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

Year 2023, , 1 - 24, 30.06.2023

Ahmet Semih Erol , Halil Anaç , Ali Olgun

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

Cited By: 2

Abstract

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.

Keywords

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

Abstract

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.

Keywords

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

References

• 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).