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THE NOVEL CONFORMABLE METHODS TO SOLVE CONFORMABLE TIME- FRACTIONAL COUPLED JAULENT-MIODEK SYSTEM

Yıl 2024, Cilt: 25 Sayı: 1, 123 - 140, 28.03.2024
https://doi.org/10.18038/estubtda.1380255

Öz

Kaynakça

  • [1] Liouville J. 1832. Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions. J Ecole Polytech 1832; 13(21): 1-69.
  • [2] Miller KS, Ross B. An introduction to the fractional calculus and fractional differential equations. Wiley, New York, 1993.
  • [3] Podlubny I. Fractional Differential Equations. Academic Press, New York, 1999.
  • [4] Baleanu D, Diethelm K, Scalas E, Trujillo JJ. Fractional calculus: models and numerical methods. World Scientific, London, 2012.
  • [5] Povstenko Y. 2015. Linear fractional diffusion-wave equation for scientists and engineers. Birkhäuser, Switzerland, 2015.
  • [6] Baleanu D, Wu GC, Zeng SD. Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations. Chaos Solit Fractals 2017; 102: 99–105.
  • [7] Sweilam NH, Abou Hasan MM, Baleanu D. New studies for general fractional financial models of awareness and trial advertising decisions. Chaos Solit Fractals 2017; 104: 772-784.
  • [8] Liu DY, Gibaru O, Perruquetti W, Laleg-Kirati TM. Fractional order differentiation by integration and error analysis in noisy environment. IEEE Trans Automat 2015; 60: 2945–2960.
  • [9] Esen A, Sulaiman TA, Bulut H, Baskonus HM. Optical solitons to the space-time fractional (1+1)-dimensional coupled nonlinear Schrödinger equation. Optik 2018; 167: 150–156.
  • [10] Caponetto R, Dongola G, Fortuna L, Gallo A. New results on the synthesis of FO-PID controllers. Commun Nonlinear Sci Numer Simul 2010; 15: 997–1007.
  • [11] Veeresha P, Prakasha DG, Baskonus HM. Novel simulations to the time-fractional Fisher’s equation. Math Sci 2019; 13(1): 33-42.
  • [12] Khalil R, Al Horani M, Yousef A, Sababheh M. A new definition of fractional derivative. J Comput Appl Math 2014; 264: 65-70.
  • [13] Aggarwal S, Chauhan R, Sharma N. 2018. Application of Elzaki transform for solving linear Volterra integral equations of first kind. Int J Res Advent Technol 2018; 6(12): 3687-3692.
  • [14] Elzaki TM. Applications of new transform “Elzaki transform” to partial differential equations. Glob J Pure Appl Math 2011; 7(1): 65-70.
  • [15] Elzaki TM. Solution of nonlinear differential equations using mixture of Elzaki transform and differential transform method. In International Mathematical Forum, 2012; 7(13): 631-638.
  • [16] Elzaki TM, Hilal EMA. Homotopy perturbation and Elzaki transform for solving nonlinear partial differential equations. Math Theory Model 2012; 2(3): 33-42.
  • [17] Elzaki TM, Kim H. The solution of radial diffusivity and shock wave equations by Elzaki variational iteration method. Int J Math Anal 2015; 9(22): 1065-1071.
  • [18] Jena RM, Chakraverty S. Solving time-fractional Navier–Stokes equations using homotopy perturbation Elzaki transform. SN Appl Sci 2019; 1: 1-16.
  • [19] Abu-Gdairi R, Al-Smadi M, Gumah G. An expansion iterative technique for handling fractional differential equations using fractional power series scheme. J Math Stat 2015; 11(2), 29–38.
  • [20] Baleanu D, Golmankhaneh AK, Baleanu MC. Fractional electromagnetic equations using fractional forms. Int J Theor Phys 2009; 48(11): 3114–3123.
