Numerical Solutions of Conformable Time-Fractional Swift-Hohenberg Equation with Proportional Delay by the Novel Methods

Yıl 2023, Cilt: 5 Sayı: 1, 1 - 24, 30.06.2023

Ahmet Semih Erol , Halil Anaç , Ali Olgun

https://doi.org/10.55213/kmujens.1221889

Cited By: 2

Öz

The conformable fractional q-Shehu homotopy analysis transform method and the conformable Shehu transform decomposition method are used to analyze the conformable time-fractional Swift-Hohenberg equations with proportional delay. The graphs of the numerical solutions to this problem are drawn. The proposed methods are effective and consistent, according to numerical simulations.

Anahtar Kelimeler

Conformable time-fractional Swift-Hohenberg equation, conformable fractional q-Shehu homotopy analysis transform method, conformable fractional Shehu transform decomposition method

Oransal Gecikmeli Uyumlu Zaman-Kesirli Swift-Hohenberg Denkleminin Yeni Yöntemlerle Sayısal Çözümleri

Öz

Uyumlu kesirli q-Shehu homotopi analizi dönüşüm yöntemi ve uyumlu Shehu dönüşümü ayrıştırma yöntemi, oransal gecikmeli uyumlu zaman-kesirli Swift-Hohenberg denklemlerini analiz etmek için kullanılmıştır. Bu problemin sayısal çözümlerinin grafikleri çizdirilmiştir. Önerilen yöntemler, sayısal simülasyonlara göre etkili ve tutarlıdır.

Anahtar Kelimeler

Uyumlu kesirli zaman kesirli mertebeden Swift-Hohenberg denklemi, uyumlu kesirli q-Shehu homotopi analiz dönüşüm metodu, uyumlu kesirli Shehu dönüşümü ayrıştırma metodu.

