Solving Fractional Oscillator Equations Via The Sumudu Transform Method
Keywords:
Fractional differential equation, Caputo fractional derivative, Equation of motion, Sumudu transform methodAbstract
The equation of motion for a driven fractional oscillator is formulated by replacing the classical second-order time derivative with a Caputo fractional derivative of order In this study, the Sumudu transform method is employed to obtain an analytical solution of the fractional differential equation. By utilizing the integral properties of the Sumudu transform, which are adapted for Caputo fractional derivatives, the system’s response is explicitly derived in the time domain. The dynamic characteristics and phase plane trajectories of the fractional oscillator are analyzed for various values of . The results demonstrate that the Sumudu transform provides an effective and accurate framework for analyzing the complex behavior of fractional oscillatory systems.
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Mainardi, F., An introduction to mathematical models. Fractional calculus and waves in linear viscoelasticity. Imperial College Press, London, 2010.
Gorenflo, R., et al., Discrete random walk models for space–time fractional diffusion. Chemical physics, 2002. 284(1-2): p. 521-541.
Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics. 1997: Springer.
Podlubny, I., Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Vol. 198. 1998: elsevier.
Li, C. and F. Zeng, Numerical methods for fractional calculus. 2015: CRC Press.
Watugala, G.K., Sumudu transform: a new integral transform to solve differential equations and control engineering problems. Integrated Education, 1993. 24(1): p. 35-43.
Mechee, M.S. and A.J. Naeemah, Sumudu transform for solving some classes of fractional differential equations. Italian Journal of Pure and Applied Mathematics: p. 740.
Kılıçman, A. and O. Altun, Some remarks on the fractional Sumudu transform and applications. Appl. Math, 2014. 8(6): p. 1-8.
Akgül, A. and G. Öztürk, Application of the sumudu transform to some equations with fractional derivatives. Sigma Journal of Engineering and Natural Sciences, 2023. 41(6): p. 1132-1143.
Singh, A. and S. Pippal, Application of Sumudu Transform and New Iterative Method to Solve Certain Nonlinear Ordinary Caputo Fractional Differential Equations. Contemporary Mathematics, 2024: p. 4614-4625.
Abdo, N., E. Ahmed, and A. Elmahdy, Some real life applications of fractional calculus. Journal of Fractional Calculus and Applications, 2019. 10(2): p. 1-2.
Kilbas, A.A., H.M. Srivastava, and J.J. Trujillo, Theory and applications of fractional differential equations. Vol. 204. 2006: elsevier.
Mainardi, F., The fundamental solutions for the fractional diffusion-wave equation. Applied Mathematics Letters, 1996. 9(6): p. 23-28.
Wang, K. and S. Liu, A new Sumudu transform iterative method for time-fractional Cauchy reaction–diffusion equation. SpringerPlus, 2016. 5: p. 1-20.
Alomari, A., Homotopy-Sumudu transforms for solving system of fractional partial differential equations. Advances in Difference Equations, 2020. 2020(1): p. 222.
Jarad, F. and K. Tas, Application of Sumudu and double Sumudu transforms to Caputo-Fractional differential equations. Journal of Computational Analysis and Applications, 2012. 14(3): p. 475-483.
Goswami, P., F. Belgacem, and S. Sivasundaram. Solving special fractional differential equations by Sumudu transform. in AIP Conference Proceedings-American Institute of Physics. 2012.
Nisar, K.S., et al., Solution of fractional kinetic equations involving class of functions and Sumudu transform. Advances in Difference Equations, 2020. 2020(1): p. 39.
Riddhi, D., Beta function and its applications. The University of Tennesse, Knoxville, USA.[online] Available from: http://sces. phys. utk. edu/moreo/mm08/Riddi. pdf, 2008.
Gupta, V. and B. Sharma, Application of Sumudu transform in reaction-diffusion systems and nonlinear waves. Applied Mathematical Sciences, 2010. 4(9-12): p. 435-446.
Baleanu, D., H. Mohammadi, and S. Rezapour, A fractional differential equation model for the COVID-19 transmission by using the Caputo–Fabrizio derivative. Advances in difference equations, 2020. 2020(1): p. 299.
Sikora, B., Remarks on the Caputo fractional derivative. Minut, 2023. 5: p. 76-84.
Al-Maaitah, A.F., Fractional Caputo analysis of Discrete systems. Eur. Sci. J, 2014. 10: p. 277-285.
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