• <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 ?

    Solving Fractional Integro-Differential Equations by Using Sumudu Transform Method and Hermite Spectral Collocation Method

    2018-03-13 02:02:26AmerMahdyandYoussef
    Computers Materials&Continua 2018年2期

    Y. A. Amer, A. M. S. Mahdy, and E. S. M. Youssef

    1 Introduction

    A lot of problems can be modeled by fractional integro-differential equations from various sciences and engineering applications. In addition to the fact that many problems cannot be found analytical solutions to them and therefore, once you get a solution is a result of a good result solutions, using numerical methods, will be very helpful. Recently, several numerical methods to solve fractional integro-differential equations (FIDEs) [Zedan,Tantawy, Sayed et al. (2017); Oyedepo, Taiwo, Abubakar et al. (2016); Wang and Zhu(2017)] have been given. Since the example collocation method for solving the nonlinear fractional Langevin equation [Bhrawy and Alghamdi (2012); Yang, Chen and Huang(2014)]. A Chebyshev polynomials method is introduced in Bhrawy et al. [Bhrawy and Alofi(2013)], Doha et al. [Doha, Bhrawy and Ezz-Eldien (2011)], Irandoust-pakchin et al.[Irandoust-pakchin, Kheiri and Abdi-mazraeh (2013)] for solving multiterm fractional orders differential equations and nonlinear Volterra and Fredholm Integro-differential equations of fractional order. The authors in Rathore et al. [Rathore, Kumar, Singh et al.(2012)] applied variational iteration method for solving fractional Integro-differential equations with the nonlocal boundary conditions and more methods in Wang et al. [Wang,Han and Xie (2012)], Lin et al. [Lin, Gu and Young (2010)].

    In this paper Sumudu transform method [Wang, Han and Xie (2012); Lin, Gu and Young(2010); Singh and Kumar (2011); Ganji (2006); Hashim, Chowdhurly and Mawa (2008);He (1999); Liao (2005); Amer, Mahdy and Youssef (2017)] and Hermite spectral collocation method [Andrews (1985); Solouma and Khader (2016); Bagherpoorfard and Ghassabzade (2013)]; Bojdi, Ahmadi-Asl and Aminataei (2013); Brill (2002); Bialecki(1993); Dyksen and Lynch (2000); He (1999)] is applied to solving fractional integro differential equations.

    In this paper, we are concerned with the numerical solution of the following linear fractional integro-differential equation [Bhrawy and Alofi(2013); Doha, Bhrawy and Ezz-Eldien (2011); Irandoust-pakchin, Kheiri and Abdi-mazraeh (2013); Mohammed (2014)]:

    with initial conditions:

    2 Basic definitions of fractional calculus

    In this section, we present the basic definitions and properties of the fractional calculus theory, which are used further in this paper

    Definition 1:A real functionis said to be in the spaceif there exists a real numbersuch thatwhereand it is said to be in the spaceif

    Definition 2:The Caputo fractional derivative operatorof orderis defined in the following form [El-Sayed and Salman (2013); El-Sayed and Salman (2013); Elsadany and Matouk (2015)]:

    Definition 3:The Sumudu transform is defined over the set of functions [Singh and Kumar(2011); Ganji (2006)]

    by the following formula:

    where

    Some special properties of the sumudu transform are as follows [Belgacem and Karaballi(2006)]:

    Definition 4:The Sumudu transform of Caputo fractional derivative is defined as follows[Amer, Mahdy and Youssef (2017); Belgacem and Karaballi (2006)]:

    Theorem:[Singh and Kumar (2011); Amer, Mahdy and Youssef (2017)]

    This theorem is very important to calculate approximate solution of the problems and for more details in Singh et al. [Singh and Kumar (2011)], Amer et al. [Amer, Mahdy and Youssef (2017)]

    Definition 5:The Hermite polynomials are given by Andrews [Andrews (1985)], Solouma et al. [Solouma and Khader (2016)], Bagherpoorfard et al. [Bagherpoorfard and Ghassabzade (2013)], Bojdi et al. [Bojdi, Ahmadi-Asl and Aminataei (2013)], Brill [Brill(2002)], Bialecki [Bialecki (1993)], Dyksen et al. [Dyksen and Lynch (2000)], He [He(1999)]:

    A lot of the properties of these polynomials are:

    The Hermite polynomials evaluated at zero argumentand are have called Hermite number as follows: [Andrews (1985); Solouma and Khader (2016)]

    3 The homotopy perturbation sumudu transform method

    In order to elucidate the solution procedure of this method, we consider a general fractional nonlinear differential equation of the form [Singh and Kumar (2011); Ganji (2006);Hashim, Chowdhurly and Mawa (2008); He (1999); Liao (2005); Amer, Mahdy and Youssef (2017)]:

    Applying the Sumudu transform (denoted throughout this paper by) on both sides of Eq.(11), we have

    Using the property of the Sumudu transform and the initial conditions in Eq. (12), we have

    Operating with the Sumudu inverse on both sides of Eq. (13) we get

    for some Adomian’s polynomials, which can be calculated with the formula [Ghorbani(2009); Jafari and Daftardar-Gejji (2006)]

    Substituting Eq. (15) and (17) in Eq. (14), we get

    Equating the terms with identical powers of, we can obtain a series of equations as the follows:

    Finally, we approximate the analytical solutionby truncated series as

    4 Basic idea of hermite collocation method

    In this section the Hermite collocation method is applied to study the numerical solution of the fractional Integro-differential (1).

    This method is based on approximating the unknown functionas

    At first by Substituting (21) into (1) we obtain

    Hence the residual equation is defined as:

    By evaluating the above equation forwe can obtain a system of (n+1)linear equations with (n+1) unknown coefficients, after calculate the coefficientwe substitute in Eq. (21) then we get the solution of)

    5 Applications

    In this section, to illustrate the method and to show the ability of the method two examples are presented.

    Example (1):Cosider the fractional integro-differential equations as

    (i)First by using Sumudu transform method

    By taking the Sumudu transform on both sides of Eq. (28), thus we get

    Using the property of the Sumudu transform and the initial condition in Eq. (30), we have

    Operating with the Sumudu inverse on both sides of Eq. (31) we get

    By substituting Eq. (33) in Eq. (32) we have

    The few components of the Adomian polynomials are given as follows:

    Then we have

    Figure 1: The behavior of y(x) by HPSTM

    (ii)By sing Hermite spectral collocation method

    First By assuming the approximate of the solution ofwith m=2 as:

    Second by Substituting (36) into (28) we obtain

    Hence the residual equation is defined as:

    Second let

    The minimum value of S is obtained by setting

    By applying (42) in (41) we have:

    From the initial condhtion y(0)=1 and from Eq. (7) we get

    By solving the Eq. (43)-(45) we get the values ofand substituting in Eq.(36) we get the solution as series:

    Figure 2: The behavior of y(x) by Hermite collocation method

    It is no doubt that the efficiency of this approach is greatly enhanced by the calculation further terms of yby using by using Sumudu transform method and Hermite spectral collocation method.As shown in Fig. 1 and Fig. 2.

    Example (2):Consider the systems of fractional integro-differential type as :

    By using the properities of Gamma function of the two Eq. (47), (48) become

    (i)First by using Sumudu transform method

    By taking the Sumudu transform on both sides of Eq. (50), thus we get

    Using the property of the Sumudu transform and the initial condition in Eq. (49), we have

    Operating with the Sumudu inverse on both sides of Eq. (52) we get

    By assuming that

    By substituting Eq. (54) in Eq. (53) we have

    Figure 3: The behavior of u(x) by HPSTM

    Figure 4: The behavior of v(x) by HPSTM

    (ii)By sing Hermite spectral collocation method

    First By assuming the approximate of the solution ofwith m=2 as:

    Second by Substituting (57) into (50) we obtain

    Hence the residual equation is defined as:

    The minimum value of S is obtained by setting

    Figure 5: The behavior of u(x) by Hermite collocation method

    Figure 6: The behavior of v(x) by Hermite collocation method

    It is no doubt that the efficiency of this approach is greatly enhanced by the calculation further terms of uby using by using Sumudu transform method and Hermite spectral collocation method. As In Fig. 3 and Fig. 4 show the The behavior of uby using Sumudu transform method and in Fig. 5 and Fig. 6. show the The behavior of uby using the Hermite collocation method.

    6 Conclusions

    The main aim of this paper is to know that the sumud transform method and Hermite spectral collocation method are of the most important and simplest methods used in solving linear and nonlinear differential equations. This method have been successfully applied to systems of fractional integro-differential equations.in this method we do not need to do the difficult computation for finding the Adomian polynomials. Generally speaking, the proposed method is promising and applicable to a broad class of linear and nonlinear problems in the theory of fractional calculus.

    Agarwal, R. P.; El-Sayed, A. M. A.; Salman, S. M.(2013): Fractional-order Chua’s system: discretization, bifurcation and chaos.Advances in Difference Equations, vol. 2013,pp. 320.

