Research Article
BibTex RIS Cite

Q-HOMOTOPY SHEHU ANALYSIS TRANSFORM METHOD OF TIME-FRACTIONAL COUPLED BURGERS EQUATIONS

Year 2023, Volume: 24 Issue: 3, 177 - 191, 22.09.2023

Umut Bektaş , Halil Anaç

https://doi.org/10.18038/estubtda.1312725

Abstract

In this study, numerical solutions to time-fractional coupled Burgers equations are obtained utilizing the q-homotopy Shehu analysis transform method. The definition of fractional derivatives in the sense of Caputo. q-homotopy Shehu analysis transform method is also used to find the numerical solutions of the time-fractional coupled Burgers equations. In addition, the MAPLE software is utilized to plot the graphs of the solutions. These results demonstrate that the presented method is accurate and simple to implement.

Keywords

q-homotopy Shehu analysis transform method coupled Burgers equations Mittag-Leffler function

References

  • [1] Hilfer R. Application of Fractional Calculus in Physics, Singapore, World Scientific Publishing Company, 2000.
  • [2] Kilbas A, Srivastava, H, Trujillo J. Theory and Applications of Fractional Differential Equations, Amsterdam, Elsevier, 2006.
  • [3] Miller KS, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations, New York, Wiley, 1993.
  • [4] Oldham KB, Spanier J. The Fractional Calculus, New York, Academic Press, 1974.
  • [5] Podlubny I. Fractional Differential Equations, New York, Academic Press, New York, 1999.
  • [6] Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives Theory and Applications, New York, Gordon and Beach, 1993.
  • [7] Metzler R, Nonnenmacher TF. Space-and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation. Chemical Physics, 2002; 284 (1-2), 67-90.
  • [8] Metzler R, Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Physics reports, 2000; 339 (1), 1-77.
  • [9] Morgado ML, Rebelo M. Numerical approximation of distributed order reaction–diffusion equations. Journal of Computational and Applied Mathematics, 2015; 275, 216-227.
  • [10] Abu-Gdairi R, Al-Smadi M, Gumah G. An expansion iterative technique for handling fractional differential equations using fractional power series scheme. Journal of Mathematics and Statistics, 2015; 11(2), 29–38.
  • [11] Baleanu D, Golmankhaneh AK, Baleanu MC. Fractional electromagnetic equations using fractional forms. International Journal of Theoretical Physics, 2009; 48(11), 3114–3123.
  • [12] Baleanu D, Jajarmi A, Hajipour M. On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag–Leffler kernel. Nonlinear Dynamics, 2018(1), 1–18.
  • [13] Baleanu D, Asad JH, Jajarmi A. New aspects of the motion of a particle in a circular cavity. Proceedings of the Romanian Academy Series A, 2018; 19(2), 143–149.
  • [14] Baleanu D, Jajarmi A, Bonyah E, Hajipour M. New aspects of poor nutrition in the life cycle within the fractional calculus. Advances in Difference Equations, 2018; 2018(1), 1-14.
  • [15] Jajarmi A, Baleanu D. Suboptimal control of fractional-order dynamic systems with delay argument. Journal of Vibration and Control, 2018; 24(12), 2430-2446.
  • [16] Jajarmi A, Baleanu D. A new fractional analysis on the interaction of HIV with CD4+ T-cells. Chaos, Solitons & Fractals, 2018; 113, 221-229.
  • [17] He JH. Addendum: new interpretation of homotopy perturbation method. International Journal of Modern Physics B, 2006; 20(18), 2561-2568.
  • [18] Laskin, N. Fractional quantum mechanics. Physical Review E, 2000; 62(3), 3135-3145.
  • [19] Mainardi F. Fractional Calculus and Waves in Linear Viscoelasticity, London, Imperial College Press, 2010.
