Homotopy methods for fractional linear/nonlinear differential equations with a local derivative operator

Yıl 2018, Cilt: 20 Sayı: 3, 75 - 89, 29.10.2018

Mehmet Yavuz , Burcu Yaşkıran

https://doi.org/10.25092/baunfbed.476608

Cited By: 4

Öz

In this paper, we consider some linear/nonlinear differential equations (DEs) containing conformable derivative operator. We obtain approximate solutions of these mentioned DEs in the form of infinite series which converges rapidly to their exact values by using and homotopy analysis method (HAM) and modified homotopy perturbation method (MHPM). Using the conformable operator in solutions of different types of DEs makes the solution steps are computable easily. Especially, the conformable operator has been used in modelling DEs and identifying particular problems such as biological, engineering, economic sciences and other some important fields of application. In this context, the aim of this study is to solve some illustrative linear/nonlinear problems as mathematically and to compare the exact solutions with the obtained solutions by considering some plots. Moreover, it is an aim to show the authenticity, applicability, and suitability of the methods constructed with the conformable operator.

Anahtar Kelimeler

Approximate solution, conformable operator, homotopy analysis method, modified homotopy perturbation method, nonlinear differential equations

Lokal türev operatörlü lineer/lineer olmayan diferansiyel denklemler için homotopi metotları

Öz

Bu çalışmada conformable (uyumlu) türev operatörü (CTO) içeren bazı lineer/lineer olmayan diferansiyel denklemler ele alınmıştır. Homotopi analiz metodunu (HAM) ve modifiyeli homotopi pertürbasyon metodunu (MHPM) kullanarak bu bahsi geçen denklemlerin sonsuz seri formunda yaklaşık çözümleri elde edilmiştir. CTO kullanılması farklı türden diferansiyel denklemlerin çözümlerini elde etmede çözüm adımlarının kolay bir şekilde hesaplanmasını sağlamaktadır. Özellikle CTO mühendislik, fiziksel bilimler, ekonomi ve diğer bazı alanlardaki problemleri modellemede kullanılmaktadır. Bu bağlamda, bu çalışmanın amacı bazı lineer/lineer olmayan diferansiyel denklemleri matematiksel olarak çözmek ve çözüm grafiklerini kullanarak elde edilen yaklaşık çözümler ile tam çözümleri karşılaştırmaktır. Ayrıca CTO ile yeniden tanımlanan HAM ve MHPM metotlarının güvenirliğini, uygulanabilirliğini ve elverişliliğini göstermektir.

Anahtar Kelimeler

Yaklaşık çözüm, uyumlu operatör, homotopi analiz metodu, modifiyeli homotopi pertürbasyon metodu, lineer olmayan diferansiyel denklemler

