The Efficient Robust Conformable Methods for Solving the Conformable Fractional Cahn-Allen Equation

Yıl 2024, Cilt: 11 Sayı: 2, 422 - 436, 29.11.2024

Özkan Avit , Halil Anaç

https://doi.org/10.35193/bseufbd.1360451

Öz

This study focuses on the novel conformable methods employed to obtain new numerical solutions for the Cahn-Allen equation with conformable fractional derivatives. One of the two distinct methods put forth is the Cq-HATM, a hybrid technique that integrates the q-homotopy analysis transform method with the Laplace transform, utilizing the definition of conformable derivative. The CHPETM is a hybrid technique that combines the homotopy perturbation method with the Elzaki transform (ET). New numerical solutions of the conformal fractional differential Cahn-Allen equation were obtained using CHPETM and Cq-HATM. The computer simulations have been conducted in order to provide validation for the efficacy and reliability of the proposed methods. Upon performing a comparative analysis between the exact solutions and the solutions obtained through the novel methods, it becomes evident that both of these approaches exhibit simplicity, efficacy, and proficiency in addressing nonlinear conformable time-fractional coupled systems.

Anahtar Kelimeler

Cahn-Allen equation, conformable homotopy perturbation Elzaki transform method, conformable Elzaki transform

Uyumlu Kesirli Mertebeden Cahn-Allen Denklemini Çözmek için Etkili Uyumlu Yöntemler

Öz

Bu çalışma, uyumlu kesirli türevli Cahn-Allen denkleminin yeni sayısal çözümlerini elde etmek için kullanılan yeni uyumlu yöntemlere odaklanmaktadır. Öne sürülen iki farklı yöntemden biri, uyumlu kesirli türev tanımını kullanarak, q-homotopi analizi dönüşüm yöntemi ile Laplace dönüşümünün birleşiminden oluşan hibrit bir yöntem olan Uq-HADY' dir. UHPEDM, homotopi pertürbasyon yöntemininin Elzaki dönüşümüyle birleşiminden oluşan hibrit bir yöntemdir. Uyumlu kesirli türevli Cahn-Allen denkleminin yeni nümerik çözümleri UHPEDM ve Uq-HADY kullanılarak elde edilmiştir. Önerilen metodların etkinliğinin ve güvenilirliğinin doğrulanmasını sağlamak amacıyla bilgisayar simülasyonları yapılmıştır. Kesin çözümler ile yeni yöntemlerden elde edilen çözümler arasında karşılaştırma analizi yapıldığında, bu yaklaşımların her ikisinin de doğrusal olmayan uyumlu zaman-kesirli bağlı sistemleri ele almada basitlik, etkinlik ve yeterlilik sergiledikleri ortaya çıkmaktadır.

Anahtar Kelimeler

Cahn-Allen denklemi, uyumlu homotopi pertürbasyon Elzaki dönüşüm metodu, uyumlu Elzaki dönüşümü