  • [20] Wazwaz AM. A reliable modification of Adomian decomposition method. Applied Mathematics and Computation, 1999; 102(1), 77-86.
  • [21] He JH. Homotopy perturbation method: a new nonlinear analytical technique. Applied Mathematics and Computation, 2003; 135(1), 73-79.
  • [22] He JH. Homotopy perturbation method for solving boundary value problems. Physics Letters, 2006; 350(1-2), 87-88.
  • [23] He JH. Addendum: new interpretation of homotopy perturbation method. International Journal of Modern Physics B, 2006; 20(18), 2561-2568.
  • [24] Alkan A. Improving homotopy analysis method with an optimal parameter for time-fractional Burgers equation, Karamanoğlu Mehmetbey Üniversitesi Mühendislik ve Doğa Bilimleri Dergisi, 2022; 4(2), 117-134.
  • [25] Turkyilmazoglu M. Convergence accelerating in the homotopy analysis method: a new approach. Advances in Applied Mathematics and Mechanics, 2018; 10(4).
  • [26] Yüzbaşı Ş. A numerical approximation for Volterra’s population growth model with fractional order. Applied Mathematical Modelling, 2013; 37(5), 3216-3227.
  • [27] Yüzbaşı Ş. Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials. Applied Mathematics and Computation, 2013; 219(11), 6328-6343.
  • [28] Yüzbaşı Ş. Numerical solutions of hyperbolic telegraph equation by using the Bessel functions of first kind and residual correction. Applied Mathematics and Computation, 2016; 287, 83-93.
  • [29] Yüzbaşı Ş. A collocation method for numerical solutions of fractional-order logistic population model. International Journal of Biomathematics, 2016; 9(2), 1650031.
  • [30] Yüzbaşı Ş. A numerical method for solving second-order linear partial differential equations under Dirichlet, Neumann and Robin boundary conditions. International Journal of Computational Methods, 2017; 14(2), 1750015.
  • [31] Yüzbaşı Ş. A collocation approach for solving two-dimensional second-order linear hyperbolic equations. Applied Mathematics and Computation, 2018; 338, 101-114.
  • [32] Merdan M, Anaç H, Kesemen T. The new Sumudu transform iterative method for studying the random component time-fractional Klein-Gordon equation. Sigma, 2019; 10(3), 343-354.
  • [33] Wang K, Liu S. A new Sumudu transform iterative method for time-fractional Cauchy reaction-diffusion equation. Springer Plus, 2016; 5(1), 865.
  • [34] Anaç H, Merdan M, Bekiryazıcı Z, Kesemen T. Bazı Rastgele Kısmi Diferansiyel Denklemlerin Diferansiyel Dönüşüm Metodu ve Laplace-Padé Metodu Kullanarak Çözümü. Gümüşhane Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 2019; 9(1), 108-118.
  • [35] Ayaz F. Solutions of the system of differential equations by differential transform method. Applied Mathematics and Computation, 2004; 147(2), 547-567.
  • [36] Kangalgil F, Ayaz F. Solitary wave solutions for the KdV and mKdV equations by differential transform method. Chaos, Solitons & Fractals, 2009; 41(1), 464-472.
  • [37] Merdan M. A new applicaiton of modified differential transformation method for modeling the pollution of a system of lakes. Selçuk Journal of Applied Mathematics, 2010; 11(2), 27-40.
  • [38] Zhou JK. Differential Transform and Its Applications for Electrical Circuits. Wuhan, Huazhong University Press, 1986.
  • [39] Maitama S, Zhao W. New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations. arXiv preprint arXiv:1904.11370. (2019).
  • [40] Akinyemi L, Iyiola OS. Exact and approximate solutions of time‐fractional models arising from physics via Shehu transform. Mathematical Methods in the Applied Sciences,2020; 43(12), 7442-7464.
