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Solutions of Time Fractional fKdV Equation Using the Residual Power Series Method

Year 2022, Volume 43, Issue 3, 468 - 476, 30.09.2022
https://doi.org/10.17776/csj.1087721

Abstract

The fifth-order Korteweg-de Vries (fKdV) equation is a nonlinear model in various long wave physical phenomena. The residual power series method (RPSM) is used to gain the approximate solutions of the time fractional fKdV equation in this study. Basic definitions of fractional derivatives are described in the Caputo sense. The solutions of the time fractional fKdV equation with easily computable components are calculated as a quick convergent series. When compared to exact solutions, the RPSM provides good accuracy for approximate solutions. The reliability of the proposed method is also illustrated with the aid of table and graphs. Cleary observed from the results that the suggested method is suitable and simple for similar type of the time fractional nonlinear differential equations.

References

  • [1] Kaya D., An Explicit and Numerical Solutions of Some Fifth-Order KdV Equation by Decomposition Method, Appl. Math. Comput., 144 (2003) 353-363.
  • [2] Handibag S., Karande B.D., Existence the Solutions of Some Fifth-Order KdV Equation by Laplace Decomposition Method, American J. Comput. Math., 3 (2013) 80-85.
  • [3] Saravi M., Nikkar A., Promising Technique for Analytic Treatment of Nonlinear Fifth-Order Equations, World J. Model. Simul., 10 (1) (2014) 27-33.
  • [4] Wazwaz A.W., A Fifth-Order Korteweg-de Vries Equation for Shallow Water with Surface Tension: Multiple Soliton Solutions, Acta Physica Polonica A, 130 (3) (2016) 679-682.
  • [5] Seadawy A.R., Lu D., Yue C., Travelling Wave Solutions of the Generalized Nonlinear Fifth-Order KdV Water Wave Equations and Its Stability, J. Taibah Uni. Sci., 11 (2017) 623-633.
  • [6] Goswami A., Singh J., Kumar D., Numerical Simulation of Fifth Order KdV Equations Occurring in Magneto-Acoustic Waves, Ain Shams Eng. J., 9 (2018) 2265-2273.
  • [7] Ahmad H., Khan T.A., Stanimirovic, P.S., Ahmad, I., Modified Variational Iteration Technique for the Numerical Solution of Fifth Order KdV-type Equations, J. Appl. Comput. Mech., 6 (2020) 1220-1227.
  • [8] Ahmad H., Khan T.A., Yao S-W., An Efficient Approach for the Numerical Solution of Fifth-Order KdV Equations, Open Math., 18 (2020) 738-748.
  • [9] Ahmad B., Nieto J.J., Existence of Solutions for Nonlocal Boundary Value Problems of Higher-Order Nonlinear Fractional Differential Equations, Hindawi, Doi:10.1155/2009/494720 (2009) 1-9.
  • [10] Wang Y., Liang S., Wang Q., Existence Results for Fractional Differential Equations with Integral and Multi-point Boundary Conditions, Boundary Val. Prob., Doi:10.1186/s13661-017-0924-4 (2018) 1-11.