    Amer, Y. A.; Mahdy, A. M. S.; Youssef, E. S. M.(2017): Solving systems of fractional differential equations using sumudu transform method.Asian Research Journal of Mathematics, vol. 7, no. 2, pp. 1-15.

    Andrews, L. C.(1985):Special functions For engineers and applied mathematical.Macmillan publishing company, New York.

    Bagherpoorfard, M.; Ghassabzade, F. A.(2013): Hermite matrix polynomial collocation method for linear complex differential equations and some comparisons.Journal of Applied Mathematics and Physics, vol. 1, pp. 58-64.

    Belgacem,F. B. M.; Karaballi, A. A.(2006): Sumudu transform fundamental properties in vestigations and applications.Journal of Applied Mathematics and Stochastic Analysis,vol. 2006, pp 1-23, doi:10.1155/JAMSA/2006/910832005.

    Bhrawy, A. H.; Alghamdi, M. A.(2012): A shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals.Boundary Value Problems, vol. 2012, pp. 62.

    Bhrawy, A. H.; Alofi, A. S.(2013): The operational matrix of fractional integration for shifted Chebyshev polynomials.Applied Mathematics Letters, vol. 26, no. 1, pp. 25-31.

    Bialecki, B.(1993): A fast domain decomposition poisson solver on a rectangle for Hermite bicubic orthogonal spline collocation.Siam Journal Numerical Analysis, vol. 30,pp. 425-434.

    Bojdi, Z. K.; Ahmadi-Asl, S.; Aminataei, A.(2013): Operational matrices with respect to Hermite polynomials and their applications in solving linear differential equations with variable coeffcients.Journal of Linear and Topological Algebra, vol. 2, no. 2, pp. 91-103.

    Brill, S. H.(2002): Analytic solution of Hermite collocation discretization of the steady state convection-diffusion equation.International Journal of Differential Equations and Applications, vol. 4, no. 2, pp. 141-155.

    Doha, E. H.; Bhrawy, A. H.; Ezz-Eldien,S. S.(2011): Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations.Applied Mathematical Modelling, vol. 35, no. 12, pp. 5662-5672.

    Dyksen, W. R. ; Lynch, R. E.(2000): A new decoupling technique for the Hermite cubic collocation equations arising from boundary value problems.Mathematics and Computers in Simulation, vol. 54, pp. 359-372.

    Elsadany, A. A.; Matouk, A. E.(2015): Dynamical behaviors of fractional-order Lotka-Voltera predator-prey model and its discretization.Applied Mathematics and Computation,vol. 49, pp. 269-283.

    El-Sayed, A. M. A.; Salman, S. M.(2013): On a discretization process of fractional order Riccati’s differential equation.Journal of Fractional Calculus and Applications, vol. 4, no.2, pp. 251-259.

    Funaro, D.(1992):Polynomial approximations of differential equations. Springer-Verlag.Ganji, D.(2006): The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer.Physics Letters A, vol. 355, pp. 337-341.

    Ghorbani, A.(2009): Beyond, Adomian polynomials: He polynomials.Chaos Solitons &Fractals, vol. 39, no. 3, pp. 1486-1492.

    Hashim, I.; Chowdhurly, M.; Mawa, S.(2008): On multistage homotopy perturbation method applied to nonlinear biochemical reaction model.Chaos, Solitons & Fractals, vol.36, pp. 823-827.

    He, J.(1999): Homotopy perturbation technique.Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257-262.

    He, J.(1999): Homotopy perturbation technique.Computer Methods in Applied Mechanics and Engineering, vol. 178, no. 3-4, pp. 257-262.

    Irandoust-pakchin, S.; Kheiri, H.; Abdi-mazraeh, S.(2013): Chebyshev cardinal functions: an effective tool for solving nonlinear Volterra and Fredholm integro differential equations of fractional order.Iranian Journal of Science and Technology Transaction A: Science, vol. 37, no. 1, pp. 53-62.

    Jafari, H.; Daftardar-Gejji, V.(2006): Solving a system of nonlinear fractional differential equations using Adomian decomposition.Journal of Computational and Applied, vol. 196, no. 2, pp. 644-651.

    Liao, S.(2005): Comparison between the homotopy analysis method and homotopy perturbation method.AppliedMathematics and Computation, vol. 169, pp. 1186-1194.

    Lin,C. Y.; Gu, M. H.; Young, D. L.(2010): The time-marching method of fundamental solutions for multi-dimensional telegraph equations.Computers, Materials & Continua,vol. 18, no. 1, pp. 43-68.

    Mohammed,D. S.(2014): Numerical solution of fractional integro-differential equations by least squares method and shifted chebyshev polynomial.Mathematical Problems in Engineering,vol. 2014.

