Atangana Konformal Türevli Modifiye Camassa-Holm ve Degasperis-Procesi Denklemlerinin Yeni Dalga Çözümlerinin Elde Edilmesi

Yıl 2024, , 819 - 828, 20.08.2024

Özlem Kırcı

https://doi.org/10.35414/akufemubid.1398202

Öz

Bu çalışmada, akışkanlar mekaniği, hidrodinamik ve fiber optik alanlarında birçok fiziksel olayı tanımlamak için sıkça kullanılması nedeni ile zaman-kesirli modifiye Camassa-Holm (mCH) ve zaman-kesirli modifiye Degasperis-Procesi (mDP) denklemlerinin yeni tam çözümlerinin elde edilmesi amaçlanmıştır. Bu kesirli denklemler, Atangana konformal türevi göz önüne alınarak nonlineer adi diferansiyel denklemlere dönüştürülmüştür. Kesirli evolüsyon denklemlerinin istenen tam çözümlerini elde etmek için bu nonlineer adi diferansiyel denklemlere (m+1/G’)-genişleme metodu uygulanmıştır. Hesaplamalar Mathematica yazılım sistemi ile gerçekleştirilmiştir. Ayrıca bu çalışmada sunulan çözümler literatürde zaman-kesirli CH ve DP denklemleri için elde edilen çözümler ile kıyaslanmış ve çözümlerin davranışları grafiksel olarak sunulmuştur.

Anahtar Kelimeler

Camassa-Holm ve Degasperis-Procesi denklemleri, (m+1/G')-genişleme yöntemi, Atangana konformal türevi

Construction of the New Wave Solutions of Modified Camassa-Holm and Degasperis-Procesi Equations with Atangana’s Conformable Derivative

Öz

In this study it is aimed to expose the new exact wave solutions of time-fractional modified Camassa-Holm (mCH) and time-fractional modified Degasperis-Procesi (mDP) equations due to being extensively used to delineate many physical phenomena in fluid mechanics, hydrodynamics and optical fibers. The aforementioned fractional equations are transformed into nonlinear ordinary differential equations (NLODE) considering the Atangana’s conformable derivative (ACD). Then the (m+1/G’)-expansion method is applied for these NLODEs to obtain the desired exact solutions of the fractional evolution equations. The evaluations are fulfilled through the software system Mathematica. Also the reported solutions in this manuscript are compared with the ones in the literature for the time-fractional CH and DP equations and the behaviors of the solutions are presented graphically.

Anahtar Kelimeler

Camassa-Holm and Degasperis-Procesi equations, (m+1/G’)-expansion method, Atangana’s conformable derivative