  • [11] Şenol M., Ata A., Approximate Solution of Time-fractional KdV Equations by Residual Power Series Method, J. Balıkesir Uni. Ins. Sci. Tech., 20 (1) (2018) 430-439.
  • [12] Hosseini, K., Ilie, M., Mirzazadeh, M., Yusuf, A., Sulaiman, T.A., Baleanu, D., and Salahshour, S., An Effective Computational Method to deal ith a Time-fractional Nonlinear Water Wave Equation in the Caputo Sense, Math. Comp. Simul., 187 (2021) 248-260.
  • [13] Hosseini, K., Sadri, K., Mirzazadeh, M., Ahmadian, A., Chu, Y-M., and Salahshour, S., Reliable Methods to Look for Analytical and Numerical Solutions of a Nonlinear Differential Equation Arising in Heat Transfer with the Conformable Derivative, Math. Methods Appl. Sci., Doi: 10.1002/mma.7582 (2021) 1-13.
  • [14] Hosseini, K., Ilie, M., Mirzazadeh, M., and D., Baleanu, An Analytic Study on the Approximate Solution of a Nonlinear Time-fractional Cauchy Reaction-diffusion Equation with the Mittag-Leffler Law, Math. Methods Appl. Sci., 44 (2021) 6247-6258.
  • [15] Tuan, N.H., Mohammadi, H., and Rezapour, S., A Mathematical Model for COVID-19 Transmission by Using the Caputo Fractional Derivative, Chaos Soliton. Fract., 140 (2020) 1-11.
  • [16] Mohammadi, H., Rezapour, S., and Jajarmi, A., On the Fractional SIRD Mathematical Model and Control for the Transmission of COVID-19: The First and the Second Waves of the Disease in Iran and Japan, ISA Trans., 124 (2022) 103–114.
  • [17] Karunakar P., Chakraverty S., Solutions of Time-fractional Third and Fifth-Order Korteweg–de-Vries equations Using Homotopy Perturbation Transform Method, Eng. Comput., 36 (7) (2019) 2309-2326.
  • [18] Chen C., Jiang Y-L., Simplest Equation Method for Some Time-fractional Partial Differential Equations with Conformable Derivative, Comp. Math. Appl., 75 (2018) 2978-2988.
  • [19] Liu T., Exact Solutions to Time-fractional Fifth Order KdV Equation by Trial Equation Method Based on Symmetry, Symmetry, 11 (742) (2019) 1-8.
  • [20] Wang G. W., Yu T.Z., Feng T., Lie Symmetry Analysis and Explicit Solutions of the Time Fractional Fifth-Order KdV Equation, Plos One, 9 (2) (2014) 1-6.
  • [21] Lu D., Yue C., Arshad M., Traveling Wave Solutions of Space-time Fractional Generalized Fifth-order KdV equation, Adv. Math. Phys., Article ID 6743276 (2017) 1-6.
  • [22] Park C., Nuruddeen R.I., Ali K.K., Muhammad L., Osman M.S., Baleanu D., Novel Hyperbolic and Exponential Ansatz Methods to the Fractional Fifth-order Korteweg-de Vries Equations, Adv. Diff. Equ., 627 (2020) 1-12.
  • [23] Arqub A., Series Solution of Fuzzy Differential Equations Under Strongly Generalized Differentiability, J. Adv. Res. Appl. Math., 5 (1) (2013) 31-52.
