Solution of time-fractional equations via Sumudu-Adomian decomposition method

Document Type : Research Paper

Authors

1 Department of Mathematics, New Arts, Commerce and Science College, Ahmednagar, Maharashtra, India.

2 Department of Mathematics, Loknete Vyankatrao Hiray Arts‚ Science and Commerce College, Nashik, Maharashtra, India

Abstract

This paper investigates the semi-analytical solutions of linear and non-linear Time Fractional Klein-Gordon equations with appropriate initial conditions to apply the New Sumudu-Adomian Decomposition method (NSADM). This paper shows the semi-analytical as well as a graphical interpretation of the solution by using mathematical software “Mathematica Wolform” and considering Caputo’s sense derivatives to semi-analytical results obtained by NSADM. 

Keywords

References

  • [1] S. Abbasbandy, Y. Tan, and S. J. Liao, Newton-homotopy analysis method for nonlinear equations, Applied Mathematics and Computation, 188(2) (2007), 1794-1800.
  • [2] S. Abbasbandy and A. Shirzadi, An unconditionally stable difference scheme for equations of conservation law form, Italian Journal of Pure and Applied Mathematics, 37 (2017), 1–4.
  • [3] M. A. Alim, M. A. Kawser, and M. M. Rahman, Asymptotic Solutions of Coupled Spring Systems with Cubic Nonlinearity using Homotopy Perturbation Method, Annals of Pure and Applied Mathematics, 18(1) (2018), 99-112.
  • [4] M. A. Asiru, Further properties of the Sumudu transform and its applications, International journal of mathemat- ical education in science and technology, 33(3) (2002), 441-449.
  • [5] M. A. Asiru, Sumudu transform and the solution of integral equations of convolution type, International Journal of Mathematical Education in Science and Technology, 32(6) (2001), 906-910.
  • [6] O. Ashiru, J. W. Polak, and R. B. Noland, Space-time user benefit and utility accessibility measures for individual activity schedules, Transportation research record, 1854(1) (2003), 62-73.
  • [7] R. K. Bairwa and K. Singh, Analytical Solution of Time-Fractional Klien-Gordon Equation by using Laplace- Adomian Decomposition Method, Annals of Pure and Applied Mathematics, 24(1) (2021), 27-35.
  • [8] V. B. L. Chaurasia, R. S. Dubey, and F. B. M. Belgacem, Fractional radial diffusion equation analytical solution via Hankel and Sumudu transforms, International Journal of Mathematics in Engineering Science and Aerospace, 3(2) (2012), 1-10.
  • [9] Y. Cherruault and G. Adomian, Decomposition methods: a new proof of convergence, Mathematical and Computer Modelling, 18(12) (1993), 103-106.
  • [10] M. Dehghan, J. Manafian, and A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numerical Methods for Partial Differential Equations: An International Journal, 26(2) (2010), 448-479.
  • [11] M. Dehghan and J. Manafian, The solution of the variable coefficients fourth-order parabolic partial differential equations by the homotopy perturbation method, Zeitschrift fu¨r Naturforschung A, 64(7-8) (2009), 420-430.
  • [12] M. Dehghan, J. Manafian, and A. Saadatmandi, Application of semi-analytic methods for the Fitzhugh–Nagumo equation, which models the transmission of nerve impulses, Mathematical Methods in the Applied Sciences, 33(11) (2010), 1384-1398.
  • [13] M. Dehghan and S. Pourghanbar, Solution of the Black- Scholes equation for pricing of barrier option, Zeitschrift fu¨r Naturforschung A, 66(5) (2011), 289–296.
  • [14] S. Faydao˘glu, The Modified Homotopy Perturbation Method For The Approximate Solution Of Nonlinear Oscil- lators, Journal of Modern Technology and Engineering, 7(1) (2022), 40-50.
  • [15] I. S. Gupta and L. Debnath, Some properties of the Mittag-Leffler functions, Integral Transforms and Special Functions, 18(5) (2007), 329-336.