    Oyedepo, T.; Taiwo, O. A.; Abubakar, J. U.; Ogunwobi, Z. O.(2016): Numerical studies for solving fractional integro-differential equations by using least squares method and bernstein polynomials.Fluid Mechanics: Open Access, vol. 3, no. 3.

    Rathore, S.; Kumar, D.; Singh, J.; Gupta, S.(2012): Homotopy analysis sumudu transform method for nonlinear equations.International Journal of Industrial Mathematics,vol. 4, no. 4, pp. 301-314.

    Singh, J.; Kumar, D.(2011): Homotopy perturbation sumudu transform method for nonlinear equations.Advances in Applied Mathematics and Mechanics,vol. 4, no. 4, pp. 165-175.

    Solouma, E. M.; Khader, M. M.(2016): Analytical and numerical simulation for solving the system of non-linear fractional dynamical model of marriage.International Mathematical Forum, vol. 11, no. 8, pp. 875-884.

    Wang, L.; Han, X.; Xie, Y.(2012): A new iterative regularization Method for solving the dynamic load identification problem.Computers, Materials & Continua, vol. 31, no. 2,pp.113-126.

    Wang, Y.; Zhu, L.(2017): Solving nonlinear Volterra integro-differential equations of fractional order by using Euler wavelet method.Advances in Difference Equations, doi:10.1186/s13662-017-1085-6.

    Yang, Y.; Chen, Y.; Huang, Y.(2014): Spectral-collocation method for fractional Fredholm integro-differential equations.Journal of the Korean Mathematical Society, vol.51, no. 1, pp. 203-224.

    Zedan, H. A.; Tantawy, S. S.; Sayed, Y. M.(2017): New solutions for system of fractional integro-differential equations and Abel’s integral equations by chebyshev spectral method.Mathematical Problems in Engineering, vol. 2017.

