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Oransal Gecikmeli Uyumlu Zaman Kesirli Genelleştirilmiş Burgers Denkleminin Yeni Yöntemlerle Sayısal Çözümü

Yıl 2023, Cilt: 13 Sayı: 2, 310 - 335, 15.06.2023

Abdullah Kartal Halil Anaç Ali Olgun

https://doi.org/10.31466/kfbd.1191870
Cited By: 2

Öz

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.

Anahtar Kelimeler

Uyumlu zaman kesirli genelleştirilmiş Burgers denklemi uyumlu q-homotopi analiz dönüşüm metodu uyumlu q-homotopi analiz dönüşüm metodu, uyumlu Shehu homotopi pertürbasyon metodu oransal gecikme

Kaynakça

  • 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.

Numerical Solution of Conformable Time Fractional Generalized Burgers Equation with Proportional Delay by New Methods

Yıl 2023, Cilt: 13 Sayı: 2, 310 - 335, 15.06.2023

Abdullah Kartal Halil Anaç Ali Olgun

https://doi.org/10.31466/kfbd.1191870
Cited By: 2

Öz

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.

Anahtar Kelimeler

Conformable time fractional generalized Burgers equation conformable q-homotopy analysis transform method conformable Shehu homotopy perturbation method proportional delay

Kaynakça

  • 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.
Toplam 39 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Abdullah Kartal 0000-0003-2763-7979

Halil Anaç 0000-0002-1316-3947

Ali Olgun 0000-0001-5365-4110

Erken Görünüm Tarihi 15 Haziran 2023
Yayımlanma Tarihi 15 Haziran 2023
Yayımlandığı Sayı Yıl 2023 Cilt: 13 Sayı: 2

Kaynak Göster

APA Kartal, A., Anaç, H., & Olgun, A. (2023). Numerical Solution of Conformable Time Fractional Generalized Burgers Equation with Proportional Delay by New Methods. Karadeniz Fen Bilimleri Dergisi, 13(2), 310-335. https://doi.org/10.31466/kfbd.1191870

Cited By

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https://doi.org/10.54286/ikjm.1335660
 Kapak Resmi İndir

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