The new numerical solutions of conformable time fractional generalized Burgers equation with proportional delay

Year 2023, Volume: 13 Issue: 4, 927 - 938, 15.10.2023

Abdullah Kartal , Halil Anaç , Ali Olgun

https://doi.org/10.17714/gumusfenbil.1281570

Abstract

The conformable time-fractional partial differential equations with proportional delay are studied using two new methods: the conformable fractional q-homotopy analysis transform method and the conformable Shehu homotopy perturbation method. The numerical solutions to this equation are graphed. Numerical simulations show that the proposed techniques are effective and trustworthy.

Keywords

Conformable q-homotopy analysis transform method, Conformable time-fractional generalized Burgers equation, Proportional delay

Oransal gecikmeli uyumlu zaman kesirli mertebeden genelleştirilmiş Burgers denkleminin yeni sayısal çözümleri

Abstract

Oransal gecikmeli uyumlu zaman-kesirli kısmi diferansiyel denklemler, iki yeni yöntem olan uyumlu kesirli q-homotopi analizi dönüşüm yöntemi ve uyumlu Shehu homotopi pertürbasyon yöntemi kullanılarak incelenir. Bu denklemin sayısal çözümleri grafiklerle gösterilmiştir. Sayısal simülasyonlar, önerilen tekniklerin etkili ve güvenilir olduğunu göstermektedir.

Keywords

Uyumlu q-homotopi analiz dönüşümü metodu, Uyumlu kesirli mertebeden genelleştirilmiş Burgers denklemi, Oransal gecikme