Kaynakça

• Liouville J. (1832). Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions. Ecole polytechnique, 13, 71-162.
• Miller, K. S., Ross, B. (1993). An introduction to the fractional calculus and fractional differential equations, Wiley, New York.
• Podlubny, I. (1999). Fractional differential equations, mathematics in science and engineering, Academic Press, New York.
• Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J. J. (2012). Fractional calculus: models and numerical methods, World Scientific, London.
• Povstenko, Y. (2015). Linear fractional diffusion-wave equation for scientists and engineers. Birkhäuser, Switzerland.
• Baleanu D., Wu G.C., Zeng S.D. (2017). Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations. Chaos Solitons Fractals, 102, 99–105.
• Sweilam, N. H., Abou Hasan, M. M., Baleanu, D. (2017). New studies for general fractional financial models of awareness and trial advertising decisions. Chaos, Solitons & Fractals, 104, 772-784.
• Liu D. Y., Gibaru O., Perruquetti W., Laleg-Kirati T. M. (2015). Fractional order differentiation by integration and error analysis in noisy environment. IEEE Transactions on Automatic Control, 60, 2945–2960.
• Esen A., Sulaiman T.A., Bulut H., Baskonus H. M. (2018). Optical solitons to the space-time fractional (1+1)-dimensional coupled nonlinear Schrödinger equation. Optik, 167, 150–156.
• Caponetto R., Dongola G., Fortuna L., Gallo A. (2010). New results on the synthesis of FO-PID controllers. Communications in Nonlinear Science and Numerical Simulation, 15, 997–1007.
• Veeresha, P., Prakasha, D.G., Baskonus, H. M. (2019). Novel simulations to the time-fractional Fisher’s equation. Mathematical Sciences, 13(1), 33-42.
• Khalil, R., Al Horani, M., Yousef, A., Sababheh, M. (2014). A new definition of fractional derivative. Journal of computational and applied mathematics, 264, 65-70.
• Aggarwal, S., Chauhan, R., Sharma, N. (2018). Application of Elzaki transform for solving linear Volterra integral equations of first kind. International Journal of Research in Advent Technology, 6(12), 3687-3692.
• Elzaki, T. M. (2011). Applications of new transform “Elzaki transform” to partial differential equations. Global Journal of Pure and Applied Mathematics, 7(1), 65-70.
• Elzaki, T. M. (2012). Solution of nonlinear differential equations using mixture of Elzaki transform and differential transform method. In International Mathematical Forum, 7(13), 631-638.
• Elzaki, T. M., Hilal, E. M. A. (2012). Homotopy perturbation and Elzaki transform for solving nonlinear partial differential equations. Mathematical Theory and Modeling, 2(3), 33-42.
• Elzaki, T. M., Kim, H. (2015). The solution of radial diffusivity and shock wave equations by Elzaki variational iteration method. International Journal of Mathematical Analysis, 9(22), 1065-1071.
• Jena, R. M., Chakraverty, S. (2019). Solving time-fractional Navier–Stokes equations using homotopy perturbation Elzaki transform. SN Applied Sciences, 1(1), 1-16.
• Abu-Gdairi, R., Al-Smadi, M., Gumah, G. (2015). An expansion iterative technique for handling fractional differential equations using fractional power series scheme. Journal of Mathematics and Statistics, 11(2), 29–38.
• Baleanu, D., Golmankhaneh, A. K., Baleanu, M. C. (2009). Fractional electromagnetic equations using fractional forms. International Journal of Theoretical Physics, 48(11), 3114–3123.
• Baleanu, D., Jajarmi, A., Hajipour, M. (2018). On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag–Leffler kernel. Nonlinear Dynamics, 2018(1), 1–18.
• Baleanu, D., Asad, J. H., Jajarmi, A. 2018. New aspects of the motion of a particle in a circular cavity. Proceedings of the Romanian Academy Series A, 19(2), 143–149.
• Baleanu, D., Jajarmi, A., Bonyah, E., Hajipour, M. (2018). New aspects of poor nutrition in the life cycle within the fractional calculus. Advances in Difference Equations, 2018(1), 1-14.
• Anaç, H., Merdan, M., Bekiryazıcı, Z., Kesemen, T. (2019). Bazı Rastgele Kısmi Diferansiyel Denklemlerin Diferansiyel Dönüşüm Metodu ve Laplace-Padé Metodu Kullanarak Çözümü. Gümüşhane Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 9(1), 108-118.
