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Zaman Değişkeninde Kesirli Türev İçeren Navier-Stokes Denklemlerinin Sayısal Çözümü

Year 2018, Volume: 39 Issue: 4, 900 - 911, 24.12.2018

İnci Çilingir Süngü , Hüseyin Demir

https://doi.org/10.17776/csj.384509
Cited By: 3

Abstract

Bu çalışmada zaman değişkenine göre kesirli türev
içeren Navier-Stokes denklemleri çözülmüştür. Denklemlerin çözümünde
genelleştirilmiş diferansiyel dönüşüm ve sonlu fark metotları beraber farklı
alt aralıklara bölünerek çok adımlı olarak kullanılmıştır. Bu melezleme ile
sonlu fark metodunun kararlılık özelliği ve diferansiyel dönüşüm metodunun
uygulama kolaylığı özelliklerinin birleştirilmesi amaçlanmıştır. Ele alınan
örneklerde karmaşık hesaplamaların getirdiği işlem yükünün azaldığı ve çok
boyutlu problemlerde ise başlangıç koşulu nedeniyle oluşan süreksizliğin
aşılabildiği görülmüştür. Zamana bağlı seri çözümünün yakınsaklığı ise çok
zaman adımlı metot kullanılarak sağlanmıştır. Yapılan çalışma melezleme
metodunun bu tür denklemlerin çözümünde etkili, güvenilir ve uygulanması kolay
olduğunu göstermiştir.

Keywords

Diferansiyel Dönüşüm/Sonlu Fark Metodu Kesirli Navier-Stokes Denklemleri Sayısal Çözüm

References

  • [1]. Ahmad, W. M., El-Khazali, R., Fractional-Order Dynamical Models of Love, Chaos, Solitons & Fractals 33 (4) (2007) 1367–1375.
  • [2]. Padovan J., Computational algorithms for FE formulations involving fractional operators, Comput. Mech. 5 (1987) 271–287.
  • [3]. Momani S., Non-perturbative analytical solutions of the space- and time-fractional Burgers equations, Chaos Solitons & Fractals 28 (4) (2006) 930–937.
  • [4]. Momani S. and Odibat Z., Analytical solution of a time-fractional Navier–Stokes equation by Adomian decomposition method, Appl. Math. Comput. 177 (2006) 488–494.
  • [5]. Odibat Z. and Momani S., Approximate solutions for boundary value problems of time-fractional wave equation, Appl. Math. Comput. 181 (2006) 1351–1358.
  • [6]. Momani S., An explicit and numerical solution of the fractional KdV equation, Math. Comput. Simulation 70 (2) (2005) 110–118.
  • [7]. Momani S. And Odibat Z., Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. Lett. A 355 (2006) 271–279.
  • [8]. Momani S. And Odibat Z., Numerical comparison of methods for solving linear differential equations of fractional order, Chaos, Solitons & Fractals, 31 (5) (2007) 1248–1255.
  • [9]. He J.H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Engrg. 167 (1998) 57–68.
  • [10]. Odibat Z. and Momani S., Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul., 7 (1) (2006) 15–27.
  • [11]. Odibat Z. and Momani S., Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order, Chaos, Solitons & Fractals 36 (2008) 167-174.
  • [12]. Momani S. and Odibat Z., Comparison between homotopy perturbation method and the variational iteration method for linear fractional partial differential equations, Comput. Math. Appl. 54 (2007) 910-919.
  • [13]. Podlubny I., Fractional Differential Equations, Academic Press, New York, 1999.
  • [14]. Baskonus, H. M.and Bulut H., On the Numerical Solutions of Some Fractional Ordinary Differential Equations by Fractional Adams-Bashforth-Moulton Method, Open Mathematics, 13(1)(2015), 547–556.
  • [15]. Odibat Z. and Momani S., A generalized differential transform method for linear partial differential equations of fractional order, Applied Mathematics Letters 21 (2) (2008) 194–199.
  • [16]. El-Shahed M. and Salem A., On the generalized Navier–Stokes equations, Appl. Math. Comput. 156 (1) (2004) 287–293.
  • [17]. Odibat Z.M. and Shawagfeh N.T., Generalized Taylor’s formula, Appl. Math. Comp. 186 (2007) 286–293.
  • [18]. Zhou J.K., Differential Transformation and its Applications for Electrical Circuits, Huazhong University Press, Wuhan, P. R., China 1986.
  • [19]. Arikoglu A. and Ozkol I., Solution of Difference Equations by Using Differential Transform Method, Appl. Math. Comput. 174(2) (2006) 1216-1228.
  • [20]. Ayaz F., Solutions of the system of differential equations by differential transform method, Appl. Math. Comput. 147 (2004) 547–567.
  • [21]. Arikoglu A. and Ozkol I., Solution of boundary value problems for integro-differential equations by using differential transform method, Appl. Math. Comput. 168(2) (2005) 1145–1158.
  • [22]. Bildik N., Konuralp A.,Orak Bek F., Kucukarslan S., Solution of different type of the partial differential equation by differential transform method and Adomian’s decomposition method, Appl. Math. Comput. 172 (2006) 551–567.
  • [23]. Abdel-Halim Hassan I.H., Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems, Chaos, Solitons & Fractals 36 (2008) 53-65.
  • [24]. Jang M.J., Chen C.L., Liy Y.C., On Solving the Initial-Value Problems Using the Differential Transformation Method, Appl. Math. Comput., 115 (2000) 145–160.
  • [25]. Rida S.Z., El-Sayed A.M.A., Arafa A.A.M., On the solutions of time-fractional reaction–diffusion equations, Communications in Nonlinear Science and Numerical Simulation 15 (2010) 3847-3854.
  • [26]. Odibat Z., Bertelle C., Aziz-Alaoui M.A., Duchamp G.H.E., A multi-step differential transform method and application to non-chaotic and chaotic systems, Comput. Math. Appl. 59 (2010) 1462–1472.
  • [27]. Zou L., Wang Z., Zong Z., Zou D., Zhang S., Solving shock wave with discontinuity by enhanced differential transform method (EDTM), Appl. Math. Mech. -Engl. Ed., 33(12) (2012) 1569–1582.
  • [28]. Liu H. and Song Y., Differential transform method applied to high index differential–algebraic equations, Appl. Math. Comp. 184 (2007) 748–753.
  • [29]. Chen X., Dai Y., Differential transform method for solving Richards’ equation, Appl. Math. Mech. -Engl. Ed., 37(2) (2016) 169–180.
  • [30]. Yu L.T. and Chen C.K., Application of the Hybrid Method to the Transient Thermal Stresses Response in Isotropic Annular Fins, J. Appl. Mech. 66 (1999) 340-347.
  • [31]. Kuo B.L. and Chen C.K., Application of a Hybrid Method to the Solution of the Nonlinear Burgers' Equation, J. Appl. Mech. 70 (2003) 926-930.
  • [32]. Chen C.K., Lai H.Y., Liu C.C., Application of hybrid differential transformation/finite difference method to nonlinear analysis of micro fixed-fixed beam, Microsyst Technol. 15 (2009) 813–820.
  • [33]. Chu H.P. and Chen C.L., Hybrid differential transform and finite difference method to solve the nonlinear heat conduction problem, Communications in Nonlinear Science and Numerical Simulation 13 (8) (2008) 1605–1614.
  • [34]. Smith G.D., Numerical Solution of Partial Differential Equations Finite Difference Methods, Oxford University Press, 1978.