  • [41] Alfaqeih S, Misirli E. On double Shehu transform and its properties with applications. International Journal of Analysis and Applications, 2020; 18(3), 381-395.
  • [42] Maitama S, Zhao W. Homotopy analysis Shehu transform method for solving fuzzy differential equations of fractional and integer order derivatives. Computational and Applied Mathematics, 2021; 40(3), 1-30.
  • [43] Kanth AR, Aruna K, Raghavendar K, Rezazadeh H, İnç M. Numerical solutions of nonlinear time fractional Klein-Gordon equation via natural transform decomposition method and iterative Shehu transform method. Journal of Ocean Engineering and Science, https://doi.org/10.1016/j.joes.2021.12.002. (2021).
  • [44] Shah R, Saad Alshehry A, Weera W. A semi-analytical method to investigate fractional-order gas dynamics equations by Shehu transform. Symmetry, 2022; 14(7), 1458.
  • [45] Abujarad ES, Jarad F, Abujarad MH, Baleanu D. Application of q-Shehu transform on q-fractional kinetic equation involving the generalizd hyper-Bessel function. Fractals, 2022; 30(05), 2240179.
  • [46] Sinha AK, Panda S. Shehu Transform in Quantum Calculus and Its Applications. International Journal of Applied and Computational Mathematics, 2022; 8(1), 1-19.
  • [47] Prakasha DG, Veeresha P, Rawashdeh MS. Numerical solution for (2+ 1)‐dimensional time‐fractional coupled Burger equations using fractional natural decomposition method. Mathematical Methods in the Applied Sciences, 2019; 42(10), 3409-3427.
  • [48] Cole JD, On a quasi-linear parabolic equation occurring in aerodynamics. Quarterly of Applied Mathematics, 1951; 9, 225–236.
  • [49] Aksan EN. Quadratic B-spline fifinite element method for numerical solution of the Burgers equation. Appl. Math. Comput., 2006; 174, 884–896.
  • [50] Kutluay S, Esen A. A lumped Galerkin method for solving the Burgers equation. Int. J. Comput. Math., 2004; 81, 1433–1444.
  • [51] Abbasbandy S, Darvishi MT. A numerical solution of Burgers equation by modifified Adomian method. Appl. Math. Comput., 2005; 163, 1265–1272.
  • [52] Jin-Ming Z, Yao-Ming Z, Abd AL-Hussein WR, Mahmood A, Shamran SNK. Exact solutions of the two-dimensional Burgers equation. J. Phys. A Math. Gen., 1999; 32, 6897–6900.
  • [53] Bateman H. Some recent researches on the motion of fluids. Monthly Weather Review, 1915; 43, 163–170.
  • [54] Hopf E. The partial differential equation ut + uux = uxx. Commun. Pure Appl. Math., 1950; 3, 201–230.
  • [55] Benton ER, Platzman GW. A table of solutions of the one-dimensional Burgers equation. Q. Appl. Math., 1972; 30, 195–212.
  • [56] Karpman VI. Non-Linear Waves in Dispersive Media: International Series of Monographs in Natural Philosophy, Amsterdam, The Netherlands, Elsevier, 2016.
  • [57] Aljahdaly NH, Agarwal RP, Shah R, Botmart T. Analysis of the time fractional-order coupled burgers equations with non-singular kernel operators. Mathematics, 2021; 9(18), 2326.
  • [58] Anaç H, Merdan M, Kesemen T. Solving for the random component time-fractional partial differential equations with the new Sumudu transform iterative method. SN Applied Sciences, 2020; 2(6), 1-11.
  • [59] Yüzbaşı Ş. Fractional Bell collocation method for solving linear fractional integro-differential equations. Mathematical Sciences, 2022; 1-12.
  • [60] Kumar D, Singh J, Baleanu D. A new analysis for fractional model of regularized long‐wave equation arising in ion acoustic plasma waves. Mathematical Methods in the Applied Sciences, 2017; 40(15), 5642-5653.
  • [61] Magreñán ÁA. A new tool to study real dynamics: The convergence plane. Applied Mathematics and Computation, 2014; 248, 215-224.