• Ayaz, F. (2004). Solutions of the system of differential equations by differential transform method. Applied Mathematics and Computation, 147(2), 547-567.
• He, J. H. (1999). Variational iteration method-a kind of non-linear analytical technique: some examples. International Journal of Non-linear Mechanics, 34(4), 699-708.
• He, J. H. (2003). Homotopy perturbation method: a new nonlinear analytical technique. Applied Mathematics and Computation, 135(1), 73-79.
• He, J. H. (2006). Homotopy perturbation method for solving boundary value problems. Physics Letters, 350(1-2), 87-88.
• He, J. H. (2006). Addendum: new interpretation of homotopy perturbation method. International Journal of Modern Physics B, 20(18), 2561-2568.
• Jajarmi, A., Baleanu, D. (2018). Suboptimal control of fractional-order dynamic systems with delay argument. Journal of Vibration and Control, 24(12), 2430-2446.
• Jajarmi, A., Baleanu, D. (2018). A new fractional analysis on the interaction of HIV with CD4+ T-cells, Chaos, Solitons & Fractals, 113, 221-229.
• Kangalgil, F., Ayaz, F. (2009). Solitary wave solutions for the KdV and mKdV equations by differential transform method. Chaos, Solitons & Fractals, 41(1), 464-472.
• Klimek, M. (2001). Fractional sequential mechanics-models with symmetric fractional derivative. Czechoslovak Journal of Physics, 51(12), 1348-1354.
• Merdan, M. (2010). A new applicaiton of modified differential transformation method for modeling the pollution of a system of lakes. Selçuk Journal of Applied Mathematics, 11(2), 27-40.
• Alkan, A. (2022). Improving Homotopy Analysis Method with An Optimal Parameter for Time-Fractional Burgers Equation. Karamanoğlu Mehmetbey Üniversitesi Mühendislik ve Doğa Bilimleri Dergisi, 4(2), 117-134.
• Wang, K., Liu, S. (2016). A new Sumudu transform iterative method for time-fractional Cauchy reaction-diffusion equation. Springer Plus, 5(1), 865.
• Wazwaz, A. M. (1999). A reliable modification of Adomian decomposition method. Applied Mathematics and Computation, 102(1), 77-86.
• Aslefallah, M., Abbasbandy, S., Yüzbaşi, Ş. (2023). Numerical Solution for a Class of Nonlinear Emden-Fowler Equations by Exponential Collocation Method. Applications and Applied Mathematics: An International Journal (AAM), 18(1), 10.
• Abdeljawad, T. (2015). On conformable fractional calculus. Journal of computational and Applied Mathematics, 279, 57-66.
• Ala, V., Demirbilek, U., Mamedov, K. R. (2020). An application of improved Bernoulli sub-equation function method to the nonlinear conformable time-fractional SRLW equation. AIMS Mathematics, 5(4), 3751-3761.
• Gözütok, U., Çoban, H., Sağıroğlu, Y. (2019). Frenet frame with respect to conformable derivative. Filomat, 33(6), 1541-1550.
• Shrinath, M., Bhadane, A. (2019). New conformable fractional Elzaki transformation: Theory and applications. Malaya Journal of Matematik, 1, 619-625.
• Ali, L., Shah, R., & Weera, W. (2022). Fractional View Analysis of Cahn–Allen Equations by New Iterative Transform Method. Fractal and Fractional, 6(6), 293.
• Yasar, E., Giresunlu, I. B. (2016). The (G’/G, 1/G)-expansion method for solving nonlinear space-time fractional differential equations. Pramana, 87, 17.
• Esen, A., Yagmurlu, N. M., Tasbozan, O. (2013). Approximate analytical solution to time-fractional damped Burger and Cahn-Allen equations. Appl. Math. Inf. Sci., 7, 1951.
• Jafari, H., Tajadodi, H., Baleanu, D. (2014). Application of a homogeneous balance method to exact solutions of nonlinear fractional evolution equations. J. Comput. Nonlinear Dyn., 9, 021019-1.
• Hariharan, G., Kannan, K. (2009). Haar wavelet method for solving Cahn-Allen equation. Appl. Math. Sci., 3, 2523–2533.
• Tascan, F., Bekir, A. (2009). Travelling wave solutions of the Cahn-Allen equation by using first integral method. Appl. Math. Comput., 207, 279–282.
• Tariq, H., Akram, G. (2017). New traveling wave exact and approximate solutions for the nonlinear Cahn-Allen equation: Evolution of a nonconserved quantity. Nonlinear Dyn., 88, 581–594.
• Bekir, A. (2012). Multisoliton solutions to Cahn-Allen equation using double exp-function method. Phys. Wave Phenom., 20, 118–121.
• Guner, O., Bekir, A., Cevikel, A.C. (2015). A variety of exact solutions for the time fractional Cahn-Allen equation. The European Physical Journal Plus, 130, 1-13.