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Oransal Gecikmeli Uyumlu Zaman-Kesirli Mertebeden Genelleştirilmiş Burgers Denklemini Çözmek için Yeni Uyumlu Metotlar

Yıl 2023, Cilt: 39 Sayı: 2, 315 - 329, 31.08.2023

Halil Hüseyin Avcı Halil Anaç

Öz

Bu makalede, uyumlu q-Mohand homotopi analiz dönüşüm yöntemi (Uq-MHADY) ve uyumlu Mohand Adomian ayrıştırma yöntemi (UMAAY) olarak adlandırılan iki yeni yöntem, oransal gecikmeli doğrusal olmayan uyumlu zaman-kesirli mertebeden genelleştirilmiş Burgers denkleminin yeni sayısal çözümlerini incelemek için kullanılmaktadır. Önerilen iki yeni yöntemden ilki olan Uq-MHADY, q-homotopi analiz dönüşüm yöntemi ile uyumlu Mohand dönüşümünün birleşiminden oluşan hibrit bir yöntemdir. Diğer yöntem olan CMADM ise Adomian ayrıştırma yöntemi ile uyumlu Mohand dönüşümünün birleşiminden oluşan hibrit bir yöntemdir. Önerilen yöntemlerin etkin çalıştığını ve güvenilir olduğunu göstermek için bilgisayar simülasyonları yapılmaktadır. Kesin çözümler bulunan çözümlerle karşılaştırıldığında, yeni tekniklerin her ikisinin de basit, güçlü ve oransal gecikmeli doğrusal olmayan uyumlu zaman-kesirli mertebeden kısmi diferansiyel denklemi çözmek için iyi çalıştığını görülmektedir.

Anahtar Kelimeler

Uyumlu zaman-kesirli mertebeden genelleştirilmiş Burgers denklemi Uyumlu Mohand Adomian ayrıştırma metodu Uyumlu Mohand dönüşümü

Kaynakça

  • [1] Miller, K. S., Ross, B. 1993. An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 376 p.
  • [2] Podlubny, I. 1999. Fractional differential equations, mathematics in science and engineering, Academic Press, New York, 365 p.
  • [3] Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J. J. 2012. Fractional calculus: models and numerical methods, World Scientific, London, 476 p.
  • [4] Povstenko, Y. 2015. Linear fractional diffusion-wave equation for scientists and engineers. Birkhäuser, Switzerland, 460 p.
  • [5] Ala, V. 2022. New exact solutions of space-time fractional Schrödinger-Hirota equation. Bulletin of the Karaganda university Mathematics series, 107(3), 17-24.
  • [6] Ala, V. 2023. Exact Solutions of Nonlinear Time Fractional Schrödinger Equation with Beta- Derivative. Fundamentals of Contemporary Mathematical Sciences, 4(1), 1-8.
  • [7] 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.
  • [8] 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.
  • [9] 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.
  • [10] 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.
  • [11] Veeresha, P., Prakasha, D. G., Baskonus, H. M. 2019. Novel simulations to the time-fractional Fisher’s equation. Mathematical Sciences, 13(1), 33-42.
  • [12] Veeresha, P., Prakasha, D. G., Baskonus, H. M. 2019. 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.
  • [13] 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.
  • [14] 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.
  • [15] Mohand, M., Mahgoub, A. 2017. The new integral transform “Mohand Transform”. Advances in Theoretical and Applied Mathematics, 12(2), 113-120.
  • [16] Aggarwal, S., Sharma, N., Chauhan, R. 2018. Solution of linear Volterra integral equations of second kind using Mohand transform. International Journal of Research in Advent Technology, 6(11), 3098-3102.
  • [17] Aggarwal, S., Gupta, A. R., Singh, D. P., Asthana, N., Kumar, N. 2018. Application of Laplace transform for solving population growth and decay problems. International Journal of Latest Technology in Engineering, Management & Applied Science, 7(9), 141-145.
  • [18] Sathya, S., Rajeswari, I. 2018. Applications of Mohand transform for solving linear partial integrodifferential equations. International Journal of Research in Advent Technology, 6(10), 2841-2843.
  • [19] Kumar, P. S., Gomathi, P., Gowri, S., Viswanathan, A. 2018. Applications of Mohand transform to mechanics and electrical circuit problems. International Journal of Research in Advent Technology, 6(10), 2838-2840.
  • [20] Kumar, P. S., Saranya, C., Gnanavel, M. G., Viswanathan, A. 2018. Applications of Mohand transform for solving linear Volterra integral equations of first kind. International Journal of Research in Advent Technology, 6(10), 2786-2789.
  • [21] 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.
  • [22] 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.
  • [23] 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.
  • [24] Mead, J., Zubik-Kowal, B. 2005. An iterated pseudospectral method for delay partial differential equations. Applied numerical mathematics, 55(2), 227-250.
  • [25] 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.
  • [26] Abazari, R., Kılıcman, A. 2014. Application of differential transform method on nonlinear integrodifferential equations with proportional delay. Neural Computing and Applications, 24, 391-397.
  • [27] Tanthanuch, J. 2012. Symmetry analysis of the nonhomogeneous inviscid Burgers equation with delay. Communications in Nonlinear Science and Numerical Simulation, 17(12), 4978-4987.
  • [28] 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.
  • [29] 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.
  • [30] 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.
  • [31] Singh, B. K., Kumar, P. Fractional variational iteration method for solving fractional partial differential equations with proportional delay. International journal of differential equations, 2017, 2017.
  • [32] 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.
  • [33] Abdeljawad, T. 2015. On conformable fractional calculus. Journal of computational and Applied Mathematics, 279, 57-66.
  • [34] 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.
  • [35] Gözütok, U., Çoban, H., Sağıroğlu, Y. 2019. Frenet frame with respect to conformable derivative. Filomat, 33(6), 1541-1550.