  • [21] Baleanu D, Jajarmi A, Hajipour M. On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag–Leffler kernel. Nonlinear Dyn 2018; 2018(1): 1–18.
  • [22] Baleanu, D., Asad, J. H., Jajarmi, A. 2018. New aspects of the motion of a particle in a circular cavity. Proc Rom Acad Ser A, 2018; 19(2): 143–149.
  • [23] Baleanu D, Jajarmi A, Bonyah E, Hajipour M. New aspects of poor nutrition in the life cycle within the fractional calculus. Adv Differ Equ 2018; 2018(1): 1-14.
  • [24] 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.
  • [25] Ayaz F. Solutions of the system of differential equations by differential transform method. Appl Math Comput 2004; 147(2): 547-567.
  • [26] He JH. Variational iteration method-a kind of non-linear analytical technique: some examples. Int J Non Linear Mech 1999; 34(4): 699-708.
  • [27] He JH. Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 2003; 135(1): 73-79.
  • [28] He JH. Homotopy perturbation method for solving boundary value problems. Phys Lett 2006; 350(1-2): 87-88.
  • [29] He JH. Addendum: new interpretation of homotopy perturbation method. Int J Mod Phys B 2006; 20(18): 2561-2568.
  • [30] Jajarmi A, Baleanu D. Suboptimal control of fractional-order dynamic systems with delay argument. J Vib Control 2018; 24(12): 2430-2446.
  • [31] Jajarmi A, Baleanu D. A new fractional analysis on the interaction of HIV with CD4+ T-cells. Chaos Solitons Fractals 2018; 113: 221-229.
  • [32] 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.
  • [33] Klimek M. Fractional sequential mechanics-models with symmetric fractional derivative. Czech J Phys 2001; 51(12): 1348-1354.
  • [34] Merdan M. A new applicaiton of modified differential transformation method for modeling the pollution of a system of lakes. Selçuk J Appl Math 2010; 11(2): 27-40.
  • [35] 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.
  • [36] Wang K, Liu S. A new Sumudu transform iterative method for time-fractional Cauchy reaction-diffusion equation. Springer Plus 2016; 5(1): 865.
  • [37] Wazwaz AM. A reliable modification of Adomian decomposition method. Appl Math Comput 1999; 102(1): 77-86.
  • [38] Aslefallah M, Abbasbandy S, Yüzbaşi Ş. (2023). Numerical Solution for a Class of Nonlinear Emden-Fowler Equations by Exponential Collocation Method. Appl Appl Math 2023; 18(1): 1-13.
  • [39] Abdeljawad, T. 2015. On conformable fractional calculus. J Comput Appl Math 2015; 279: 57-66.
  • [40] Ala V, Demirbilek U, Mamedov KR. An application of improved Bernoulli sub-equation function method to the nonlinear conformable time-fractional SRLW equation. AIMS Math 2020; 5(4): 3751-3761.
  • [41] Gözütok U, Çoban H, Sağıroğlu Y. Frenet frame with respect to conformable derivative. Filomat 2019; 33(6): 1541-1550.
  • [42] Shrinath M, Bhadane A. New conformable fractional Elzaki transformation: Theory and applications. Malaya J Mat 2019; 1: 619-625.
  • [43] Kaya D, El-Sayed SM. A numerical method for solving Jaulent-Miodek equation. Phys Lett A, 2003; 318(4-5): 345-353.
  • [44] Cinar M, Onder I, Secer A, Bayram M, Abdulkadir Sulaiman T, Yusuf A. Solving the fractional Jaulent–Miodek system via a modified Laplace decomposition method. Waves Random Complex Media 2022; 1-14.
  • [45] Silva FS, Moreira DM, Moret MA. Conformable Laplace transform of fractional differential equations. Axioms, 2018; 7(3): 55.