    亚洲欧美成人精品一区二区| 久久精品国产亚洲av香蕉五月| 欧美一区二区精品小视频在线| 91午夜精品亚洲一区二区三区| 在线观看一区二区三区| 女生性感内裤真人,穿戴方法视频| videossex国产| 晚上一个人看的免费电影| 青春草视频在线免费观看| 亚洲色图av天堂| 亚洲真实伦在线观看| 精品国产三级普通话版| 十八禁网站免费在线| 久久欧美精品欧美久久欧美| 伊人久久精品亚洲午夜| 男人狂女人下面高潮的视频| 高清日韩中文字幕在线| 欧美日韩国产亚洲二区| 成人无遮挡网站| 日本免费一区二区三区高清不卡| 国产精品日韩av在线免费观看| 国产av在哪里看| av中文乱码字幕在线| 国产三级在线视频| 欧美日韩精品成人综合77777| 少妇丰满av| 欧美性猛交╳xxx乱大交人| 日韩亚洲欧美综合| 大又大粗又爽又黄少妇毛片口| 白带黄色成豆腐渣| 九九久久精品国产亚洲av麻豆| 深爱激情五月婷婷| 乱系列少妇在线播放| 国产色婷婷99| 日韩一本色道免费dvd| 国产精品久久久久久精品电影| 亚洲av第一区精品v没综合| 在线看三级毛片| 看片在线看免费视频| 精品一区二区三区视频在线观看免费| 亚州av有码| 俺也久久电影网| 精品久久久久久久久久久久久| 人妻制服诱惑在线中文字幕| 成人国产麻豆网| 男女之事视频高清在线观看| 精品午夜福利视频在线观看一区| 久久久久精品国产欧美久久久| 秋霞在线观看毛片| av在线天堂中文字幕| 亚洲精品亚洲一区二区| 午夜影院日韩av| 午夜免费男女啪啪视频观看 | 最新中文字幕久久久久| av女优亚洲男人天堂| 一级毛片我不卡| av在线天堂中文字幕| 国产精品爽爽va在线观看网站| 亚洲性夜色夜夜综合| 男女之事视频高清在线观看| 亚洲婷婷狠狠爱综合网| 麻豆成人午夜福利视频| 毛片女人毛片| 美女cb高潮喷水在线观看| 亚洲av二区三区四区| 中文字幕久久专区| 国产伦精品一区二区三区四那| 性插视频无遮挡在线免费观看| 久久久精品94久久精品| 国产精品一区二区三区四区久久| 国产不卡一卡二| 搡老妇女老女人老熟妇| 久久久久国内视频| 午夜精品在线福利| 久久九九热精品免费| 久久韩国三级中文字幕| 女人十人毛片免费观看3o分钟| 少妇丰满av| 日本黄色片子视频| 舔av片在线| 深夜精品福利| 97超碰精品成人国产| 一区二区三区免费毛片| 国产三级中文精品| 免费av观看视频| 国产探花在线观看一区二区| 大又大粗又爽又黄少妇毛片口| 直男gayav资源| 成人三级黄色视频| 草草在线视频免费看| 亚洲av美国av| 色综合站精品国产| 综合色丁香网| 日韩欧美免费精品| 免费人成视频x8x8入口观看| 禁无遮挡网站| 成年免费大片在线观看| 男人狂女人下面高潮的视频| 黄色欧美视频在线观看| 国产毛片a区久久久久| 日韩精品青青久久久久久| 99国产极品粉嫩在线观看| 日韩 亚洲 欧美在线| 国产一区二区激情短视频| 久久精品久久久久久噜噜老黄 | av国产免费在线观看| 国产成人freesex在线 | a级毛片a级免费在线| 国产精品一及| 一进一出好大好爽视频| 日韩三级伦理在线观看| 国产精品女同一区二区软件| 久久久久国内视频| 欧美丝袜亚洲另类| 女的被弄到高潮叫床怎么办| 99久久成人亚洲精品观看| 美女免费视频网站| 成年女人永久免费观看视频| 久久久久久久久久黄片| 嫩草影院新地址| 国产成人aa在线观看| 久久久久久久亚洲中文字幕| 国产午夜精品论理片| 国产毛片a区久久久久| 国产女主播在线喷水免费视频网站 | 亚洲最大成人中文| 午夜视频国产福利| 精品久久久久久久久久免费视频| 亚洲在线自拍视频| 麻豆一二三区av精品| 亚洲色图av天堂| 日韩精品青青久久久久久| av免费在线看不卡| 免费观看人在逋| 成年免费大片在线观看| 插逼视频在线观看| 国产成人福利小说| 国产69精品久久久久777片| 人妻制服诱惑在线中文字幕| 五月玫瑰六月丁香| 亚洲精品456在线播放app| 国产黄片美女视频| 精品久久久久久久久av| 中文字幕精品亚洲无线码一区| 亚洲性久久影院| 精华霜和精华液先用哪个| 亚洲国产精品成人综合色| 乱码一卡2卡4卡精品| 欧美日韩综合久久久久久| 97超视频在线观看视频| 在线看三级毛片| 久久久久国产网址| 一进一出好大好爽视频| 在线免费十八禁| 日韩欧美在线乱码| 热99在线观看视频| 