A Computational Method for the Time-Fractional Navier-Stokes Equation

Year 2018, Volume: 39 Issue: 4, 900 - 911, 24.12.2018

İnci Çilingir Süngü , Hüseyin Demir

https://doi.org/10.17776/csj.384509
Cited By: 3

Abstract

In this
study, Navier-Stokes equations with fractional derivate are solved according to
time variable. To solve these equations, hybrid generalized differential
transformation and finite difference methods are used in various subdomains.
The aim of this hybridization is to combine the stability of the difference
method and simplicity of the differential transformation method in use. It has
been observed that the computational intensity of complex calculations is
reduced and also discontinuity due to initial conditions can be overcome when
the size increased in the study. The convergence of the time-dependent series
solution is ensured by multi-time-stepping method. This study has shown that
the hybridization method is effective, reliable and easy to apply for solving
such type of equations.



2010 Mathematics Subject Classification: 35Q30, 35R11, 65M06, 65N55.

Keywords

Hybrid Differential Transform/Finite Difference Method Numerical solution Time-Fractional Navier-Stokes Equations

References

  • [1]. Ahmad, W. M., El-Khazali, R., Fractional-Order Dynamical Models of Love, Chaos, Solitons & Fractals 33 (4) (2007) 1367–1375.
  • [2]. Padovan J., Computational algorithms for FE formulations involving fractional operators, Comput. Mech. 5 (1987) 271–287.
  • [3]. Momani S., Non-perturbative analytical solutions of the space- and time-fractional Burgers equations, Chaos Solitons & Fractals 28 (4) (2006) 930–937.
  • [4]. Momani S. and Odibat Z., Analytical solution of a time-fractional Navier–Stokes equation by Adomian decomposition method, Appl. Math. Comput. 177 (2006) 488–494.
  • [5]. Odibat Z. and Momani S., Approximate solutions for boundary value problems of time-fractional wave equation, Appl. Math. Comput. 181 (2006) 1351–1358.
  • [6]. Momani S., An explicit and numerical solution of the fractional KdV equation, Math. Comput. Simulation 70 (2) (2005) 110–118.
  • [7]. Momani S. And Odibat Z., Analytical approach to linear fractional partial differential equations arising in fluid mechanics, Phys. Lett. A 355 (2006) 271–279.
  • [8]. Momani S. And Odibat Z., Numerical comparison of methods for solving linear differential equations of fractional order, Chaos, Solitons & Fractals, 31 (5) (2007) 1248–1255.
  • [9]. He J.H., Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput. Methods Appl. Mech. Engrg. 167 (1998) 57–68.
  • [10]. Odibat Z. and Momani S., Application of variational iteration method to nonlinear differential equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul., 7 (1) (2006) 15–27.
  • [11]. Odibat Z. and Momani S., Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order, Chaos, Solitons & Fractals 36 (2008) 167-174.
  • [12]. Momani S. and Odibat Z., Comparison between homotopy perturbation method and the variational iteration method for linear fractional partial differential equations, Comput. Math. Appl. 54 (2007) 910-919.
  • [13]. Podlubny I., Fractional Differential Equations, Academic Press, New York, 1999.
  • [14]. Baskonus, H. M.and Bulut H., On the Numerical Solutions of Some Fractional Ordinary Differential Equations by Fractional Adams-Bashforth-Moulton Method, Open Mathematics, 13(1)(2015), 547–556.
  • [15]. Odibat Z. and Momani S., A generalized differential transform method for linear partial differential equations of fractional order, Applied Mathematics Letters 21 (2) (2008) 194–199.
  • [16]. El-Shahed M. and Salem A., On the generalized Navier–Stokes equations, Appl. Math. Comput. 156 (1) (2004) 287–293.