  • [24] Şenol M., Alquran M., Kasmaei H.D., On the Comparison of Perturbation-iteration Algorithm and Residual Power Series Method to Solve Fractional Zakharov-Kuznetsov Equation, Results Phys., 9 (2018) 321-327.
  • [25] Körpınar Z., The Residual Power Series Method for Solving Fractional Klein-Gordon Equation, Sakarya Uni. J. Sci., 21 (3) (2017) 285-293.
  • [26] Kumar S., Kumar A., Baleanu D., Two Analytical Methods for Time-fractional Nonlinear Coupled Boussinesq-Burger’s Equations Arise in Propagation of Shallow Water Waves, Nonlinear Dyn, 85 (2016) 699-715.
  • [27] Alquran M., Analytical Solutions of Fractional Foam Drainage Equation by Residual Power Series Method, Math. Sci., 8 (2014) 153-160.
  • [28] Prakasha D.G, Veeresha P., Baskonus H.M., Residual Power Series Method for Fractional Swift-Hohenberg Equation, Fractal and Fractional, 3 (9) (2019) 1-16.
  • [29] Kumar A., Kumar S., Singh M., Residual Power Series Method for Fractional Sharma-Tasso-Olever Equation, Comm. Numer. Analy., 1 (2016) 1-10.
  • [30] Qurashi M.M.A., Korpinar Z., Baleanu D., Inc, M., A New Iterative Algorithm on the Time-fractional Fisher Equation: Residual Power Series Method, Adv. Mech. Eng., 9 (9) (2017) 1-8.
  • [31] Jena R.M., Chakraverty S., Residual Power series Method for Solving Time-fractional Model of Vibration Equation of Large Membranes, J. Appl. Comput. Mech., 5 (4) (2019) 603-615.
  • [32] Jaber K.K., Ahmad R.S., Analytical Solution of the Time Fractional Navier-Stokes Equation, Ain Shams Eng. J., 9 (4) (2018) 1917-1927.
  • [33] Zhang, J., Chen, X, and Li, L., and Zhou, C., Elzaki Transform Residual Power Series Method for the Fractional Population Diffusion Equations, Eng. Let., 29 (4) (2021) 1-12.
  • [34] Podlubny I., Fractional differential equations, New York: Academic Press, (1999).
  • [35] El-Ajou A., Arqub O.A., Zhour Z.A., Momani S., New Results on Fractional Power Series: Theories and Applications, Entropy, 15 (2013) 5305-5323.
  • [36] Arqub O.A., Abo-Hammour Z., Al-Badarneh R., Momani S., A Reliable Analytical Method for Solving Higher-order Initial Value Problems, Hindawi, Doi:10.1155/2013/673829 (2013) 1-12.
  • [37] Arqub O.A., El-Ajou A., Zhour Z.A., Momani S., Multiple Solutions of Nonlinear Boundary Value Problems of Fractional Order: A New Analytic Iterative Technique, Entropy, 16 (2014) 471-493.
  • [38] Arqub O.A., El-Ajou A., Bataineh A.S., Hashim I., A Representation of the Exact Solution of eGneralized Lane-Emden Equations Using a New Analytical method, Hindawi, Doi:10.1155/2013/378593 (2013) 1-10.
  • [39] El-Ajou A., Arqub O.A., Momani S., Approximate Analytical Solution of the Nonlinear Fractional KdV-Burgers Equation: A New Iterative Algorithm, J. Comput. Phys., 293 (2015) 81-95.