Kaynakça

• Avci, D., Iskender Eroglu, B. B. and Ozdemir, N., Conformable heat equation on a radial symmetric plate, Thermal Science, 21, 2, 819-826, (2017).
• Çenesiz, Y., Baleanu, D., Kurt, A. and Tasbozan, O., New exact solutions of burgers’ type equations with conformable derivative, Waves in Random and Complex Media, 27, 1, 103-116, (2017).
• Ilie, M., Biazar, J. and Ayati, Z., Optimal homotopy asymptotic method for first-order conformable fractional differential equations, Journal of Fractional Calculus and Applications, 10, 1, 33-45, (2019).
• Yavuz, M., Novel solution methods for initial boundary value problems of fractional order with conformable differentiation, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 8, 1, 1-7, (2018).
• Bildik, N., Konuralp, A., Bek, F. O. and Küçükarslan, S., Solution of different type of the partial differential equation by differential transform method and Adomian’s decomposition method, Applied Mathematics and Computation, 172, 1, 551-567, (2006).
• Morales-Delgado, V. F., Gómez-Aguilar, J. F., Yépez-Martínez, H., Baleanu, D., Escobar-Jimenez, R. F. and Olivares-Peregrino, V. H., Laplace homotopy analysis method for solving linear partial differential equations using a fractional derivative with and without kernel singular, Advances in Difference Equations, 2016, 1, 164, (2016).
• Özdemir, N. and Yavuz, M., Numerical solution of fractional black-scholes equation by using the multivariate Padé approximation, Acta Physica Polonica A, 132, 3, 1050-1053, (2017).
• Turut, V. and Güzel, N., On solving partial differential equations of fractional order by using the variational iteration method and multivariate Padé approximations, European Journal of Pure and Applied Mathematics, 6, 2, 147-171, (2013).
• Yokus, A., Sulaiman, T. A. and Bulut, H., On the analytical and numerical solutions of the Benjamin–Bona–Mahony equation, Optical and Quantum Electronics, 50, 1, 31, (2018).
• Akgül, A., Khan, Y., Akgül, E. K., Baleanu, D. and Al Qurashi, M. M., Solutions of nonlinear systems by reproducing kernel method, The Journal of Nonlinear Sciences and Applications, 10, 4408-4417, (2017).
• Khalil, R., Al Horani, M., Yousef, A. and Sababheh, M., A new definition of fractional derivative, Journal of Computational and Applied Mathematics, 264, 65-70, (2014).
• Yavuz, M. and Özdemir, N., New numerical techniques for solving fractional partial differential equations in conformable sense in: Ostalczyk P., Sankowski D., Nowakowski J. (eds) Non-Integer Order Calculus and its Applications. Lecture Notes in Electrical Engineering, vol 496. Springer, Cham, 49-62, (2019).
• Evirgen, F., Conformable fractional gradient based dynamic system for constrained optimization problem, Acta Physica Polonica A, 132, 1066-1069, (2017).
• Eroğlu, B. İ., Avcı, D. and Özdemir, N., Optimal control problem for a conformable fractional heat conduction equation, Acta Physica Polonica A, 132, 3, 658-662, (2017).
• Ekici, M., Mirzazadeh, M., Eslami, M., Zhou, Q., Moshokoa, S. P., Biswas, A. and Belic, M., Optical soliton perturbation with fractional-temporal evolution by first integral method with conformable fractional derivatives, Optik-International Journal for Light and Electron Optics, 127, 22, 10659-10669, (2016).
• Atangana, A., Baleanu, D. and Alsaedi, A., New properties of conformable derivative, Open Mathematics, 13, 1, (2015).
• Abdeljawad, T., AL Horani, M. and Khalil, R., Conformable fractional semigroups of operators, Journal of Semigroup Theory and Applications, 2015, Article ID 7, (2015).
• Usta, F. and Sarıkaya, M. Z., Explicit bounds on certain integral inequalities via conformable fractional calculus, Cogent Mathematics, 4, 1, 1277505, (2017).
• Usta, F., A conformable calculus of radial basis functions and its applications, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 8, 2, 176-182, (2018).
• Kumar, S., Singh, J., Kumar, D. and Kapoor, S., New homotopy analysis transform algorithm to solve volterra integral equation, Ain Shams Engineering Journal, 5, 1, 243-246, (2014).
• Arqub, O. A. and El-Ajou, A., Solution of the fractional epidemic model by homotopy analysis method, Journal of King Saud University-Science, 25, 1, 73-81, (2013).
• Sakar, M. G. and Erdogan, F., The homotopy analysis method for solving the time-fractional Fornberg–Whitham equation and comparison with Adomian’s decomposition method, Applied Mathematical Modelling, 37, 20-21, 8876-8885, (2013).
• Inc, M., Gencoglu, M. T. and Akgül, A., Application of extended Adomian decomposition method and extended variational iteration method to Hirota-Satsuma coupled kdv equation, Journal of Advanced Physics, 6, 2, 216-222, (2017).
• Evirgen, F. and Özdemir, N., Multistage Adomian decomposition method for solving nlp problems over a nonlinear fractional dynamical system, Journal of Computational and Nonlinear Dynamics, 6, 2, 021003, (2011).
• Yavuz, M. and Özdemir, N., A quantitative approach to fractional option pricing problems with decomposition series, Konuralp Journal of Mathematics, 6, 1, 102-109, (2018).
• Yavuz, M., Ozdemir, N. and Okur, Y. Y., Generalized differential transform method for fractional partial differential equation from finance, Proceedings, International Conference on Fractional Differentiation and its Applications, Novi Sad, Serbia, pp 778-785, (2016).
• Yang, X.-J., Machado, J. T. and Srivastava, H., A new numerical technique for solving the local fractional diffusion equation: Two-dimensional extended differential transform approach, Applied Mathematics and Computation, 274, 143-151, (2016).
• Khan, Y., Akgul, A., Faraz, N., Inc, M., Akgul, E. K. and Baleanu, D., A homotopy perturbation solution for solving highly nonlinear fluid flow problem arising in mechanical engineering, Proceedings, AIP Conference Proceedings: AIP Publishing, pp 130004, (2018).
• Evirgen, F. and Özdemir, N., A fractional order dynamical trajectory approach for optimization problem with hpm in: Baleanu, D., Machado, J.A.T., Luo, A. (eds) Fractional Dynamics and Control, Springer, 145-155, (2012).
• Yavuz, M. and Özdemir, N., A different approach to the European option pricing model with new fractional operator, Mathematical Modelling of Natural Phenomena, 13, 1, 12, (2018).
• Yavuz, M. and Yaskıran, B., Approximate-analytical solutions of cable equation using conformable fractional operator, New Trends in Mathematical Science, 5, 209-219, (2017).
• Tasbozan, O., Şenol, M., Kurt, A. and Özkan, O., New solutions of fractional Drinfeld-Sokolov-Wilson system in shallow water waves, Ocean Engineering, 161, 62-68, (2018).
• Yavuz, M., Özdemir, N., On the solutions of fractional Cauchy problem featuring conformable derivative, Proceedings, ITM Web of Conferences, EDP Sciences, Vol. 22, p. 01045, (2018).
• Momani, S. and Odibat, Z., Homotopy perturbation method for nonlinear partial differential equations of fractional order, Physics Letters A, 365, 5, 345-350, (2007).
• Hemeda, A. A., Modified homotopy perturbation method for solving fractional differential equations, Journal of Applied Mathematics, 2014, (2014).
• Javidi, M. and Ahmad, B., Numerical solution of fractional partial differential equations by numerical Laplace inversion technique, Advances in Difference Equations, 2013, 1, 375, (2013).