Kaynakça

• Abdeljawad T., On conformable fractional calculus, Journal of Computational and Applied Mathematics, Vol. 279, pp. 57-66, (2015).
• Baleanu D., Diethelm K., Scalas E., Trujillo JJ., Fractional Calculus: Models and Numerical Methods, Boston, MA, USA, World Scientific, (2012).
• Baleanu D., Wu GC., Zeng SD., Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations, Chaos Solitons Fractals, Vol. 102, pp. 99–105, (2017).
• Benattia ME., Belghaba K., Shehu conformable fractional transform, theories and applications, Cankaya University Journal of Science and Engineering, Vol. 18, No. 1, pp. 24-32, (2021).
• Caponetto R., Dongola G., Fortuna L., Gallo A., New results on the synthesis of FO-PID controllers, Communications in Nonlinear Science and Numerical Simulation, Vol. 15, pp. 997–1007, (2010).
• Caputo M., Elasticità e Dissipazione, Bologna, Italy, Zanichelli, (1969).
• Chen X., Wang L., The variational iteration method for solving a neutral functional-differential equation with proportional delays, Computers and Mathematics with Applications, Vol. 59, No. 8, pp. 2696-2702, (2010).
• Cross MC., Hohenberg PC., Pattern formation outside of equilibrium, Reviews of modern physics, Vol. 65, No. 3, pp. 851, (1993).
• Debnath L., Recent applications of fractional calculus to science and engineering, International Journal of Mathematics and Mathematical Sciences, Vol. 2003, No. 54, pp. 3413-3442, (2003).
• Esen A., Sulaiman TA., Bulut H., Baskonus HM., Optical solitons to the space-time fractional (1+1)-dimensional coupled nonlinear Schrödinger equation, Optik, Vol. 167, pp. 150–156, (2018).
• Gao F., Chi C., Improvement on conformable fractional derivative and its applications in fractional differential equations, Journal of Function Spaces, 2020, 5852414, (2020).
• Gözütok NY., Gözütok U., Multivariable conformable fractional calculus, Filomat, Vol. 32, No. 1, pp. 45-53, (2017).
• Hohenberg PC., Swift JB., Effects of additive noise at the onset of Rayleigh-Bénard convection, Physical Review A, Vol. 46, No. 8, pp. 4773, (1992).
• Hoyle RB., Pattern Formation, Cambridge, UK, Cambridge University Press, (2006).
• Keller AA., Contribution of the delay differential equations to the complex economic macrodynamics, WSEAS Transactions on Systems, Vol. 9, No. 4, pp. 358–371, (2010).
• Khalil R., Al Horani M., Yousef A., Sababheh M., A new definition of fractional derivative, Journal of Computational and Applied Mathematics, Vol. 264, pp. 65-70, (2014).
• Khan NA., Khan NU., Ayaz M., Mahmood A., Analytical methods for solving the time-fractional Swift–Hohenberg (S–H) equation, Comput. Math. Appl. Vol. 2011, No 61, pp. 2181–2185, (2011).
• Khan A., Liaqat MI., Alqudah MA., Abdeljawad T., Analysis of the Conformable Temporal-Fractional Swift-Hohenberg Equation Using a Novel Computational Technique, Fractals, (2023).
• Kilbas AA., Srivastava HM., Trujillo JJ, Theory and applications of fractional differential equations, Amsterdam, The Netherlands, Elsevier B.V., (2006).
• Lega J., Moloney JV., Newell AC., Swift-Hohenberg equation for lasers, Physical review letters, Vol. 73, No. 22, pp. 2978, (1994).
• Liaqat MI., Okyere E., The Fractional Series Solutions for the Conformable Time-Fractional Swift-Hohenberg Equation through the Conformable Shehu Daftardar-Jafari Approach with Comparative Analysis, Journal of Mathematics, 2022, Article ID 3295076, 20 pages, (2022).
• Liouville J., Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions, Ecole polytechnique, Vol. 13, pp. 71-162, (1832).
• Liu DY., Gibaru O., Perruquetti W., Laleg-Kirati TM., Fractional order differentiation by integration and error analysis in noisy environment, IEEE Transactions on Automatic Control, Vol. 60, pp. 2945–2960, (2015).
• Merdan M., A numeric–analytic method for time-fractional Swift–Hohenberg (S-H) equation with modified Riemann–Liouville derivative, Appl. Math. Model. Vol. 2013, No. 37, pp. 4224–4231, (2013).
• Miller KS, Ross B., An Introduction to Fractional Calculus and Fractional Differential Equations, New York, NY, USA, Wiley, (1993).
• Mittag-Leffler GM., Sur la nouvelle fonction E_α (x), Comptes Rendus de l’Academie des Sciences, Vol. 137, pp. 554-558, (1903).
• Omorodion SS., Conformable fractional reduced differential transform method for solving linear and nonlinear time-fractional Swift-Hohenberg (SH) equation, International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol. 8, No. 6, pp. 20-29, (2021).
• Peletier LA., Rottschäfer V., Large time behaviour of solutions of the Swift–Hohenberg equation, Comptes Rendus Mathematique, Vol. 336, No. 3, pp. 225-230, (2003).
• Podlubny I., Fractional Differential Equations, New York, NY, USA, Academic Press, (1999).
• Pomeau Y., Zaleski S., Manneville P., Dislocation motion in cellular structures, Physical Review A, Vol. 27, No. 5, pp. 2710, (1983).
• Povstenko Y., Linear Fractional Diffusion-Wave Equation for Scientists and Engineers, New York, NY, USA, Birkhäuser, (2015).
• Prakash A., Veeresha P., Prakasha DG., Goyal M., A homotopy technique for fractional order multi-dimensional telegraph equation via Laplace transform, The European Physical Journal Plus, Vol. 134, pp. 1–18, (2019).
• Prakasha DG., Veeresha P., Baskonus HM., Residual power series method for fractional Swift–Hohenberg equation, Fractal and Fractional, Vol. 3, No. 1, pp. 9, (2019).
• Riemann GFB, Versuch einer allgemeinen Auffassung der Integration und Differentiation, Leipzig, Germany, Gesammelte Mathematische Werke, (1896).
• Singh BK., Kumar P., Fractional variational iteration method for solving fractional partial differential equations with proportional delay, International Journal of Differential. Equations, 5206380, (2017).
• Sweilam NH., Hasan MMA., Baleanu D., New studies for general fractional financial models of awareness and trial advertising decisions, Chaos Solitons Fractals, Vol. 104, pp. 772–784, (2017).
• Swift J., Hohenberg PC., Hydrodynamic fluctuations at the convective instability, Physical Review A, Vol. 15, No. 1, pp. 319, (1977).
• Veeresha P., Prakasha DG., Baskonus HM., New numerical surfaces to the mathematical model of cancerchemotherapy effect in Caputo fractional derivatives, Chaos, 29, 013119, (2019).
• Veeresha P., Prakasha DG., Baskonus HM., Novel simulations to the time-fractional Fisher’s equation, Mathematical Sciences, Vol. 13, No. 1, pp.33-42, (2019).
• Veeresha P., Prakasha DG., Baleanu D., Analysis of fractional Swift‐Hohenberg equation using a novel computational technique, Mathematical Methods in the Applied Sciences, Vol. 43, No. 4, pp. 1970-1987, (2020).
• Vishal K., Kumar S., Das S., Application of homotopy analysis method for fractional Swift-Hohenberg equation revisited, Appl. Math. Model. Vol. 2012, No. 36, pp. 3630–3637, (2012).
• Vishal K., Das S., Ong SH., Ghosh P., On the solutions of fractional Swift-Hohenberg equation with dispersion, Appl. Math. Comput. Vol. 2013, No. 219, pp. 5792–5801, (2013).
• Wu J., Theory and Applications of Partial Functional Differential Equations, New York, NY, USA, Springer, (1996).