成人毛片| 99久久精品一区二区三区| 麻豆久久精品国产亚洲av| 高清日韩中文字幕在线| 亚洲国产精品成人综合色| 1024手机看黄色片| 欧美另类亚洲清纯唯美| 中国美白少妇内射xxxbb| 久久精品91蜜桃| 午夜a级毛片| 长腿黑丝高跟| 亚洲高清免费不卡视频| 婷婷色av中文字幕| 久久久欧美国产精品| 国产精品一区二区性色av| 国产成人91sexporn| 一夜夜www| 亚洲精品国产av成人精品| 欧美成人精品欧美一级黄| 尾随美女入室| 日韩一本色道免费dvd| 色综合亚洲欧美另类图片| 国产精品精品国产色婷婷| 美女cb高潮喷水在线观看| 女的被弄到高潮叫床怎么办| 91精品一卡2卡3卡4卡| 国产极品精品免费视频能看的| 久久久成人免费电影| 亚洲av一区综合| 在线免费观看不下载黄p国产| 十八禁国产超污无遮挡网站| 成年版毛片免费区| 中文欧美无线码| 亚洲精品国产成人久久av| 最近手机中文字幕大全| 亚洲三级黄色毛片| 亚洲中文字幕日韩| 亚洲欧洲国产日韩| 嫩草影院新地址| 秋霞在线观看毛片| 国产高清三级在线| 国产精品久久久久久久电影| 久久这里只有精品中国| 欧美日韩乱码在线| 久久久久性生活片| 看黄色毛片网站| 黄片无遮挡物在线观看| 一个人免费在线观看电影| 精品人妻一区二区三区麻豆| av.在线天堂| 少妇的逼好多水| 午夜精品在线福利| 老女人水多毛片| 全区人妻精品视频| a级毛片免费高清观看在线播放| 国产精品久久久久久精品电影小说 | 秋霞在线观看毛片| 一本精品99久久精品77| 亚洲人成网站在线播放欧美日韩| 免费不卡的大黄色大毛片视频在线观看 | 欧美成人精品欧美一级黄| 赤兔流量卡办理| 中文亚洲av片在线观看爽| 色吧在线观看| 给我免费播放毛片高清在线观看| 大又大粗又爽又黄少妇毛片口| 天天躁夜夜躁狠狠久久av| 少妇裸体淫交视频免费看高清| 免费看日本二区| 一级黄色大片毛片| 欧美精品国产亚洲| 日韩一区二区三区影片| 成人永久免费在线观看视频| 少妇丰满av| 免费观看的影片在线观看| 国产伦精品一区二区三区四那| 欧美+亚洲+日韩+国产| 欧美激情在线99| 国产中年淑女户外野战色| 老师上课跳d突然被开到最大视频| 国产伦理片在线播放av一区 | 村上凉子中文字幕在线| 在线观看美女被高潮喷水网站| 欧美色视频一区免费| 男女那种视频在线观看| 插逼视频在线观看| 日日撸夜夜添| 亚洲av免费高清在线观看| 国产色婷婷99| 国产亚洲5aaaaa淫片| 日日摸夜夜添夜夜爱| 黄色一级大片看看| 国产蜜桃级精品一区二区三区| 亚洲va在线va天堂va国产| 99热精品在线国产| 99国产极品粉嫩在线观看| 少妇猛男粗大的猛烈进出视频 | 久久久久性生活片| kizo精华| 午夜福利高清视频| 成人特级av手机在线观看| 国产人妻一区二区三区在| 99久久精品一区二区三区| 此物有八面人人有两片| 青春草亚洲视频在线观看| 干丝袜人妻中文字幕| 久久鲁丝午夜福利片| 久久婷婷人人爽人人干人人爱| 插阴视频在线观看视频| av在线老鸭窝| 国产高潮美女av| 国产成人福利小说| 非洲黑人性xxxx精品又粗又长| 欧美+日韩+精品| 国产一区二区在线观看日韩| 六月丁香七月| 国产亚洲av片在线观看秒播厂 | 亚洲欧美成人综合另类久久久 | 日韩成人av中文字幕在线观看| 好男人在线观看高清免费视频| 久久久久久久久久久丰满| 99视频精品全部免费 在线| 国产乱人视频| 综合色丁香网| 国产91av在线免费观看| 夫妻性生交免费视频一级片| 欧美+亚洲+日韩+国产| 久久精品影院6| 免费搜索国产男女视频| 免费在线观看成人毛片| 伊人久久精品亚洲午夜| 美女被艹到高潮喷水动态| 性欧美人与动物交配| 熟妇人妻久久中文字幕3abv| 丰满的人妻完整版| 精品人妻熟女av久视频| 国产探花极品一区二区| 中文精品一卡2卡3卡4更新| 免费看av在线观看网站| 91午夜精品亚洲一区二区三区| 超碰av人人做人人爽久久| 亚洲国产精品国产精品| 悠悠久久av| 一个人免费在线观看电影| 99精品在免费线老司机午夜| 亚洲美女视频黄频| 又爽又黄无遮挡网站| 12—13女人毛片做爰片一| 亚洲欧美日韩卡通动漫| 日韩在线高清观看一区二区三区| 亚洲一级一片aⅴ在线观看| 久久99热6这里只有精品| 寂寞人妻少妇视频99o| 两个人视频免费观看高清| 亚洲av成人精品一区久久| av天堂在线播放| 欧美一区二区亚洲| 欧美成人精品欧美一级黄| av福利片在线观看| 哪里可以看免费的av片| 麻豆精品久久久久久蜜桃| 国产探花极品一区二区| www.