Q-HOMOTOPY SHEHU ANALYSIS TRANSFORM METHOD OF TIME-FRACTIONAL COUPLED BURGERS EQUATIONS

Year 2023, Volume: 24 Issue: 3, 177 - 191, 22.09.2023

Umut Bektaş , Halil Anaç

https://doi.org/10.18038/estubtda.1312725

Abstract

References

  • [1] Hilfer R. Application of Fractional Calculus in Physics, Singapore, World Scientific Publishing Company, 2000.
  • [2] Kilbas A, Srivastava, H, Trujillo J. Theory and Applications of Fractional Differential Equations, Amsterdam, Elsevier, 2006.
  • [3] Miller KS, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations, New York, Wiley, 1993.
  • [4] Oldham KB, Spanier J. The Fractional Calculus, New York, Academic Press, 1974.
  • [5] Podlubny I. Fractional Differential Equations, New York, Academic Press, New York, 1999.
  • [6] Samko SG, Kilbas AA, Marichev OI. Fractional Integrals and Derivatives Theory and Applications, New York, Gordon and Beach, 1993.
  • [7] Metzler R, Nonnenmacher TF. Space-and time-fractional diffusion and wave equations, fractional Fokker-Planck equations, and physical motivation. Chemical Physics, 2002; 284 (1-2), 67-90.
  • [8] Metzler R, Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Physics reports, 2000; 339 (1), 1-77.
  • [9] Morgado ML, Rebelo M. Numerical approximation of distributed order reaction–diffusion equations. Journal of Computational and Applied Mathematics, 2015; 275, 216-227.
  • [10] Abu-Gdairi R, Al-Smadi M, Gumah G. An expansion iterative technique for handling fractional differential equations using fractional power series scheme. Journal of Mathematics and Statistics, 2015; 11(2), 29–38.
  • [11] Baleanu D, Golmankhaneh AK, Baleanu MC. Fractional electromagnetic equations using fractional forms. International Journal of Theoretical Physics, 2009; 48(11), 3114–3123.
  • [12] Baleanu D, Jajarmi A, Hajipour M. On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag–Leffler kernel. Nonlinear Dynamics, 2018(1), 1–18.
  • [13] Baleanu D, Asad JH, Jajarmi A. New aspects of the motion of a particle in a circular cavity. Proceedings of the Romanian Academy Series A, 2018; 19(2), 143–149.
  • [14] Baleanu D, Jajarmi A, Bonyah E, Hajipour M. New aspects of poor nutrition in the life cycle within the fractional calculus. Advances in Difference Equations, 2018; 2018(1), 1-14.
  • [15] Jajarmi A, Baleanu D. Suboptimal control of fractional-order dynamic systems with delay argument. Journal of Vibration and Control, 2018; 24(12), 2430-2446.
  • [16] Jajarmi A, Baleanu D. A new fractional analysis on the interaction of HIV with CD4+ T-cells. Chaos, Solitons & Fractals, 2018; 113, 221-229.
  • [17] He JH. Addendum: new interpretation of homotopy perturbation method. International Journal of Modern Physics B, 2006; 20(18), 2561-2568.
  • [18] Laskin, N. Fractional quantum mechanics. Physical Review E, 2000; 62(3), 3135-3145.
  • [19] Mainardi F. Fractional Calculus and Waves in Linear Viscoelasticity, London, Imperial College Press, 2010.
  • [20] Wazwaz AM. A reliable modification of Adomian decomposition method. Applied Mathematics and Computation, 1999; 102(1), 77-86.
  • [21] He JH. Homotopy perturbation method: a new nonlinear analytical technique. Applied Mathematics and Computation, 2003; 135(1), 73-79.
  • [22] He JH. Homotopy perturbation method for solving boundary value problems. Physics Letters, 2006; 350(1-2), 87-88.
  • [23] He JH. Addendum: new interpretation of homotopy perturbation method. International Journal of Modern Physics B, 2006; 20(18), 2561-2568.