References

• Abazari, R., & Ganji, M. (2011). Extended two-dimensional DTM and its application on nonlinear PDEs with proportional delay. International Journal of Computer Mathematics, 88(8), 1749-1762. https://doi.org/10.1080/00207160.2010.526704
• Abazari, R., & Kılıcman, A. (2014). Application of differential transform method on nonlinear integro-differential equations with proportional delay. Neural Computing and Applications, 24, 391-397. https://doi.org/10.1007/s00521-012-1235-4
• Abdeljawad, T. (2015). On conformable fractional calculus. Journal of Computational and Applied Mathematics, 279, 57-66. https://doi.org/10.1016/j.cam.2014.10.016
• 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. https://doi.org/10.3934/math.2020243
• 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. https://doi.org/10.55213/kmujens.1206517 Atangana, A., Akgül, A., Khan, M. A., & Ibrahim, R. W. (2022). Conformable derivative: A derivative associated to the Riemann-Stieltjes integral. Progress in Fractional Differentiation and Applications, 8(2), 321-348. http://dx.doi.org/10.18576/pfda/080211
• Baleanu, D., Diethelm, K., Scalas, E., & Trujillo, J. J. (2012). Fractional calculus: models and numerical methods (Vol. 3). World Scientific.
• 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. https://doi.org/10.1016/j.chaos.2017.02.007
• Benattia, M. E., & Belghaba, K. (2021). Shehu conformable fractional transform, theories and applications. Cankaya University Journal of Science and Engineering, 18(1), 24-32. https://dergipark.org.tr/en/pub/cankujse/issue/61974/852208
• Biazar, J., & Ghanbari, B. (2012). The homotopy perturbation method for solving neutral functional–differential equations with proportional delays. Journal of King Saud University-Science, 24(1), 33-37. https://doi.org/10.1016/j.jksus.2010.07.026
• 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(4), 997-1007. https://doi.org/10.1016/j.cnsns.2009.05.040
• Caputo, M. (1969). Elasticitá e dissipazione (Elasticity and anelastic dissipation). Zanichelli, Bologna, 4, 98.
• Chen, X., & Wang, L. (2010). The variational iteration method for solving a neutral functional-differential equation with proportional delays. Computers & Mathematics with Applications, 59(8), 2696-2702. https://doi.org/10.1016/j.camwa.2010.01.037
• 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. https://doi.org/10.1016/j.ijleo.2018.04.015
• Gözütok, U., Çoban, H., & SAĞIROĞLU, Y. (2019). Frenet frame with respect to conformable derivative. Filomat, 33(6), 1541-1550. https://doi.org/10.2298/fil1906541g
• Jackiewicz, Z., & Zubik-Kowal, B. (2006). Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations. Applied Numerical Mathematics, 56(3-4), 433-443. https://doi.org/10.1016/j.apnum.2005.04.021
• Hasan, A., Akgül, A., Farman, M., Chaudhry, F., Sultan, M., & De la Sen, M. (2023). Epidemiological Analysis of Symmetry in Transmission of the Ebola Virus with Power Law Kernel. Symmetry, 15(3), 665. https://doi.org/10.3390/sym15030665
• 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. https://doi.org/10.1016/j.cam.2014.01.002
• Iqbal, M. S., Yasin, M. W., Ahmed, N., Akgül, A., Rafiq, M., & Raza, A. (2023). Numerical simulations of nonlinear stochastic Newell-Whitehead-Segel equation and its measurable properties. Journal of Computational and Applied Mathematics, 418, 114618. https://doi.org/10.1016/j.cam.2022.114618
• Iyanda, F. K., Rezazadeh, H., Inc, M., Akgül, A., Bashiru, I. M., Hafeez, M. B., & Krawczuk, M. (2023). Numerical simulation of temperature distribution of heat flow on reservoir tanks connected in a series. Alexandria Engineering Journal, 66, 785-795. https://doi.org/10.1016/j.aej.2022.10.062
• Liaqat, M. I., Akgül, A., De la Sen, M., & Bayram, M. (2023). Approximate and exact solutions in the sense of conformable derivatives of quantum mechanics models using a novel algorithm. Symmetry, 15(3), 744. https://doi.org/10.3390/sym15030744
• Liouville, J. (1832). Mémoire sur quelques questions de géométrie et de mécanique, et sur un nouveau genre de calcul pour résoudre ces questions.
• 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(11), 2945-2960. https://doi.org/10.1109/TAC.2015.2417852
• Mead, J., & Zubik-Kowal, B. (2005). An iterated pseudospectral method for delay partial differential equations. Applied Numerical Mathematics, 55(2), 227-250. https://doi.org/10.1016/j.apnum.2005.02.010
• Miller, K. S., & Ross, B. (1993). An introduction to the fractional calculus and fractional differential equations. Wiley.
• Podlubny, I. (1999). Fractional differential equations, mathematics in science and engineering. Academic Press.
• Povstenko, Y. (2015). Linear fractional diffusion-wave equation for scientists and engineers (p. 460). Cham: Springer International Publishing.
• Prakash, A., Veeresha, P., Prakasha, D. G., & Goyal, M. (2019). A homotopy technique for a fractional order multi-dimensional telegraph equation via the Laplace transform. The European Physical Journal Plus, 134, 1-18. https://doi.org/10.1140/epjp/i2019-12411-y
• Riemann, G. F. B. (1896). Versuch Einer Allgemeinen Auffassung der Integration und Differentiation. Gesammelte Mathematische Werke. Teubner, Leipzig.
• Sakar, M. G., Uludag, F., & Erdogan, F. (2016). Numerical solution of time-fractional nonlinear PDEs with proportional delays by homotopy perturbation method. Applied Mathematical Modelling, 40(13-14), 6639-6649. https://doi.org/10.1016/j.apm.2016.02.005
• Shahzad, A., Imran, M., Tahir, M., Khan, S. A., Akgül, A., Abdullaev, S., ... & Yahia, I. S. (2023). Brownian motion and thermophoretic diffusion impact on Darcy-Forchheimer flow of bioconvective micropolar nanofluid between double disks with Cattaneo-Christov heat flux. Alexandria Engineering Journal, 62, 1-15. https://doi.org/10.1016/j.aej.2022.07.023
• Singh, B. K., & Kumar, P. (2017). Fractional variational iteration method for solving fractional partial differential equations with proportional delay. International Journal of Differential Equations, 2017. https://doi.org/10.1155/2017/5206380
• 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. https://doi.org/10.1016/j.chaos.2017.09.013
• Tanthanuch, J. (2012). Symmetry analysis of the nonhomogeneous inviscid Burgers equation with delay. Communications in Nonlinear Science and Numerical Simulation, 17(12), 4978-4987. https://doi.org/10.1016/j.cnsns.2012.05.031
• Veeresha, P., Prakasha, D. G., & Baskonus, H. M. (2019a). Novel simulations to the time-fractional Fisher’s equation. Mathematical Sciences, 13(1), 33-42. https://doi.org/10.1007/s40096-019-0276-6
• Veeresha, P., Prakasha, D. G., & Baskonus, H. M. (2019b). New numerical surfaces to the mathematical model of cancer chemotherapy effect in Caputo fractional derivatives. Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(1), 013119. https://doi.org/10.1063/1.5074099
• Zubik-Kowal, B. (2000). Chebyshev pseudospectral method and waveform relaxation for differential and differential–functional parabolic equations. Applied Numerical Mathematics, 34(2-3), 309-328. https://doi.org/10.1016/S0168-9274(99)00135-X