色视频.com| 国产淫片久久久久久久久| 久久久欧美国产精品| 少妇被粗大猛烈的视频| 99riav亚洲国产免费| 麻豆一二三区av精品| av免费观看日本| 99riav亚洲国产免费| 国产av麻豆久久久久久久| 少妇人妻一区二区三区视频| 日韩欧美精品免费久久| 亚洲av中文字字幕乱码综合| 少妇被粗大猛烈的视频| 国产淫片久久久久久久久| 欧美+日韩+精品| 国产成人91sexporn| 国产麻豆成人av免费视频| 熟妇人妻久久中文字幕3abv| 高清日韩中文字幕在线| 午夜激情欧美在线| 精品不卡国产一区二区三区| 亚洲精品粉嫩美女一区| 成人鲁丝片一二三区免费| 亚洲av二区三区四区| 我的老师免费观看完整版| 校园人妻丝袜中文字幕| 久久久a久久爽久久v久久| 亚洲精品日韩在线中文字幕 | 成人综合一区亚洲| 欧美日本亚洲视频在线播放| 亚洲aⅴ乱码一区二区在线播放| 老女人水多毛片| 2021天堂中文幕一二区在线观| 18禁裸乳无遮挡免费网站照片| 国产成人午夜福利电影在线观看| 亚洲成人中文字幕在线播放| 久久九九热精品免费| 亚洲,欧美,日韩| 我的女老师完整版在线观看| 日本撒尿小便嘘嘘汇集6| 欧美三级亚洲精品| 欧美日韩乱码在线| 成人美女网站在线观看视频| 欧美+亚洲+日韩+国产| 六月丁香七月| 18+在线观看网站| 国产蜜桃级精品一区二区三区| 久久人人爽人人片av| 一个人看的www免费观看视频| 一级毛片电影观看 | 亚洲aⅴ乱码一区二区在线播放| 啦啦啦韩国在线观看视频| 午夜a级毛片| 国产黄色视频一区二区在线观看 | 免费看光身美女| 久久精品影院6| 亚洲不卡免费看| 国产亚洲5aaaaa淫片| 亚洲av免费高清在线观看| 亚洲最大成人中文| 丰满乱子伦码专区| 国产又黄又爽又无遮挡在线| 亚洲成人精品中文字幕电影| 老司机影院成人| 大香蕉久久网| 最近视频中文字幕2019在线8| 国内精品久久久久精免费| 亚洲成人中文字幕在线播放| 美女 人体艺术 gogo| 国产午夜福利久久久久久| 丝袜美腿在线中文| 亚洲国产日韩欧美精品在线观看| 日韩三级伦理在线观看| 91精品国产九色| 亚洲av中文av极速乱| 久久久色成人| 国产精品一区二区在线观看99 | 久久久国产成人精品二区| 边亲边吃奶的免费视频| 亚洲国产精品成人综合色| 欧美日韩国产亚洲二区| 哪里可以看免费的av片| 精品久久久久久久久久久久久| 久久国产乱子免费精品| 热99re8久久精品国产| 亚洲第一区二区三区不卡| 综合色丁香网| 国产在线男女| 又粗又硬又长又爽又黄的视频 | 国语自产精品视频在线第100页| 青春草亚洲视频在线观看| 国产精品电影一区二区三区| 99热只有精品国产| 人妻系列 视频| 男的添女的下面高潮视频| 亚洲欧美清纯卡通| 波野结衣二区三区在线| 亚洲av一区综合| 国产成人午夜福利电影在线观看| 中文精品一卡2卡3卡4更新| 真实男女啪啪啪动态图| 久久精品国产自在天天线| 国产单亲对白刺激| 日韩av不卡免费在线播放| 日本一二三区视频观看| 欧美激情久久久久久爽电影| 美女xxoo啪啪120秒动态图| av专区在线播放| 亚洲av成人av| 日本三级黄在线观看| 日韩中字成人| 成人二区视频| 国内少妇人妻偷人精品xxx网站| 插阴视频在线观看视频| 婷婷精品国产亚洲av| 国产黄片视频在线免费观看| 久久久久久久久中文| 国产高清视频在线观看网站| 中文字幕人妻熟人妻熟丝袜美| 欧美bdsm另类| 一级av片app| 日产精品乱码卡一卡2卡三| 精品无人区乱码1区二区| 国产乱人视频| 国产成人精品一,二区 | 热99re8久久精品国产| 三级国产精品欧美在线观看| 国内精品宾馆在线| 亚洲国产高清在线一区二区三| 一边摸一边抽搐一进一小说| 伦理电影大哥的女人|