Oransal Gecikmeli Uyumlu Zaman Kesirli Genelleştirilmiş Burgers Denkleminin Yeni Yöntemlerle Sayısal Çözümü
Year 2023,
Volume: 13 Issue: 2, 310 - 335, 15.06.2023
Abdullah Kartal
,
Halil Anaç
,
Ali Olgun
Abstract
Uyumlu kesirli q-homotopi analiz dönüşümü yöntemi ve uyumlu Shehu homotopi pertürbasyon yöntemi olarak adlandırılan iki yeni yöntem kullanılarak, orantılı gecikmeli uyumlu zaman-fraksiyonel kısmi diferansiyel denklemler analiz edilmiştir. Bu denklemin sayısal çözümlerinin grafikleri çizilir. Sayısal simülasyonlara göre önerilen yöntemler etkili ve güvenilirdir.
References
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- Abazari, R., and Kılıcman, A. (2014). Application of differential transform method on nonlinear integro–differential equations with proportional delay. Neural Computing and Applications, 24(2), 391–397.
- Abdeljawad, T. (2015). On conformable fractional calculus. Journal of Computational and Applied Mathematics, 279, 57-66.
- Ala, V. (2022). New exact solutions of space-time fractional Schrödinger-Hirota equation. Bulletin of the Karaganda University Mathematics Series, 107(3).
- Ala, V., and Shaikhova, G. (2022). Analytical Solutions of Nonlinear Beta Fractional Schrödinger Equation Via Sine-Cosine Method. Lobachevskii Journal of Mathematics, 43(11), 3033-3038.
- 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.
- Baleanu, D., Diethelm, K., Scalas, E., and Trujillo, J. J., (2012). Fractional Calculus: Models and Numerical Methods. Boston, USA: World Scientific.
- Baleanu, D., Wu, G. C., and Zeng, S. D., (2017). Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations. Chaos Solitons Fractals, 102, 99–105.
- Benattia, M. E., and Belghaba, K. (2021). Shehu conformable fractional transform, theories and applications. Cankaya University Journal of Science and Engineering, 18(1), 24-32.
- Biazar, J., and 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.
- Caponetto, R., Dongola, G., Fortuna, L., and Gallo, A., (2010). New results on the synthesis of FO-PID controllers. Communications in Nonlinear Science and Numerical Simulation, 15, 997–1007.
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- Debnath, L. (2003). Recent applications of fractional calculus to science and engineering. International Journal of Mathematics and Mathematical Sciences, 2003(54), 3413-3442.
- Esen, A., Sulaiman, T. A., Bulut, H., and Baskonus, H. M., (2018). Optical solitons to the space-time fractional (1+1)-dimensional coupled nonlinear Schrödinger equation. Optik, 167, 150–156.
- Gao, F., and Chi, C. (2020). Improvement on conformable fractional derivative and its applications in fractional differential equations. Journal of Function Spaces, 2020, 5852414.
- Gözütok, N. Y., and Gözütok, U. (2017). Multivariable conformable fractional calculus. arXiv preprint arXiv:1701.00616.
- Keller, A. A. (2010). Contribution of the delay differential equations to the complex economic macrodynamics. WSEAS Transactions on Systems, 9(4), 358–371.
- Kilbas, A. A., Srivastava, H. M., and Trujillo, J. J., (2006). Theory and applications of fractional differential equations. Amsterdam: Elsevier B.V.
- Khalil, R., Al Horani, M., Yousef, A., and Sababheh, M., (2014). A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264, 65-70.
- 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.
- Liu, D. Y., Gibaru, O., Perruquetti, W., and Laleg-Kirati, T. M., (2015). Fractional order differentiation by integration and error analysis in noisy environment. IEEE Transactions on Automatic Control, 60, 2945–2960.
- Maitama, S., and Zhao, W., (2019). New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations. arXiv preprint arXiv:1904.11370.
- Mead, J., and Zubik-Kowal, B., (2005). An iterated pseudospectral method for delay partial differential equations. Applied Numerical Mathematics, 55(2), 227–250.
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- Podlubny, I., (1999). Fractional Differential Equations. New York, NY: Academic Press.