Kaynakça

• Alesemi, M., 2023. Numerical Analysis of Fractional-Order Camassa–Holm and Degasperis–Procesi Models. Symmetry, 15(2), 269. https://doi.org/10.3390/sym15020269
• Ali, I., Khan, S.U., 2023. A Dynamic Competition Analysis of Stochastic Fractional Differential Equation Arising in Finance via Pseudospectral Method. Mathematics, 11(6), 1328. https://doi.org/10.3390/math11061328
• Alquran, M., 2023. The amazing fractional Maclaurin series for solving different types of fractional mathematical problems that arise in physics and engineering. Partial Differential Equations in Applied Mathematics, 7, 100506. https://doi.org/10.1016/j.padiff.2023.100506
• Alquran, M., Ali, M., Jaradat, I., Al-Ali, N., 2021. Changes in the physical structures for new versions of the Degasperis-Procesi-Camassa-Holm model. Chinese Journal of Physics, 71, 85-94. https://doi.org/10.1016/j.cjph.2020.11.010
• Atangana, A., 2017. Fractional operators with constant and variable order with application to geo-hydrology. Academic Press, 79-112.
• Atangana, A., 2015. Derivative with a new parameter: theory, methods and applications. Academic Press, 25-40.
• Bas, E., Metin Turk, F., Ozarslan, R., Ercan, A., 2021. Spectral data of conformable Sturm–Liouville direct problems. Analysis and Mathematical Physics, 11(1), 8. https://doi.org/10.1007/s13324-020-00428-6
• Ercan, A., 2020. Adomian decomposition method for solving nonlinear fractional sturm-liouville problem. Cumhuriyet Science Journal, 41(1), 169-175. http://dx.doi.org/10.17776/csj.632415
• Ercan, A., 2022. Comparative analysis for fractional nonlinear Sturm-Liouville equations with singular and non-singular kernels. AIMS Mathematics, 7(7), 13325-13343. https://doi.org/10.3934/math.2022736
• Ercan, A., Bas, E., 2021. Regular spectral problem for conformable Dirac system with simulation analysis. Journal of Interdisciplinary Mathematics, 24(6), 1497-1514. https://doi.org/10.1080/09720502.2020.1827507
• Fang, J., Nadeem, M., Wahash, H.A., 2022. A semianalytical approach for the solution of nonlinear modified camassa–holm equation with fractional order. Journal of Mathematics, 5665766. https://doi.org/10.1155/2022/5665766
• Farman, M., Jamil, S., Riaz, M.B., Azeem, M., Saleem, M.U., 2023. Numerical and quantitative analysis of HIV/AIDS model with modified Atangana-Baleanu in Caputo sense derivative. Alexandria Engineering Journal, 66, 31-42. https://doi.org/10.1016/j.aej.2022.11.034
• Ganji, D.D., Sadeghi, E.M.M., Rahmat, M.G., 2008. Modified Camassa–Holm and Degasperis–Procesi Equations Solved by Adomian’s Decomposition Method and Comparison with HPM and Exact Solutions. Acta applicandae mathematicae, 104, 303-311. https://doi.org/10.1007/s10440-008-9258-7
• Ghanbari, B., 2023. A new model for investigating the transmission of infectious diseases in a prey‐predator system using a non‐singular fractional derivative. Mathematical Methods in the Applied Sciences, 46(7), 8106-8125. https://doi.org/10.1002/mma.7412
• Jawarneh, Y., Yasmin, H., Al-Sawalha, M.M., Khan, A., 2023. Fractional comparative analysis of Camassa-Holm and Degasperis-Procesi equations. AIMS Mathematics, 8(11), 25845-62. https://doi.org/10.3934/math.20231318
• 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
• Khatun, M.M., Akbar, M.A., (2024). Analytical soliton solutions of the beta time-fractional simplified modified Camassa-Holm equation in shallow water wave propagation. Journal of Umm Al-Qura University for Applied Sciences, 10, 120-128. https://doi.org/10.1007/s43994-023-00085-y
• Kumar, P., Erturk, V.S., 2023. The analysis of a time delay fractional COVID-19 model via Caputo type fractional derivative. Mathematical Methods in the Applied Sciences, 46(7), 7618-7631. https://doi.org/10.1002/mma.6935
• Ozarslan, R., Ercan, A., Bas, E., 2019. Novel fractional models compatible with real world problems. Fractal and Fractional, 3(2), 15. https://doi.org/10.3390/fractalfract3020015
• Peter, O.J., Yusuf, A., Ojo, M.M., Kumar, S., Kumari, N., Oguntolu, F.A., 2022. A mathematical model analysis of meningitis with treatment and vaccination in fractional derivatives. International Journal of Applied and Computational Mathematics, 8(3), 117. https://doi.org/10.1007/s40819-022-01317-1
• Shloof, A.M., Senu, N., Ahmadian, A., Pakdaman, M., Salahshour, S., 2023. A new iterative technique for solving fractal-fractional differential equations based on artificial neural network in the new generalized Caputo sense. Engineering with Computers, 39(1), 505-515. https://doi.org/10.1007/s00366-022-01607-8
• Singh, J., Gupta, A., 2022. Computational analysis of fractional modified Degasperis-Procesi equation with Caputo-Katugampola derivative. AIMS Mathematics, 8(1), 194-212. https://doi.org/10.3934/math.2023009
• Vellappandi, M., Kumar, P., Govindaraj, V., 2023. Role of fractional derivatives in the mathematical modeling of the transmission of Chlamydia in the United States from 1989 to 2019. Nonlinear Dynamics, 111(5), 4915-4929. https://doi.org/10.1007/s11071-022-08073-3
• Veeresha, P., Prakasha, D.G., 2020. Novel approach for modified forms of Camassa–Holm and Degasperis–Procesi equations using fractional operator. Communications in Theoretical Physics, 72(10), 105002. https://doi.org/10.1088/1572-9494/aba24b
• Wazwaz, A.M., 2007. New solitary wave solutions to the modified forms of Degasperis–Procesi and Camassa–Holm equations. Applied Mathematics and Computation, 186(1), 130-141. https://doi.org/10.1016/j.amc.2006.07.092
• Wazwaz, A.M., 2006. Solitary wave solutions for modified forms of Degasperis–Procesi and Camassa–Holm equations. Physics Letters A, 352(6), 500-504. https://doi.org/10.1016/j.physleta.2005.12.036
• Zhang, K., Alshehry, A.S., Aljahdaly, N.H., Shah, R., Shah, N.A., Ali, M.R., 2023. Efficient computational approaches for fractional-order Degasperis-Procesi and Camassa–Holm equations. Results in Physics, 50, 106549. https://doi.org/10.1016/j.rinp.2023.106549