  • [24] Alkan A. Improving homotopy analysis method with an optimal parameter for time-fractional Burgers equation, Karamanoğlu Mehmetbey Üniversitesi Mühendislik ve Doğa Bilimleri Dergisi, 2022; 4(2), 117-134.
  • [25] Turkyilmazoglu M. Convergence accelerating in the homotopy analysis method: a new approach. Advances in Applied Mathematics and Mechanics, 2018; 10(4).
  • [26] Yüzbaşı Ş. A numerical approximation for Volterra’s population growth model with fractional order. Applied Mathematical Modelling, 2013; 37(5), 3216-3227.
  • [27] Yüzbaşı Ş. Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials. Applied Mathematics and Computation, 2013; 219(11), 6328-6343.
  • [28] Yüzbaşı Ş. Numerical solutions of hyperbolic telegraph equation by using the Bessel functions of first kind and residual correction. Applied Mathematics and Computation, 2016; 287, 83-93.
  • [29] Yüzbaşı Ş. A collocation method for numerical solutions of fractional-order logistic population model. International Journal of Biomathematics, 2016; 9(2), 1650031.
  • [30] Yüzbaşı Ş. A numerical method for solving second-order linear partial differential equations under Dirichlet, Neumann and Robin boundary conditions. International Journal of Computational Methods, 2017; 14(2), 1750015.
  • [31] Yüzbaşı Ş. A collocation approach for solving two-dimensional second-order linear hyperbolic equations. Applied Mathematics and Computation, 2018; 338, 101-114.
  • [32] Merdan M, Anaç H, Kesemen T. The new Sumudu transform iterative method for studying the random component time-fractional Klein-Gordon equation. Sigma, 2019; 10(3), 343-354.
  • [33] Wang K, Liu S. A new Sumudu transform iterative method for time-fractional Cauchy reaction-diffusion equation. Springer Plus, 2016; 5(1), 865.
  • [34] Anaç H, Merdan M, Bekiryazıcı Z, Kesemen T. Bazı Rastgele Kısmi Diferansiyel Denklemlerin Diferansiyel Dönüşüm Metodu ve Laplace-Padé Metodu Kullanarak Çözümü. Gümüşhane Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 2019; 9(1), 108-118.
  • [35] Ayaz F. Solutions of the system of differential equations by differential transform method. Applied Mathematics and Computation, 2004; 147(2), 547-567.
  • [36] Kangalgil F, Ayaz F. Solitary wave solutions for the KdV and mKdV equations by differential transform method. Chaos, Solitons & Fractals, 2009; 41(1), 464-472.
  • [37] Merdan M. A new applicaiton of modified differential transformation method for modeling the pollution of a system of lakes. Selçuk Journal of Applied Mathematics, 2010; 11(2), 27-40.
  • [38] Zhou JK. Differential Transform and Its Applications for Electrical Circuits. Wuhan, Huazhong University Press, 1986.
  • [39] Maitama S, Zhao W. New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations. arXiv preprint arXiv:1904.11370. (2019).
  • [40] Akinyemi L, Iyiola OS. Exact and approximate solutions of time‐fractional models arising from physics via Shehu transform. Mathematical Methods in the Applied Sciences,2020; 43(12), 7442-7464.
  • [41] Alfaqeih S, Misirli E. On double Shehu transform and its properties with applications. International Journal of Analysis and Applications, 2020; 18(3), 381-395.
  • [42] Maitama S, Zhao W. Homotopy analysis Shehu transform method for solving fuzzy differential equations of fractional and integer order derivatives. Computational and Applied Mathematics, 2021; 40(3), 1-30.
  • [43] Kanth AR, Aruna K, Raghavendar K, Rezazadeh H, İnç M. Numerical solutions of nonlinear time fractional Klein-Gordon equation via natural transform decomposition method and iterative Shehu transform method. Journal of Ocean Engineering and Science, https://doi.org/10.1016/j.joes.2021.12.002. (2021).