  • [16] G. Hariharan, R. Rajaraman, and M. Mahalakshmi, Wavelet method for a class of space and time fractional telegraph equations, International Journal of Physical Sciences, 7(10) (2012), 1591-1598.
  • [17] M. Inokuti, H. Sekine, and T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics, Variational method in the mechanics of solids, 33(5) (1978), 156-162.
  • [18] H. Jafari and V. Daftardar-Gejji, Solving a system of nonlinear fractional differential equations using Adomian decomposition, Journal of Computational and Applied Mathematics, 196(2) (2006), 644-651.
  • [19] Q. D. Kataetbeh and F. B. M. Belgacem, Applications of the Sumudu transform to differential equations, Nonlinear Studies, 18(1) (2011), 99-112.
  • [20] Y. Khan and Q. Wu, Homotopy perturbation transform method for nonlinear equations using He’s polynomials, Computers & Mathematics with Applications, 61(8) (2011), 1963-1967.
  • [21] S. A. Khuri, A Laplace decomposition algorithm applied to a class of nonlinear differential equations, Journal of Applied Mathematics, 1(4) (2001), 141-155.
  • [22] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 204 (2006).
  • [23] J. Manafian, Novel solitary wave solutions for the (3+1)-dimensional extended Jimbo–Miwa equations, Computers & Mathematics with Applications, 76(5) (2018), 1246-1260.
  • [24] J. Manafian and N. Allahverdiyeva, An analytical analysis to solve the fractional differential equations, Advanced Mathematical Models & Application, 6 (2021), 128-161.
  • [25] I. Podlubny, Fractional-order systems and PI/sup/spl lambda//D/sup/spl mu//-controllers, IEEE Transactions on automatic control, 44(1) (1999), 208-214.
  • [26] S. Pourghanbar, J. Manafian, M. Ranjbar, A. Aliyeva, and Y. S. Gasimov, An efficient alternating direction explicit method for solving a nonlinear partial differential equation, Mathematical Problems in Engineering, 2020 (2020), Article ID 9647416.
  • [27] S. G. Samko and B. Ross, Integration and differentiation to a variable fractional order, Integral transforms and special functions, 1(4) (1993), 277-300.
  • [28] N. Smaui and M. Zribi, Dynamics and control of the sevenmode truncation system of the 2-d Navier Stokes equations, Communications in Nonlinear Science and Numerical Simulation, 32 (2016), 169–189.
  • [29] S. A. Tarate, A. P. Bhadane, S. B. Gaikwad, and K. A. Kshirsagar, Sumudu-iteration transform method for fractional telegraph equations, J. Math. Comput. Sci., 12 (2022), Article-ID.
  • [30] G. Watugala, Sumudu transform: a new integral transform to solve differential equations and control engineering problems, Integrated Education, 24(1) (1993), 35-43.
  • [31] A. M. Wazwaz, The combined Laplace transform–Adomian decomposition method for handling nonlinear Volterra integro–differential equations, Applied Mathematics and Computation, 216(4) (2010), 1304-1309.
  • [32] A. Yildirim, S. T. Mohyud-Din, and D. H. Zhang, Analytical solutions to the pulsed Klein–Gordon equation using modified variational iteration method (MVIM) and Boubaker polynomials expansion scheme (BPES), Computers & Mathematics with Applications, 59(8) (2010), 2473-2477.
  • [33] B. Yildiz, O. Kilicoglu, and G. Yagubov, Optimal control problem for nonstationary Schr odinger equation, Numerical Methods for Partial Differential Equations, 25(5) (2009), 1195–1203.
  • [34] X. Zhong, J. Vrijmoed, E. Moulas, and L. Tajˇcmanov´a, A coupled model for intragranular deformation and chemical diffusion, Earth and Planetary Science Letters, 474 (2017), 387–396.
Computational Methods for Differential Equations
Volume 11, Issue 2
April 2023
Pages 345-356

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History

  • Receive Date: 04 May 2022
  • Revise Date: 14 June 2022
  • Accept Date: 27 July 2022

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