The New Conformable Methods to Solve Conformable Time- Fractional Generalized Burgers Equation with Proportional Delay

Yıl 2023, Cilt: 39 Sayı: 2, 315 - 329, 31.08.2023

Halil Hüseyin Avcı Halil Anaç

Öz

In this article, two novel methods called conformable q-Mohand homotopy analysis transform method (Cq-MHATM) and conformable Mohand Adomian decomposition method (CMADM) are utilized to examine the novel numerical solutions for nonlinear conformable time-fractional generalized Burgers equation with proportional delay. The first of the two new methods proposed, Cq-MHATM, is a hybrid method that combines q-homotopy analysis transform method and Mohand transform in the sense of comformable derivative. The other method, CMADM is also a hybrid method that combines Adomian decomposition method and Mohand transform in the sense of comformable derivative. The computer simulations were worked out to prove that the proposed methods work and are trusted. When the exact solutions are compared to the solutions that were found, it is seen that both of the new techniques are simple, powerful, and work well to solve nonlinear conformable time-fractional partial differential equation with proportional delay.

Anahtar Kelimeler

Conformable time-fractional generalized Burgers equation Conformable Mohand Adomian decomposition method Conformable Mohand transform

Kaynakça

  • [1] Miller, K. S., Ross, B. 1993. An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 376 p.
  • [2] Podlubny, I. 1999. Fractional differential equations, mathematics in science and engineering, Academic Press, New York, 365 p.
  • [3] Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J. J. 2012. Fractional calculus: models and numerical methods, World Scientific, London, 476 p.
  • [4] Povstenko, Y. 2015. Linear fractional diffusion-wave equation for scientists and engineers. Birkhäuser, Switzerland, 460 p.
  • [5] Ala, V. 2022. New exact solutions of space-time fractional Schrödinger-Hirota equation. Bulletin of the Karaganda university Mathematics series, 107(3), 17-24.
  • [6] Ala, V. 2023. Exact Solutions of Nonlinear Time Fractional Schrödinger Equation with Beta- Derivative. Fundamentals of Contemporary Mathematical Sciences, 4(1), 1-8.
  • [7] 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.
  • [8] 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.
  • [9] 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.
  • [10] 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.
  • [11] Veeresha, P., Prakasha, D. G., Baskonus, H. M. 2019. Novel simulations to the time-fractional Fisher’s equation. Mathematical Sciences, 13(1), 33-42.
  • [12] Veeresha, P., Prakasha, D. G., Baskonus, H. M. 2019. 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.
  • [13] 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.
  • [14] 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.
  • [15] Mohand, M., Mahgoub, A. 2017. The new integral transform “Mohand Transform”. Advances in Theoretical and Applied Mathematics, 12(2), 113-120.
  • [16] Aggarwal, S., Sharma, N., Chauhan, R. 2018. Solution of linear Volterra integral equations of second kind using Mohand transform. International Journal of Research in Advent Technology, 6(11), 3098-3102.
  • [17] Aggarwal, S., Gupta, A. R., Singh, D. P., Asthana, N., Kumar, N. 2018. Application of Laplace transform for solving population growth and decay problems. International Journal of Latest Technology in Engineering, Management & Applied Science, 7(9), 141-145.
  • [18] Sathya, S., Rajeswari, I. 2018. Applications of Mohand transform for solving linear partial integrodifferential equations. International Journal of Research in Advent Technology, 6(10), 2841-2843.
  • [19] Kumar, P. S., Gomathi, P., Gowri, S., Viswanathan, A. 2018. Applications of Mohand transform to mechanics and electrical circuit problems. International Journal of Research in Advent Technology, 6(10), 2838-2840.
  • [20] Kumar, P. S., Saranya, C., Gnanavel, M. G., Viswanathan, A. 2018. Applications of Mohand transform for solving linear Volterra integral equations of first kind. International Journal of Research in Advent Technology, 6(10), 2786-2789.