  • [44] Shah R, Saad Alshehry A, Weera W. A semi-analytical method to investigate fractional-order gas dynamics equations by Shehu transform. Symmetry, 2022; 14(7), 1458.
  • [45] Abujarad ES, Jarad F, Abujarad MH, Baleanu D. Application of q-Shehu transform on q-fractional kinetic equation involving the generalizd hyper-Bessel function. Fractals, 2022; 30(05), 2240179.
  • [46] Sinha AK, Panda S. Shehu Transform in Quantum Calculus and Its Applications. International Journal of Applied and Computational Mathematics, 2022; 8(1), 1-19.
  • [47] Prakasha DG, Veeresha P, Rawashdeh MS. Numerical solution for (2+ 1)‐dimensional time‐fractional coupled Burger equations using fractional natural decomposition method. Mathematical Methods in the Applied Sciences, 2019; 42(10), 3409-3427.
  • [48] Cole JD, On a quasi-linear parabolic equation occurring in aerodynamics. Quarterly of Applied Mathematics, 1951; 9, 225–236.
  • [49] Aksan EN. Quadratic B-spline fifinite element method for numerical solution of the Burgers equation. Appl. Math. Comput., 2006; 174, 884–896.
  • [50] Kutluay S, Esen A. A lumped Galerkin method for solving the Burgers equation. Int. J. Comput. Math., 2004; 81, 1433–1444.
  • [51] Abbasbandy S, Darvishi MT. A numerical solution of Burgers equation by modifified Adomian method. Appl. Math. Comput., 2005; 163, 1265–1272.
  • [52] Jin-Ming Z, Yao-Ming Z, Abd AL-Hussein WR, Mahmood A, Shamran SNK. Exact solutions of the two-dimensional Burgers equation. J. Phys. A Math. Gen., 1999; 32, 6897–6900.
  • [53] Bateman H. Some recent researches on the motion of fluids. Monthly Weather Review, 1915; 43, 163–170.
  • [54] Hopf E. The partial differential equation ut + uux = uxx. Commun. Pure Appl. Math., 1950; 3, 201–230.
  • [55] Benton ER, Platzman GW. A table of solutions of the one-dimensional Burgers equation. Q. Appl. Math., 1972; 30, 195–212.
  • [56] Karpman VI. Non-Linear Waves in Dispersive Media: International Series of Monographs in Natural Philosophy, Amsterdam, The Netherlands, Elsevier, 2016.
  • [57] Aljahdaly NH, Agarwal RP, Shah R, Botmart T. Analysis of the time fractional-order coupled burgers equations with non-singular kernel operators. Mathematics, 2021; 9(18), 2326.
  • [58] Anaç H, Merdan M, Kesemen T. Solving for the random component time-fractional partial differential equations with the new Sumudu transform iterative method. SN Applied Sciences, 2020; 2(6), 1-11.
  • [59] Yüzbaşı Ş. Fractional Bell collocation method for solving linear fractional integro-differential equations. Mathematical Sciences, 2022; 1-12.
  • [60] Kumar D, Singh J, Baleanu D. A new analysis for fractional model of regularized long‐wave equation arising in ion acoustic plasma waves. Mathematical Methods in the Applied Sciences, 2017; 40(15), 5642-5653.
  • [61] Magreñán ÁA. A new tool to study real dynamics: The convergence plane. Applied Mathematics and Computation, 2014; 248, 215-224.
There are 61 citations in total.

Details

Primary Language English
Subjects Applied Mathematics (Other)
Journal Section Articles
Authors

Umut Bektaş 0000-0002-7383-2604

Halil Anaç 0000-0002-1316-3947

Publication Date September 22, 2023
Published in Issue Year 2023 Volume: 24 Issue: 3

Cite

AMA Bektaş U, Anaç H. Q-HOMOTOPY SHEHU ANALYSIS TRANSFORM METHOD OF TIME-FRACTIONAL COUPLED BURGERS EQUATIONS. Estuscience - Se. September 2023;24(3):177-191. doi:10.18038/estubtda.1312725