Year 2022, Volume 43, Issue 3, 468 - 476, 30.09.2022
https://doi.org/10.17776/csj.1087721

Abstract

References

  • [1] Kaya D., An Explicit and Numerical Solutions of Some Fifth-Order KdV Equation by Decomposition Method, Appl. Math. Comput., 144 (2003) 353-363.
  • [2] Handibag S., Karande B.D., Existence the Solutions of Some Fifth-Order KdV Equation by Laplace Decomposition Method, American J. Comput. Math., 3 (2013) 80-85.
  • [3] Saravi M., Nikkar A., Promising Technique for Analytic Treatment of Nonlinear Fifth-Order Equations, World J. Model. Simul., 10 (1) (2014) 27-33.
  • [4] Wazwaz A.W., A Fifth-Order Korteweg-de Vries Equation for Shallow Water with Surface Tension: Multiple Soliton Solutions, Acta Physica Polonica A, 130 (3) (2016) 679-682.
  • [5] Seadawy A.R., Lu D., Yue C., Travelling Wave Solutions of the Generalized Nonlinear Fifth-Order KdV Water Wave Equations and Its Stability, J. Taibah Uni. Sci., 11 (2017) 623-633.
  • [6] Goswami A., Singh J., Kumar D., Numerical Simulation of Fifth Order KdV Equations Occurring in Magneto-Acoustic Waves, Ain Shams Eng. J., 9 (2018) 2265-2273.
  • [7] Ahmad H., Khan T.A., Stanimirovic, P.S., Ahmad, I., Modified Variational Iteration Technique for the Numerical Solution of Fifth Order KdV-type Equations, J. Appl. Comput. Mech., 6 (2020) 1220-1227.
  • [8] Ahmad H., Khan T.A., Yao S-W., An Efficient Approach for the Numerical Solution of Fifth-Order KdV Equations, Open Math., 18 (2020) 738-748.
  • [9] Ahmad B., Nieto J.J., Existence of Solutions for Nonlocal Boundary Value Problems of Higher-Order Nonlinear Fractional Differential Equations, Hindawi, Doi:10.1155/2009/494720 (2009) 1-9.
  • [10] Wang Y., Liang S., Wang Q., Existence Results for Fractional Differential Equations with Integral and Multi-point Boundary Conditions, Boundary Val. Prob., Doi:10.1186/s13661-017-0924-4 (2018) 1-11.
  • [11] Şenol M., Ata A., Approximate Solution of Time-fractional KdV Equations by Residual Power Series Method, J. Balıkesir Uni. Ins. Sci. Tech., 20 (1) (2018) 430-439.
  • [12] Hosseini, K., Ilie, M., Mirzazadeh, M., Yusuf, A., Sulaiman, T.A., Baleanu, D., and Salahshour, S., An Effective Computational Method to deal ith a Time-fractional Nonlinear Water Wave Equation in the Caputo Sense, Math. Comp. Simul., 187 (2021) 248-260.
  • [13] Hosseini, K., Sadri, K., Mirzazadeh, M., Ahmadian, A., Chu, Y-M., and Salahshour, S., Reliable Methods to Look for Analytical and Numerical Solutions of a Nonlinear Differential Equation Arising in Heat Transfer with the Conformable Derivative, Math. Methods Appl. Sci., Doi: 10.1002/mma.7582 (2021) 1-13.
  • [14] Hosseini, K., Ilie, M., Mirzazadeh, M., and D., Baleanu, An Analytic Study on the Approximate Solution of a Nonlinear Time-fractional Cauchy Reaction-diffusion Equation with the Mittag-Leffler Law, Math. Methods Appl. Sci., 44 (2021) 6247-6258.
  • [15] Tuan, N.H., Mohammadi, H., and Rezapour, S., A Mathematical Model for COVID-19 Transmission by Using the Caputo Fractional Derivative, Chaos Soliton. Fract., 140 (2020) 1-11.
  • [16] Mohammadi, H., Rezapour, S., and Jajarmi, A., On the Fractional SIRD Mathematical Model and Control for the Transmission of COVID-19: The First and the Second Waves of the Disease in Iran and Japan, ISA Trans., 124 (2022) 103–114.
  • [17] Karunakar P., Chakraverty S., Solutions of Time-fractional Third and Fifth-Order Korteweg–de-Vries equations Using Homotopy Perturbation Transform Method, Eng. Comput., 36 (7) (2019) 2309-2326.
  • [18] Chen C., Jiang Y-L., Simplest Equation Method for Some Time-fractional Partial Differential Equations with Conformable Derivative, Comp. Math. Appl., 75 (2018) 2978-2988.
  • [19] Liu T., Exact Solutions to Time-fractional Fifth Order KdV Equation by Trial Equation Method Based on Symmetry, Symmetry, 11 (742) (2019) 1-8.