THE NOVEL CONFORMABLE METHODS TO SOLVE CONFORMABLE TIME- FRACTIONAL COUPLED JAULENT-MIODEK SYSTEM

Yıl 2024, Cilt: 25 Sayı: 1, 123 - 140, 28.03.2024
https://doi.org/10.18038/estubtda.1380255

Öz

This research utilizes two novel methods, specifically the conformable q-homotopy analysis transform method (Cq-HATM) and the conformable Elzaki Adomian decomposition method (CEADM), to examine the numerical solutions for the conformable time-fractional coupled Jaulent-Miodek system. One of the two unique methods proposed is the Cq-HATM, which is a hybrid approach that combines the q-homotopy analysis transform method with the Laplace transform, employing the concept of conformable derivative. The CEADM method, similar to the aforementioned approach, is a hybrid technique that combines the Adomian decomposition method with Elzaki transform through the utilization of the concept of conformable derivative. The computer simulations were performed to offer validation for the effectiveness and dependability of the suggested approaches. After conducting a comparison between the exact solutions and the solutions acquired using the unique methods, it is apparent that both of these approaches demonstrate simplicity, effectiveness, and competency in tackling nonlinear conformable time-fractional coupled systems.

Kaynakça

  • [1] Liouville J. 1832. Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions. J Ecole Polytech 1832; 13(21): 1-69.
  • [2] Miller KS, Ross B. An introduction to the fractional calculus and fractional differential equations. Wiley, New York, 1993.
  • [3] Podlubny I. Fractional Differential Equations. Academic Press, New York, 1999.
  • [4] Baleanu D, Diethelm K, Scalas E, Trujillo JJ. Fractional calculus: models and numerical methods. World Scientific, London, 2012.
  • [5] Povstenko Y. 2015. Linear fractional diffusion-wave equation for scientists and engineers. Birkhäuser, Switzerland, 2015.
  • [6] Baleanu D, Wu GC, Zeng SD. Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations. Chaos Solit Fractals 2017; 102: 99–105.
  • [7] Sweilam NH, Abou Hasan MM, Baleanu D. New studies for general fractional financial models of awareness and trial advertising decisions. Chaos Solit Fractals 2017; 104: 772-784.
  • [8] Liu DY, Gibaru O, Perruquetti W, Laleg-Kirati TM. Fractional order differentiation by integration and error analysis in noisy environment. IEEE Trans Automat 2015; 60: 2945–2960.
  • [9] Esen A, Sulaiman TA, Bulut H, Baskonus HM. Optical solitons to the space-time fractional (1+1)-dimensional coupled nonlinear Schrödinger equation. Optik 2018; 167: 150–156.
  • [10] Caponetto R, Dongola G, Fortuna L, Gallo A. New results on the synthesis of FO-PID controllers. Commun Nonlinear Sci Numer Simul 2010; 15: 997–1007.
  • [11] Veeresha P, Prakasha DG, Baskonus HM. Novel simulations to the time-fractional Fisher’s equation. Math Sci 2019; 13(1): 33-42.
  • [12] Khalil R, Al Horani M, Yousef A, Sababheh M. A new definition of fractional derivative. J Comput Appl Math 2014; 264: 65-70.
  • [13] Aggarwal S, Chauhan R, Sharma N. 2018. Application of Elzaki transform for solving linear Volterra integral equations of first kind. Int J Res Advent Technol 2018; 6(12): 3687-3692.
  • [14] Elzaki TM. Applications of new transform “Elzaki transform” to partial differential equations. Glob J Pure Appl Math 2011; 7(1): 65-70.
  • [15] Elzaki TM. Solution of nonlinear differential equations using mixture of Elzaki transform and differential transform method. In International Mathematical Forum, 2012; 7(13): 631-638.
  • [16] Elzaki TM, Hilal EMA. Homotopy perturbation and Elzaki transform for solving nonlinear partial differential equations. Math Theory Model 2012; 2(3): 33-42.
  • [17] Elzaki TM, Kim H. The solution of radial diffusivity and shock wave equations by Elzaki variational iteration method. Int J Math Anal 2015; 9(22): 1065-1071.
  • [18] Jena RM, Chakraverty S. Solving time-fractional Navier–Stokes equations using homotopy perturbation Elzaki transform. SN Appl Sci 2019; 1: 1-16.
  • [19] Abu-Gdairi R, Al-Smadi M, Gumah G. An expansion iterative technique for handling fractional differential equations using fractional power series scheme. J Math Stat 2015; 11(2), 29–38.