久久精品夜色国产| 白带黄色成豆腐渣| 国产精品日韩av在线免费观看| 麻豆av噜噜一区二区三区| 色综合色国产| 久久人人爽人人爽人人片va| 99热全是精品| 亚洲欧美日韩东京热| 国产精品久久久久久精品电影| 亚洲欧美日韩高清专用| 国产精品国产高清国产av| 美女黄网站色视频| 97超级碰碰碰精品色视频在线观看| 一个人看视频在线观看www免费| 搡女人真爽免费视频火全软件 | 看黄色毛片网站| 97人妻精品一区二区三区麻豆| 毛片一级片免费看久久久久| 国产激情偷乱视频一区二区| 草草在线视频免费看| 99久久无色码亚洲精品果冻| 亚洲av不卡在线观看| 久久久精品大字幕| 欧美一区二区国产精品久久精品| 高清日韩中文字幕在线| 久久人人爽人人爽人人片va| av在线天堂中文字幕| 男人的好看免费观看在线视频| 色综合站精品国产| 亚洲欧美成人综合另类久久久 | 欧美成人一区二区免费高清观看| 黄色视频,在线免费观看| 十八禁网站免费在线| 超碰av人人做人人爽久久| 色播亚洲综合网| 99久国产av精品国产电影| 一级av片app| 免费一级毛片在线播放高清视频| av福利片在线观看| 无遮挡黄片免费观看| 中国国产av一级| 在线看三级毛片| 国产白丝娇喘喷水9色精品| 久久精品影院6| 最近的中文字幕免费完整| 日日摸夜夜添夜夜添小说| av福利片在线观看| 热99在线观看视频| 免费人成视频x8x8入口观看| 国产单亲对白刺激| 亚洲最大成人中文| 日本与韩国留学比较| a级毛片免费高清观看在线播放| 99riav亚洲国产免费| 直男gayav资源| 九九在线视频观看精品| 日日撸夜夜添| av视频在线观看入口| 日韩欧美精品免费久久| 99热6这里只有精品| 精品久久久噜噜| 亚洲国产高清在线一区二区三| 久久婷婷人人爽人人干人人爱| 乱人视频在线观看| 毛片女人毛片| 午夜精品国产一区二区电影 | 精品午夜福利在线看| 亚洲精品国产av成人精品 | 成年女人永久免费观看视频| 一进一出抽搐动态| 美女被艹到高潮喷水动态| 久久精品国产鲁丝片午夜精品| 国产不卡一卡二| 欧美日韩一区二区视频在线观看视频在线 | 久久久欧美国产精品| 亚洲av二区三区四区| 蜜臀久久99精品久久宅男| 欧美性猛交黑人性爽| 99热6这里只有精品| 成人美女网站在线观看视频| 国产极品精品免费视频能看的| 欧美bdsm另类| 少妇猛男粗大的猛烈进出视频 | 嫩草影院入口| 亚洲不卡免费看| 91在线观看av| 精品久久久久久久久亚洲| 性色avwww在线观看| 中文字幕人妻熟人妻熟丝袜美| 国产精品一区二区三区四区免费观看 | 久久精品夜夜夜夜夜久久蜜豆| 在线观看午夜福利视频| 亚洲精品乱码久久久v下载方式| 日韩精品中文字幕看吧| 中文字幕av成人在线电影| 一夜夜www| 女的被弄到高潮叫床怎么办| 欧美激情久久久久久爽电影| 亚洲美女视频黄频| 性插视频无遮挡在线免费观看| 国产精品永久免费网站| 亚洲色图av天堂| 麻豆国产97在线/欧美| 99久久精品国产国产毛片| 成人精品一区二区免费| 波野结衣二区三区在线| 十八禁网站免费在线| 亚洲成人久久爱视频| 人妻丰满熟妇av一区二区三区| 免费观看在线日韩| 丰满乱子伦码专区| 久久精品国产亚洲网站| 中文字幕久久专区| 亚洲欧美日韩无卡精品| 免费观看人在逋| 又爽又黄无遮挡网站| 免费一级毛片在线播放高清视频| 精品人妻视频免费看| 听说在线观看完整版免费高清| 国产中年淑女户外野战色| 乱码一卡2卡4卡精品| 99国产极品粉嫩在线观看| 搡老妇女老女人老熟妇| 国产精品永久免费网站| 欧美日韩一区二区视频在线观看视频在线 | 欧美成人a在线观看| 国产av一区在线观看免费| 又黄又爽又免费观看的视频| 色在线成人网| 中文字幕熟女人妻在线| 别揉我奶头~嗯~啊~动态视频| av专区在线播放| 精品人妻偷拍中文字幕| 噜噜噜噜噜久久久久久91| 亚洲久久久久久中文字幕| 国产爱豆传媒在线观看| 日韩欧美一区二区三区在线观看| 天天躁日日操中文字幕| 午夜爱爱视频在线播放| 天堂av国产一区二区熟女人妻| av天堂在线播放| 亚洲国产欧美人成| 国产精品久久电影中文字幕| 国产老妇女一区| 国国产精品蜜臀av免费| 久久精品国产99精品国产亚洲性色| 看黄色毛片网站| 久久精品人妻少妇| 美女黄网站色视频| 秋霞在线观看毛片| 少妇熟女aⅴ在线视频| 