  • [20] Wang G. W., Yu T.Z., Feng T., Lie Symmetry Analysis and Explicit Solutions of the Time Fractional Fifth-Order KdV Equation, Plos One, 9 (2) (2014) 1-6.
  • [21] Lu D., Yue C., Arshad M., Traveling Wave Solutions of Space-time Fractional Generalized Fifth-order KdV equation, Adv. Math. Phys., Article ID 6743276 (2017) 1-6.
  • [22] Park C., Nuruddeen R.I., Ali K.K., Muhammad L., Osman M.S., Baleanu D., Novel Hyperbolic and Exponential Ansatz Methods to the Fractional Fifth-order Korteweg-de Vries Equations, Adv. Diff. Equ., 627 (2020) 1-12.
  • [23] Arqub A., Series Solution of Fuzzy Differential Equations Under Strongly Generalized Differentiability, J. Adv. Res. Appl. Math., 5 (1) (2013) 31-52.
  • [24] Şenol M., Alquran M., Kasmaei H.D., On the Comparison of Perturbation-iteration Algorithm and Residual Power Series Method to Solve Fractional Zakharov-Kuznetsov Equation, Results Phys., 9 (2018) 321-327.
  • [25] Körpınar Z., The Residual Power Series Method for Solving Fractional Klein-Gordon Equation, Sakarya Uni. J. Sci., 21 (3) (2017) 285-293.
  • [26] Kumar S., Kumar A., Baleanu D., Two Analytical Methods for Time-fractional Nonlinear Coupled Boussinesq-Burger’s Equations Arise in Propagation of Shallow Water Waves, Nonlinear Dyn, 85 (2016) 699-715.
  • [27] Alquran M., Analytical Solutions of Fractional Foam Drainage Equation by Residual Power Series Method, Math. Sci., 8 (2014) 153-160.
  • [28] Prakasha D.G, Veeresha P., Baskonus H.M., Residual Power Series Method for Fractional Swift-Hohenberg Equation, Fractal and Fractional, 3 (9) (2019) 1-16.
  • [29] Kumar A., Kumar S., Singh M., Residual Power Series Method for Fractional Sharma-Tasso-Olever Equation, Comm. Numer. Analy., 1 (2016) 1-10.
  • [30] Qurashi M.M.A., Korpinar Z., Baleanu D., Inc, M., A New Iterative Algorithm on the Time-fractional Fisher Equation: Residual Power Series Method, Adv. Mech. Eng., 9 (9) (2017) 1-8.
  • [31] Jena R.M., Chakraverty S., Residual Power series Method for Solving Time-fractional Model of Vibration Equation of Large Membranes, J. Appl. Comput. Mech., 5 (4) (2019) 603-615.
  • [32] Jaber K.K., Ahmad R.S., Analytical Solution of the Time Fractional Navier-Stokes Equation, Ain Shams Eng. J., 9 (4) (2018) 1917-1927.
  • [33] Zhang, J., Chen, X, and Li, L., and Zhou, C., Elzaki Transform Residual Power Series Method for the Fractional Population Diffusion Equations, Eng. Let., 29 (4) (2021) 1-12.
  • [34] Podlubny I., Fractional differential equations, New York: Academic Press, (1999).
  • [35] El-Ajou A., Arqub O.A., Zhour Z.A., Momani S., New Results on Fractional Power Series: Theories and Applications, Entropy, 15 (2013) 5305-5323.
  • [36] Arqub O.A., Abo-Hammour Z., Al-Badarneh R., Momani S., A Reliable Analytical Method for Solving Higher-order Initial Value Problems, Hindawi, Doi:10.1155/2013/673829 (2013) 1-12.
  • [37] Arqub O.A., El-Ajou A., Zhour Z.A., Momani S., Multiple Solutions of Nonlinear Boundary Value Problems of Fractional Order: A New Analytic Iterative Technique, Entropy, 16 (2014) 471-493.
  • [38] Arqub O.A., El-Ajou A., Bataineh A.S., Hashim I., A Representation of the Exact Solution of eGneralized Lane-Emden Equations Using a New Analytical method, Hindawi, Doi:10.1155/2013/378593 (2013) 1-10.
  • [39] El-Ajou A., Arqub O.A., Momani S., Approximate Analytical Solution of the Nonlinear Fractional KdV-Burgers Equation: A New Iterative Algorithm, J. Comput. Phys., 293 (2015) 81-95.

Details

Primary Language English
Subjects Mathematics
Journal Section Natural Sciences
Authors

Sevil ÇULHA ÜNAL> (Primary Author)
SULEYMAN DEMIREL UNIVERSITY
0000-0001-7447-9219
Türkiye

Publication Date September 30, 2022
Application Date March 14, 2022
Acceptance Date August 2, 2022
Published in Issue Year 2022, Volume 43, Issue 3

Cite

APA Çulha Ünal, S. (2022). Solutions of Time Fractional fKdV Equation Using the Residual Power Series Method . Cumhuriyet Science Journal , 43 (3) , 468-476 . DOI: 10.17776/csj.1087721