Kesirli Mertebeden Pseudo Hiperbolik Diferansiyel Denklemlerin Sonlu Fark Metodu ile Nümerik Çözümleri

Year 2022, Volume: 22 Issue: 5, 998 - 1004, 27.10.2022

Fatih Özbağ , Mahmut Modanlı

https://doi.org/10.35414/akufemubid.1124445

Abstract

Kesirli mertebeden diferansiyel denklemler mühendislik, fizik ve biyoloji gibi alanlarda matematiksel problemlerin modellenmesinde önemli yer almaktadır. Bu makalede kesirli mertebeden pseudo hiperbolik diferansiyel denklemler için bir başlangıç sınır değer probleminin sonlu fark metodu ile yaklaşık çözümleri araştırılmıştır. İlk olarak başlangıç sınır değer problemi için birinci mertebeden sonlu fark şeması oluşturulmuştur. Daha sonra bu sonlu fark şeması için kararlılık analizi yapılmıştır. Elde edilen teorik sonuçları desteklemek için örnek bir problemin farklı kesirli mertebeden türevlerinde gerçek ve yaklaşık çözümler için hata değerleri hesaplanmıştır. Uygulanan çözüm metodunun etkinliğini göstermek için bazı nümerik simülasyonlar verilmiştir.

Keywords

Kesirli Mertebeden Pseudo Hiperbolik Denklem, Sonlu Fark Metodu, Nümerik Çözüm, Kararlık

Numerical Solutions of Fractional Order Pseudo Hyperbolic Differential Equations by Finite Difference Method

Abstract

Fractional differential equations are useful for modelling mathematical issues in fields including engineering, physics, and biology. In this article, approximate solutions of an initial boundary value problem for fractional pseudo hyperbolic differential equations are investigated using the finite difference method. First, a first-order finite difference scheme is created for the initial boundary value problem. Then, stability analysis was performed for this finite difference scheme. In order to support the theoretical results obtained, error values were calculated for precise and approximate solutions in different fractional order derivatives of a sample problem. Some numerical simulations are also given to show the effectiveness of the applied solution method.

Keywords

Fractional Order Pseudo Hyperbolic Equation, Finite Difference Method, Numerical Solutions, Stability

References

• Abdulazeez, S. T. and Modanli, M., 2022. Solutions of fractional order pseudo-hyperbolic telegraph partial differential equations using finite difference method. Alexandria Engineering Journal, 61(12), 12443-12451.
• Almeida, R., Brito da Cruz, A., Martins, N. and Monteiro, M. T. T., 2019. An epidemiological MSEIR model described by the Caputo fractional derivative. International journal of dynamics and control, 7(2), 776-784.
• Baleanu, D., Jajarmi, A., Mohammadi, H. and Rezapour, S., 2020. A new study on the mathematical modelling of human liver with Caputo–Fabrizio fractional derivative. Chaos, Solitons & Fractals, 134, 109705.
• Chen, G. and Yang, Z., 1993. Initial value problem for a class of nonlinear pseudo-hyperbolic equations. Acta Mathematicae Applicatae Sinica, 9(2), 166-173.
• Çiçek, H. ve Modanlı, M., 2022. Kesirli mertebeden pseudo hiperbolik kısmi diferansiyel denkleminin homotopi pertürbasyon yöntemiyle yaklaşık çözümü. Uşak Üniversitesi Fen ve Doğa Bilimleri Dergisi, (Basım aşamasında).
• Fedotov, I., Shatalov, M. and Marais, J., 2016. Hyperbolic and pseudo-hyperbolic equations in the theory of vibration. Acta Mechanica, 227(11), 3315-3324.
• Ghanbari, B. 2021. A new model for investigating the transmission of infectious diseases in a preypredator system using a nonsingular fractional derivative. Mathematical Methods in the Applied Sciences, 1-20.
• Hilfer, R., 2000. Applications of Fractional Calculus in Physics, Rudolf Hilfer, World Scientific Publishing, 1-85.
• Kilbas, A. A., Srivastava, H. M. and Trujillo J. J., 2006. Theory and applications of fractional differential equation, 204, Jan van Mill, Elsevier, 1-463.
• Krutitskii, P. A., 1997. An initial-boundary value problem for the pseudo-hyperbolic equation of gravity-gyroscopic waves. Journal of Mathematics of Kyoto University, 37(2), 343-365.
• Liu, Y., Wang, J., Li, H., Gao, W. and He, S., 2011. A new splitting H1-Galerkin mixed method for pseudo-hyperbolic equations. International Journal of Mathematical and Computational Sciences, 5(3), 413-418.
• Modanli, M., Abdulazeez, S. T. and Husien, A. M., 2021. A residual power series method for solving pseudo hyperbolic partial differential equations with nonlocal conditions. Numerical Methods for Partial Differential Equations, 37(3), 2235-2243.
• Modanli, M., Göktepe, E., Akgül, A., Alsallami, S. A., and Khalil, E. M., 2022. Two approximation methods for fractional order Pseudo-Parabolic differential equations. Alexandria Engineering Journal, 61(12), 10333-10339.
• Ozbag, F. and Modanli, M., 2021. On the stability estimates and numerical solution of fractional order telegraph integro-differential equation. Physica Scripta, 96(9), 094008.
• Podlubny, I., 1998. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, 198, Elsevier, 1-340.
• Potapova, S. V., 2012. Boundary value problems for pseudohyperbolic equations with a variable time direction. Journal of Pure Applied Mathematics, 3(1), 73-91.
• Qureshi, S. and Yusuf, A., 2019. Modeling chickenpox disease with fractional derivatives: From caputo to atangana-baleanu. Chaos, Solitons & Fractals, 122, 111-118.
• Zhang, Y., Niu, Y. and Shi, D., 2012. Nonconforming H1 -Galerkin mixed finite element method for pseudo-hyperbolic equations. American Journal of Computational Mathematics, 2, 269-273.
• Zhao, Z., and Li, H., 2019. A continuous Galerkin method for pseudo-hyperbolic equations with variable coefficients. Journal of Mathematical Analysis and Applications, 473(2), 1053-1072.