亚洲美女黄片视频| 亚洲aⅴ乱码一区二区在线播放| 中文字幕免费在线视频6| 国产午夜福利久久久久久| 欧美精品国产亚洲| a级毛色黄片| 欧美成人精品欧美一级黄| 国产久久久一区二区三区| 国产av一区在线观看免费| 国产高潮美女av| 丰满人妻一区二区三区视频av| 最近的中文字幕免费完整| 国产又黄又爽又无遮挡在线| 网址你懂的国产日韩在线| 亚洲熟妇熟女久久| 国产人妻一区二区三区在| 淫妇啪啪啪对白视频| 神马国产精品三级电影在线观看| a级毛片a级免费在线| 别揉我奶头 嗯啊视频| 欧美日韩综合久久久久久| 日韩欧美三级三区| 干丝袜人妻中文字幕| 91狼人影院| 亚洲综合色惰| 久99久视频精品免费| 人妻制服诱惑在线中文字幕| 亚洲欧美成人综合另类久久久 | 日韩欧美三级三区| 亚洲一区高清亚洲精品| 国产午夜福利久久久久久| 偷拍熟女少妇极品色| 久久精品国产鲁丝片午夜精品| 久久欧美精品欧美久久欧美| 亚洲四区av| 国产精品野战在线观看| 国产 一区精品| 欧美色视频一区免费| 久久久久国产精品人妻aⅴ院| 久久久久久久午夜电影| 又爽又黄a免费视频| 欧美日本亚洲视频在线播放| 免费大片18禁| 久久精品国产鲁丝片午夜精品| 在线国产一区二区在线| 热99在线观看视频| 国产精品嫩草影院av在线观看| 日韩大尺度精品在线看网址| 亚洲精品日韩在线中文字幕 | 欧美色视频一区免费| 欧美不卡视频在线免费观看| 美女黄网站色视频| 成年免费大片在线观看| 精品一区二区三区视频在线观看免费| 欧美性感艳星| 天堂影院成人在线观看| 国产免费一级a男人的天堂| 91久久精品国产一区二区成人| 日本一本二区三区精品| ponron亚洲| 欧美成人a在线观看| 久久久久久久久中文| 久久亚洲精品不卡| 最近的中文字幕免费完整| 天堂av国产一区二区熟女人妻| 少妇人妻一区二区三区视频| 亚洲最大成人手机在线| 成人永久免费在线观看视频| 久久这里只有精品中国| 别揉我奶头~嗯~啊~动态视频| 日韩,欧美,国产一区二区三区 | 高清毛片免费看| 久久精品人妻少妇| 精品欧美国产一区二区三| 十八禁国产超污无遮挡网站| 欧美日本亚洲视频在线播放| 在线观看美女被高潮喷水网站| 免费看日本二区| 国产精品久久久久久亚洲av鲁大| 中国国产av一级| 老师上课跳d突然被开到最大视频| 午夜福利在线在线| 欧美性感艳星| 中出人妻视频一区二区| 国产极品精品免费视频能看的| 天美传媒精品一区二区| 人妻丰满熟妇av一区二区三区| 热99在线观看视频| 久久久久性生活片| 亚洲成人av在线免费| 69人妻影院| 中文字幕人妻熟人妻熟丝袜美| 国产精品久久久久久亚洲av鲁大| 一区二区三区免费毛片| 亚洲性久久影院| 国产成人91sexporn| 精品一区二区三区视频在线| 亚洲,欧美,日韩| 国产亚洲91精品色在线| 美女高潮的动态| 国产精品一区二区性色av| av视频在线观看入口| 国产伦精品一区二区三区四那| 亚洲综合色惰| 久久草成人影院| 亚洲欧美精品自产自拍| 日韩成人av中文字幕在线观看 | 国产高清视频在线观看网站| 乱码一卡2卡4卡精品| 久久精品久久久久久噜噜老黄 | 国产久久久一区二区三区| 老熟妇仑乱视频hdxx| 午夜福利在线观看吧| 国产伦一二天堂av在线观看| 精品国产三级普通话版| 亚洲最大成人手机在线| 国产精品不卡视频一区二区| 久久中文看片网| 成年免费大片在线观看| 久久精品91蜜桃| 亚洲欧美日韩卡通动漫| 看黄色毛片网站| 免费不卡的大黄色大毛片视频在线观看 | 熟女人妻精品中文字幕| 亚洲经典国产精华液单| 欧美性感艳星| 免费人成在线观看视频色| 久久中文看片网| 老司机福利观看| 日韩人妻高清精品专区| 一区二区三区免费毛片| 日本熟妇午夜| 丰满乱子伦码专区| 99热这里只有是精品50| 欧美bdsm另类| 在线看三级毛片| 麻豆久久精品国产亚洲av| 女人十人毛片免费观看3o分钟| 看黄色毛片网站| 亚洲内射少妇av| 国产大屁股一区二区在线视频| 国产午夜精品久久久久久一区二区三区 | 精品一区二区三区视频在线| 亚洲av二区三区四区| 免费观看人在逋| 亚洲精华国产精华液的使用体验 | 欧美性猛交黑人性爽| 亚洲av成人精品一区久久| avwww免费| 国产69精品久久久久777片| 综合色av麻豆| 久久久久久久久久久丰满| 亚洲自拍偷在线| 热99re8久久精品国产| 