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- Prakash, A., Veeresha, P., Prakasha, D. G., and Goyal, M., (2019). A homotopy technique for fractional ordermulti-dimensional telegraph equation via Laplace transform. The European Physical Journal Plus, 134, 1–18.
- Riemann, G. F. B., (1896). Versuch einer allgemeinen Auffassung der Integration und Differentiation. Leipzig, Germany: Gesammelte Mathematische Werke.
- Sakar, M. G., Uludag, F., and 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.
- Singh, B. K., and Kumar, P., (2017). Fractional variational iteration method for solving fractional partial differential equations with proportional delay. International Journal of Differential. Equations, 2017, 5206380.
- Sweilam, N. H., Hasan, M. M. A., and Baleanu, D., (2017). New studies for general fractional financial models of awareness and trial advertising decisions. Chaos Solitons Fractal, 104, 772–784.
- Tanthanuch, J. (2012). Symmetry analysis of the nonhomogeneous inviscid Burgers equation with delay. Communications in Nonlinear Science and Numerical Simulation, 17(12), 4978–4987.
- Veeresha, P., Prakasha, D. G., and Baskonus, H. M., (2019a). Novel simulations to the time-fractional Fisher’s equation. Mathematical Sciences, 13(1), 33-42.
- Veeresha, P., Prakasha, D. G., and Baskonus, H. M., (2019b). New numerical surfaces to the mathematical model of cancerchemotherapy effect in Caputo fractional derivatives. Chaos, 29, 013119.
- Wu, J., (1996). Theory and Applications of Partial Functional Differential Equations. New York, NY: Springer.
- Zubik-Kawal, B. (2000). Chebyshev pseudospectral method and waveform relaxation for differential and differential-functional parabolic equations. Applied Numerical Mathematics, 34(2-3), 309-328.
- Zubik-Kawal, B., and Jackiewicz, Z., (2006). Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations. Applied Numerical Mathematics, 56(3-4), 433–443.
Numerical Solution of Conformable Time Fractional Generalized Burgers Equation with Proportional Delay by New Methods
Year 2023,
Volume: 13 Issue: 2, 310 - 335, 15.06.2023
Abdullah Kartal
,
Halil Anaç
,
Ali Olgun
Abstract
By using two new methods, called the conformable fractional q-homotopy analysis transform method and the conformable Shehu homotopy perturbation method, the conformable time-fractional partial differential equations with proportional delay is analysed. The graphs of this equation's numerical solutions are drawn. According to numerical simulations, the proposed methods are effective and reliable.
References
- Abazari, R., and 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.
- Abazari, R., and Kılıcman, A. (2014). Application of differential transform method on nonlinear integro–differential equations with proportional delay. Neural Computing and Applications, 24(2), 391–397.
- Abdeljawad, T. (2015). On conformable fractional calculus. Journal of Computational and Applied Mathematics, 279, 57-66.
- Ala, V. (2022). New exact solutions of space-time fractional Schrödinger-Hirota equation. Bulletin of the Karaganda University Mathematics Series, 107(3).
- Ala, V., and Shaikhova, G. (2022). Analytical Solutions of Nonlinear Beta Fractional Schrödinger Equation Via Sine-Cosine Method. Lobachevskii Journal of Mathematics, 43(11), 3033-3038.
- 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.
- Baleanu, D., Diethelm, K., Scalas, E., and Trujillo, J. J., (2012). Fractional Calculus: Models and Numerical Methods. Boston, USA: World Scientific.
- Baleanu, D., Wu, G. C., and Zeng, S. D., (2017). Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations. Chaos Solitons Fractals, 102, 99–105.
- Benattia, M. E., and Belghaba, K. (2021). Shehu conformable fractional transform, theories and applications. Cankaya University Journal of Science and Engineering, 18(1), 24-32.
- Biazar, J., and 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.
- Caponetto, R., Dongola, G., Fortuna, L., and Gallo, A., (2010). New results on the synthesis of FO-PID controllers. Communications in Nonlinear Science and Numerical Simulation, 15, 997–1007.
- Caputo, M., (1969). Elasticità e Dissipazione. Bologna, Italy: Zanichelli.
- Chen, X., and Wang, L. (2010). The variational iteration method for solving a neutral functional-differential equation with proportional delays. Computers and Mathematics with Applications, 59(8), 2696-2702.