  • [17]. Odibat Z.M. and Shawagfeh N.T., Generalized Taylor’s formula, Appl. Math. Comp. 186 (2007) 286–293.
  • [18]. Zhou J.K., Differential Transformation and its Applications for Electrical Circuits, Huazhong University Press, Wuhan, P. R., China 1986.
  • [19]. Arikoglu A. and Ozkol I., Solution of Difference Equations by Using Differential Transform Method, Appl. Math. Comput. 174(2) (2006) 1216-1228.
  • [20]. Ayaz F., Solutions of the system of differential equations by differential transform method, Appl. Math. Comput. 147 (2004) 547–567.
  • [21]. Arikoglu A. and Ozkol I., Solution of boundary value problems for integro-differential equations by using differential transform method, Appl. Math. Comput. 168(2) (2005) 1145–1158.
  • [22]. Bildik N., Konuralp A.,Orak Bek F., Kucukarslan S., Solution of different type of the partial differential equation by differential transform method and Adomian’s decomposition method, Appl. Math. Comput. 172 (2006) 551–567.
  • [23]. Abdel-Halim Hassan I.H., Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems, Chaos, Solitons & Fractals 36 (2008) 53-65.
  • [24]. Jang M.J., Chen C.L., Liy Y.C., On Solving the Initial-Value Problems Using the Differential Transformation Method, Appl. Math. Comput., 115 (2000) 145–160.
  • [25]. Rida S.Z., El-Sayed A.M.A., Arafa A.A.M., On the solutions of time-fractional reaction–diffusion equations, Communications in Nonlinear Science and Numerical Simulation 15 (2010) 3847-3854.
  • [26]. Odibat Z., Bertelle C., Aziz-Alaoui M.A., Duchamp G.H.E., A multi-step differential transform method and application to non-chaotic and chaotic systems, Comput. Math. Appl. 59 (2010) 1462–1472.
  • [27]. Zou L., Wang Z., Zong Z., Zou D., Zhang S., Solving shock wave with discontinuity by enhanced differential transform method (EDTM), Appl. Math. Mech. -Engl. Ed., 33(12) (2012) 1569–1582.
  • [28]. Liu H. and Song Y., Differential transform method applied to high index differential–algebraic equations, Appl. Math. Comp. 184 (2007) 748–753.
  • [29]. Chen X., Dai Y., Differential transform method for solving Richards’ equation, Appl. Math. Mech. -Engl. Ed., 37(2) (2016) 169–180.
  • [30]. Yu L.T. and Chen C.K., Application of the Hybrid Method to the Transient Thermal Stresses Response in Isotropic Annular Fins, J. Appl. Mech. 66 (1999) 340-347.
  • [31]. Kuo B.L. and Chen C.K., Application of a Hybrid Method to the Solution of the Nonlinear Burgers' Equation, J. Appl. Mech. 70 (2003) 926-930.
  • [32]. Chen C.K., Lai H.Y., Liu C.C., Application of hybrid differential transformation/finite difference method to nonlinear analysis of micro fixed-fixed beam, Microsyst Technol. 15 (2009) 813–820.
  • [33]. Chu H.P. and Chen C.L., Hybrid differential transform and finite difference method to solve the nonlinear heat conduction problem, Communications in Nonlinear Science and Numerical Simulation 13 (8) (2008) 1605–1614.
  • [34]. Smith G.D., Numerical Solution of Partial Differential Equations Finite Difference Methods, Oxford University Press, 1978.
There are 34 citations in total.

Details

Primary Language English
Journal Section Natural Sciences
Authors

İnci Çilingir Süngü 0000-0001-7788-181X

Hüseyin Demir 0000-0002-3552-0511

Publication Date December 24, 2018
Submission Date January 26, 2018
Acceptance Date November 23, 2018
Published in Issue Year 2018Volume: 39 Issue: 4

Cite

APA Çilingir Süngü, İ., & Demir, H. (2018). A Computational Method for the Time-Fractional Navier-Stokes Equation. Cumhuriyet Science Journal, 39(4), 900-911. https://doi.org/10.17776/csj.384509

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