国产精品一区二区性色av| 性插视频无遮挡在线免费观看| 国产精品乱码一区二三区的特点| 少妇的逼好多水| 丝袜喷水一区| 日韩精品有码人妻一区| 国产美女午夜福利| 寂寞人妻少妇视频99o| 99久久无色码亚洲精品果冻| 国产精品99久久久久久久久| 精品久久国产蜜桃| 丝袜喷水一区| 精品一区二区免费观看| 晚上一个人看的免费电影| 欧美+亚洲+日韩+国产| 日韩亚洲欧美综合| 插逼视频在线观看| 性插视频无遮挡在线免费观看| 内射极品少妇av片p| 国产精品爽爽va在线观看网站| 欧美一级a爱片免费观看看| 一进一出好大好爽视频| 欧美绝顶高潮抽搐喷水| 国产成人福利小说| 人妻丰满熟妇av一区二区三区| 99热只有精品国产| 久久精品国产鲁丝片午夜精品| 久久亚洲精品不卡| 内地一区二区视频在线| 一a级毛片在线观看| 久久久国产成人精品二区| 神马国产精品三级电影在线观看| 国产精品一区www在线观看| 免费在线观看影片大全网站| 亚洲精品国产av成人精品 | 亚洲婷婷狠狠爱综合网| 白带黄色成豆腐渣| 18禁在线播放成人免费| 看片在线看免费视频| 国产欧美日韩精品亚洲av| 男女边吃奶边做爰视频| 99热这里只有是精品在线观看| 久久热精品热| 别揉我奶头~嗯~啊~动态视频| 三级经典国产精品| 一个人免费在线观看电影| 91在线观看av| 有码 亚洲区| 国产综合懂色| 你懂的网址亚洲精品在线观看 | 国产真实乱freesex| 黄色日韩在线| 日韩成人av中文字幕在线观看 | 成人特级av手机在线观看| 欧美一级a爱片免费观看看| www.色视频.com| 精品99又大又爽又粗少妇毛片| 国产精品福利在线免费观看| 能在线免费观看的黄片| 国产精品一区二区性色av| 日本一二三区视频观看| 一个人看的www免费观看视频| 两个人视频免费观看高清| 色5月婷婷丁香| 我要搜黄色片| 久久精品国产亚洲av涩爱 | 免费观看的影片在线观看| 亚洲国产高清在线一区二区三| 亚洲精品国产av成人精品 | 日韩亚洲欧美综合| 精品一区二区三区人妻视频| 春色校园在线视频观看| 99视频精品全部免费 在线| 久久久久九九精品影院| 中文亚洲av片在线观看爽| 一本久久中文字幕| 欧美三级亚洲精品| 国内精品宾馆在线| 色哟哟哟哟哟哟| 在现免费观看毛片| 精品熟女少妇av免费看| 春色校园在线视频观看| 永久网站在线| 久久久国产成人免费| 国产爱豆传媒在线观看| 日韩 亚洲 欧美在线| 此物有八面人人有两片| 少妇人妻精品综合一区二区 | 插逼视频在线观看| 春色校园在线视频观看| 国内精品美女久久久久久| 午夜精品在线福利| 91久久精品电影网| 中文字幕精品亚洲无线码一区| 久久欧美精品欧美久久欧美| 99热这里只有是精品在线观看| 国产 一区精品| 一级av片app| 国产三级在线视频| 人妻久久中文字幕网| 久久精品夜夜夜夜夜久久蜜豆| 尤物成人国产欧美一区二区三区| 日韩欧美三级三区| av专区在线播放| 舔av片在线| 又黄又爽又免费观看的视频| 成人无遮挡网站| 你懂的网址亚洲精品在线观看 | 国产精品一及| 日韩大尺度精品在线看网址| 亚洲精品国产成人久久av| 成人永久免费在线观看视频| 丝袜美腿在线中文| 高清午夜精品一区二区三区 | 欧美丝袜亚洲另类| 深夜a级毛片| 色尼玛亚洲综合影院| 国产精品人妻久久久影院| 中文字幕av在线有码专区| 搡老妇女老女人老熟妇| 在线观看66精品国产| 亚洲欧美日韩高清专用| 婷婷精品国产亚洲av在线| 成年女人毛片免费观看观看9| 少妇被粗大猛烈的视频| 国产又黄又爽又无遮挡在线| 亚洲国产欧美人成| 国产成人freesex在线 | 在线观看免费视频日本深夜| 亚洲最大成人手机在线| 午夜福利在线在线| 免费黄网站久久成人精品| 香蕉av资源在线| 亚洲三级黄色毛片| 欧美性猛交╳xxx乱大交人| 黄色配什么色好看| 给我免费播放毛片高清在线观看| 免费在线观看影片大全网站| 99热这里只有是精品在线观看| 日韩精品有码人妻一区| 精品久久久噜噜| 国产精品女同一区二区软件| 亚洲精品一卡2卡三卡4卡5卡| 麻豆成人午夜福利视频| 国产精品无大码| 国产在视频线在精品| 国国产精品蜜臀av免费| 亚洲欧美日韩东京热| 午夜免费激情av| 国产三级中文精品| 人妻久久中文字幕网| 亚洲自偷自拍三级| av在线亚洲专区| 欧美xxxx黑人xx丫x性爽|