- Debnath, L. (2003). Recent applications of fractional calculus to science and engineering. International Journal of Mathematics and Mathematical Sciences, 2003(54), 3413-3442.
- Esen, A., Sulaiman, T. A., Bulut, H., and Baskonus, H. M., (2018). Optical solitons to the space-time fractional (1+1)-dimensional coupled nonlinear Schrödinger equation. Optik, 167, 150–156.
- Gao, F., and Chi, C. (2020). Improvement on conformable fractional derivative and its applications in fractional differential equations. Journal of Function Spaces, 2020, 5852414.
- Gözütok, N. Y., and Gözütok, U. (2017). Multivariable conformable fractional calculus. arXiv preprint arXiv:1701.00616.
- Keller, A. A. (2010). Contribution of the delay differential equations to the complex economic macrodynamics. WSEAS Transactions on Systems, 9(4), 358–371.
- Kilbas, A. A., Srivastava, H. M., and Trujillo, J. J., (2006). Theory and applications of fractional differential equations. Amsterdam: Elsevier B.V.
- Khalil, R., Al Horani, M., Yousef, A., and Sababheh, M., (2014). A new definition of fractional derivative. Journal of Computational and Applied Mathematics, 264, 65-70.
- 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.
- Liu, D. Y., Gibaru, O., Perruquetti, W., and Laleg-Kirati, T. M., (2015). Fractional order differentiation by integration and error analysis in noisy environment. IEEE Transactions on Automatic Control, 60, 2945–2960.
- Maitama, S., and Zhao, W., (2019). New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations. arXiv preprint arXiv:1904.11370.
- Mead, J., and Zubik-Kowal, B., (2005). An iterated pseudospectral method for delay partial differential equations. Applied Numerical Mathematics, 55(2), 227–250.
- Miller, K. S., and Ross, B., (1993). An Introduction to Fractional Calculus and Fractional Differential Equations. New York, NY: Wiley.
- Mittag-Leffler, G. M. (1903). Sur la nouvelle fonction E_α (x). Comptes Rendus de l’Academie des Sciences, 137, 554-558.
- Podlubny, I., (1999). Fractional Differential Equations. New York, NY: Academic Press.
- Povstenko, Y., (2015). Linear Fractional Diffusion-Wave Equation for Scientists and Engineers. New York, NY: Birkhäuser.
- Prakash, A., Veeresha, P., Prakasha, D. G., and Goyal, M., (2019). A homotopy technique for fractional ordermulti-dimensional telegraph equation via Laplace transform. The European Physical Journal Plus, 134, 1–18.
- Riemann, G. F. B., (1896). Versuch einer allgemeinen Auffassung der Integration und Differentiation. Leipzig, Germany: Gesammelte Mathematische Werke.
- Sakar, M. G., Uludag, F., and 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.
- Singh, B. K., and Kumar, P., (2017). Fractional variational iteration method for solving fractional partial differential equations with proportional delay. International Journal of Differential. Equations, 2017, 5206380.
- Sweilam, N. H., Hasan, M. M. A., and Baleanu, D., (2017). New studies for general fractional financial models of awareness and trial advertising decisions. Chaos Solitons Fractal, 104, 772–784.
- Tanthanuch, J. (2012). Symmetry analysis of the nonhomogeneous inviscid Burgers equation with delay. Communications in Nonlinear Science and Numerical Simulation, 17(12), 4978–4987.
- Veeresha, P., Prakasha, D. G., and Baskonus, H. M., (2019a). Novel simulations to the time-fractional Fisher’s equation. Mathematical Sciences, 13(1), 33-42.
- Veeresha, P., Prakasha, D. G., and Baskonus, H. M., (2019b). New numerical surfaces to the mathematical model of cancerchemotherapy effect in Caputo fractional derivatives. Chaos, 29, 013119.
- Wu, J., (1996). Theory and Applications of Partial Functional Differential Equations. New York, NY: Springer.
- Zubik-Kawal, B. (2000). Chebyshev pseudospectral method and waveform relaxation for differential and differential-functional parabolic equations. Applied Numerical Mathematics, 34(2-3), 309-328.
- Zubik-Kawal, B., and Jackiewicz, Z., (2006). Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations. Applied Numerical Mathematics, 56(3-4), 433–443.