  • [21] 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.
  • [22] 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.
  • [23] 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.
  • [24] Mead, J., Zubik-Kowal, B. 2005. An iterated pseudospectral method for delay partial differential equations. Applied numerical mathematics, 55(2), 227-250.
  • [25] 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.
  • [26] Abazari, R., Kılıcman, A. 2014. Application of differential transform method on nonlinear integrodifferential equations with proportional delay. Neural Computing and Applications, 24, 391-397.
  • [27] Tanthanuch, J. 2012. Symmetry analysis of the nonhomogeneous inviscid Burgers equation with delay. Communications in Nonlinear Science and Numerical Simulation, 17(12), 4978-4987.
  • [28] 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.
  • [29] 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.
  • [30] 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.
  • [31] Singh, B. K., Kumar, P. Fractional variational iteration method for solving fractional partial differential equations with proportional delay. International journal of differential equations, 2017, 2017.
  • [32] 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.
  • [33] Abdeljawad, T. 2015. On conformable fractional calculus. Journal of computational and Applied Mathematics, 279, 57-66.
  • [34] 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.
  • [35] Gözütok, U., Çoban, H., Sağıroğlu, Y. 2019. Frenet frame with respect to conformable derivative. Filomat, 33(6), 1541-1550.
Toplam 35 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Kısmi Diferansiyel Denklemler
Bölüm Makaleler
Yazarlar

Halil Hüseyin Avcı 0009-0007-6734-0236

Halil Anaç 0000-0002-1316-3947

Yayımlanma Tarihi 31 Ağustos 2023
Yayımlandığı Sayı Yıl 2023 Cilt: 39 Sayı: 2

Kaynak Göster

APA Avcı, H. H., & Anaç, H. (2023). The New Conformable Methods to Solve Conformable Time- Fractional Generalized Burgers Equation with Proportional Delay. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi, 39(2), 315-329.
AMA Avcı HH, Anaç H. The New Conformable Methods to Solve Conformable Time- Fractional Generalized Burgers Equation with Proportional Delay. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi. Ağustos 2023;39(2):315-329.
Chicago Avcı, Halil Hüseyin, ve Halil Anaç. “The New Conformable Methods to Solve Conformable Time- Fractional Generalized Burgers Equation With Proportional Delay”. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi 39, sy. 2 (Ağustos 2023): 315-29.
EndNote Avcı HH, Anaç H (01 Ağustos 2023) The New Conformable Methods to Solve Conformable Time- Fractional Generalized Burgers Equation with Proportional Delay. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi 39 2 315–329.
IEEE H. H. Avcı ve H. Anaç, “The New Conformable Methods to Solve Conformable Time- Fractional Generalized Burgers Equation with Proportional Delay”, Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi, c. 39, sy. 2, ss. 315–329, 2023.
ISNAD Avcı, Halil Hüseyin - Anaç, Halil. “The New Conformable Methods to Solve Conformable Time- Fractional Generalized Burgers Equation With Proportional Delay”. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi 39/2 (Ağustos 2023), 315-329.
JAMA Avcı HH, Anaç H. The New Conformable Methods to Solve Conformable Time- Fractional Generalized Burgers Equation with Proportional Delay. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi. 2023;39:315–329.
MLA Avcı, Halil Hüseyin ve Halil Anaç. “The New Conformable Methods to Solve Conformable Time- Fractional Generalized Burgers Equation With Proportional Delay”. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi, c. 39, sy. 2, 2023, ss. 315-29.
Vancouver Avcı HH, Anaç H. The New Conformable Methods to Solve Conformable Time- Fractional Generalized Burgers Equation with Proportional Delay. Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi. 2023;39(2):315-29.
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