Zaman-Kesirli Mertebeden Burgers Denklemi İçin Optimal Bir Parametre ile Homotopi Analiz Yönteminin Geliştirilmesi

Year 2022, , 117 - 134, 29.12.2022

Aslı Alkan

https://doi.org/10.55213/kmujens.1206517

Cited By: 8

Abstract

Çalışmanın amacı h keyfi parametresinin seçimi ile ilgili artık hata fonksiyonunu kullanarak bu parametrenin optimal değerini belirleyerek mutlak hatayı azaltmaktır. Bazı sayısal örnekler çözülmüş ve mevcut sonuçlarla karşılaştırılmıştır. Homotopi analiz yöntemi, seri çözümler elde etmek için Burgers denklemine başarıyla uygulanmıştır. Gerekli denklemler için elde edilen çözümlere dayanarak, bu yöntemin zaman-kesirli kısmi diferansiyel denklemlere uygulanabileceği gösterilmiştir.

Keywords

Homotopi analiz metodu, Burgers denklemi, Caputo kesirli türevi

Improving Homotopy Analysis Method with An Optimal Parameter for Time-Fractional Burgers Equation

Abstract

The aim of the study is to reduce the absolute error by determining the optimal value of this arbitrary parameter using the residual error function related to the selection of the arbitrary parameter h. Some numerical examples are solved and compared to existing results. The homotopy analysis method has been successfully implemented to Burgers equation to obtain serial solutions. On the base of the solutions obtained for the required equations, it has been shown that this method is applicable to time-fractional partial differential equations.

Keywords

Homotopy analysis method, Burgers equation, Caputo fractional derivative, Homotopi analiz metodu, Burgers denklemi, Caputo kesirli türevi

References

• Abbasbandy S., Homotopy analysis method for generalized Benjamin-Bona-Mahony equation, Z. Angew. Math. Phys., 59, 51-62, (2008).
• Abdulaziz O., Hashim I., Saif A., Series Solutions of Time-Fractional PDEs by Homotopy Analysis Method, Differential Equations and Nonlinear Mechanics, 13, 16, (2008).
• Arafa A.A.M., Rida S.Z., Mohamed H., Approximate analytical solutions of Schnakenberg systems by homotopy analysis method, Applied Mathematical Modelling, 36, 4789–4796, (2012).
• Aslanov A., A Homotopy-Analysis Approach for Nonlinear Wave-Like Equations with Variable Coefficients, Abstract and Applied Analysis, 7, (2015).
• Dehghan M., Manafian J., Saadatmandi A., Solving Nonlinear Fractional Partial Differential Equations Using the Homotopy Analysis Method, Numerical Methods for Partial Differential Equations, 26, 448-479, (2009).
• Elsaid A., Homotopy analysis method for solving a class of fractional partial differential equations, Commun. Nonlinear Sci. Numer. Simulat., 16, 3655–3664, (2011).
• Fan T., You X., Optimal homotopy analysis method for nonlinear differential equations in the boundary layer, Numerical Algorithms, 62(2), 337–354, (2013).
• Freihat A. A., Zurigat M., Handam A. H., The multi-step homotopy analysis method for modified epidemiological model for computer viruses, Afr. Mat. 26, 585–596, (2013).
• Hariharan G., A homotopy analysis method for the nonlinear partial differential equations arising in engineering, International Journal for Computational Methods in Engineering Science and Mechanics, 18(2-3), 191-200, (2017).
• Jia V., He X., Guo L., The Optimal Homotopy Analysis Method for solving linear optimal control problems, Applied Mathematical Modelling, 45, 865-880, (2017).
• Liao S. J., The Proposed Homotopy Analysis Technique for the Solution of Non-linear Problems, PhD, Shanghai Jiao Tong University, Shanghai, China, (1992).
• Liao S. J., An explicit, totally analytic approximate solution for Blasius’ viscous flow problems, International Journal of Non-Linear Mechanics, 34(4), 759-778, (1999).
• Liao S.J., Beyond Perturbation: Introduction to the Homotopy Analysis Method, Boca Raton, FL, USA, Chapman and Hall/CRC Press, (2003).
• Liao S.J., Comparison between the homotopy analysis method and homotopy perturbation method, Appl. Math. Cumput., 169: 1186-1194, (2005).
• Liao S.J., An optimal homotopy-analysis approach for strongly nonlinear differential equations, Commun. Nonlinear Sci. Numer. Simulat, 15, 2003-2016, (2010).
• Lu D., Liu J., Application of the Homotopy Analysis Method for Solving the Variable Coefficient KdV-Burgers Equation, Abstract and Applied Analysis, 4, (2014).
• Niu Z., Wang C., A one-step optimal homotopy analysis method for nonlinear differential equations, Communications in Nonlinear Science and Numerical Simulation, 15(8), 2026-2036, (2010).
• Odibat Z., Bataineh A. S., An adaptation of homotopy analysis method for reliable treatment of strongly nonlinear problems: construction of homotopy polynomials, Math. Meth. Appl. Sci., 38, 991–1000, (2015).
• Odibat Z., On the optimal selection of the linear operator and the initial approximation in the application of the homotopy analysis method to nonlinear fractional differential equations, Applied Numerical Mathematics, 137, 203-212, (2018).
• Pandey R.K., Mishra H.R., Homotopy analysis Sumudu transform method for time-fractional third order dispersive partial differential equation, Adv. Comput. Math., 43, 365–383, (2017).
• Sakar M.G., Erdoğan F., The homotopy analysis method for solving the time-fractional Fornberg-Whitham equation and comparison with Adomian's decomposition method, Applied Mathematical Modelling, 37, 8876-8885, (2013).
• Sakar M.G., Numerical solution of neutral functional-differential equations with proportional delays, An International Journal of Optimization and Control: Theories & Applications, 7(2), 186-194, (2017).
• Shaiq M.S., Iqbal Z., Mohyud-Din S.T., Homotopy analysıs method for time-fractional wave-like equatıons, Computational Mathematics and Modeling, 24(4), 592-603, (2013).
• Song L., Zhang H., Application of homotopy analysis method to fractional KdV-Burgers-Kuramoto equation, Physics Letters A, 367, 88-94, (2007).
• Sun Q., Solving the Klein–Gordon equation by means of the homotopy analysis method, Applied Mathematics and Computation, 169, 355-365, (2004).
• Turkyilmazoğlu M., An effective approach for evaluation of the optimal convergence control parameter in the homotopy analysis method, Filomat, 30(6), 1633 -1650, (2016).
• Van Gorder R.A., Vajravelu K., On the selection of auxiliary functions, operators, and convergence control parameters in the application of the Homotopy Analysis Method to nonlinear differential equations: A general approach, Communications in Nonlinear Science and Numerical Simulation, 14(12), 4078-4089, (2009).
• Van Gorder R.A., Optimal homotopy analysis and control of error for implicitly defined fully nonlinear differential equations, Numerical Algorithms, 81(1), 181-196, (2019).
• Vishal K., Kumar S., Das S., Application of homotopy analysis method for fractional Swift-Hohenberg equation-revisited, Applied Mathematical Modelling, 36(8), 3630-3637, (2012).
• Yusufoğlu E., Selam C., The homotopy analysis method to solve the modified Equal Width wave equation, Numerical Methods Partial Differential Equations, 26, 1434-1442, (2010).