  • [20] Baleanu D, Golmankhaneh AK, Baleanu MC. Fractional electromagnetic equations using fractional forms. Int J Theor Phys 2009; 48(11): 3114–3123.
  • [21] Baleanu D, Jajarmi A, Hajipour M. On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag–Leffler kernel. Nonlinear Dyn 2018; 2018(1): 1–18.
  • [22] Baleanu, D., Asad, J. H., Jajarmi, A. 2018. New aspects of the motion of a particle in a circular cavity. Proc Rom Acad Ser A, 2018; 19(2): 143–149.
  • [23] Baleanu D, Jajarmi A, Bonyah E, Hajipour M. New aspects of poor nutrition in the life cycle within the fractional calculus. Adv Differ Equ 2018; 2018(1): 1-14.
  • [24] 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.
  • [25] Ayaz F. Solutions of the system of differential equations by differential transform method. Appl Math Comput 2004; 147(2): 547-567.
  • [26] He JH. Variational iteration method-a kind of non-linear analytical technique: some examples. Int J Non Linear Mech 1999; 34(4): 699-708.
  • [27] He JH. Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 2003; 135(1): 73-79.
  • [28] He JH. Homotopy perturbation method for solving boundary value problems. Phys Lett 2006; 350(1-2): 87-88.
  • [29] He JH. Addendum: new interpretation of homotopy perturbation method. Int J Mod Phys B 2006; 20(18): 2561-2568.
  • [30] Jajarmi A, Baleanu D. Suboptimal control of fractional-order dynamic systems with delay argument. J Vib Control 2018; 24(12): 2430-2446.
  • [31] Jajarmi A, Baleanu D. A new fractional analysis on the interaction of HIV with CD4+ T-cells. Chaos Solitons Fractals 2018; 113: 221-229.
  • [32] 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.
  • [33] Klimek M. Fractional sequential mechanics-models with symmetric fractional derivative. Czech J Phys 2001; 51(12): 1348-1354.
  • [34] Merdan M. A new applicaiton of modified differential transformation method for modeling the pollution of a system of lakes. Selçuk J Appl Math 2010; 11(2): 27-40.
  • [35] 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.
  • [36] Wang K, Liu S. A new Sumudu transform iterative method for time-fractional Cauchy reaction-diffusion equation. Springer Plus 2016; 5(1): 865.
  • [37] Wazwaz AM. A reliable modification of Adomian decomposition method. Appl Math Comput 1999; 102(1): 77-86.
  • [38] Aslefallah M, Abbasbandy S, Yüzbaşi Ş. (2023). Numerical Solution for a Class of Nonlinear Emden-Fowler Equations by Exponential Collocation Method. Appl Appl Math 2023; 18(1): 1-13.
  • [39] Abdeljawad, T. 2015. On conformable fractional calculus. J Comput Appl Math 2015; 279: 57-66.
  • [40] Ala V, Demirbilek U, Mamedov KR. An application of improved Bernoulli sub-equation function method to the nonlinear conformable time-fractional SRLW equation. AIMS Math 2020; 5(4): 3751-3761.
  • [41] Gözütok U, Çoban H, Sağıroğlu Y. Frenet frame with respect to conformable derivative. Filomat 2019; 33(6): 1541-1550.
  • [42] Shrinath M, Bhadane A. New conformable fractional Elzaki transformation: Theory and applications. Malaya J Mat 2019; 1: 619-625.
  • [43] Kaya D, El-Sayed SM. A numerical method for solving Jaulent-Miodek equation. Phys Lett A, 2003; 318(4-5): 345-353.
  • [44] Cinar M, Onder I, Secer A, Bayram M, Abdulkadir Sulaiman T, Yusuf A. Solving the fractional Jaulent–Miodek system via a modified Laplace decomposition method. Waves Random Complex Media 2022; 1-14.
  • [45] Silva FS, Moreira DM, Moret MA. Conformable Laplace transform of fractional differential equations. Axioms, 2018; 7(3): 55.
Toplam 45 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Sayısal ve Hesaplamalı Matematik (Diğer)
Bölüm Makaleler
Yazarlar

Özkan Avit 0009-0003-7503-1012

Halil Anaç 0000-0002-1316-3947

Yayımlanma Tarihi 28 Mart 2024
Gönderilme Tarihi 23 Ekim 2023
Kabul Tarihi 26 Ocak 2024
Yayımlandığı Sayı Yıl 2024 Cilt: 25 Sayı: 1

Kaynak Göster

AMA Avit Ö, Anaç H. THE NOVEL CONFORMABLE METHODS TO SOLVE CONFORMABLE TIME- FRACTIONAL COUPLED JAULENT-MIODEK SYSTEM. Eskişehir Technical University Journal of Science and Technology A - Applied Sciences and Engineering. Mart 2024;25(1):123-140. doi:10.18038/estubtda.1380255