Numerical solutions of Troesch and Duffing equations by Taylor wavelets

Year 2023, Volume: 52 Issue: 2, 292 - 302, 31.03.2023

Gökçe Özaltun Sevin Gümgüm

https://doi.org/10.15672/hujms.1063791

Cited By: 1

Abstract

The aim of this study is to obtain accurate numerical results for the Troesch and Duffing equations by using Taylor wavelets. Important features of the method include easy implementation and simple calculation. The effectiveness and accuracy of the applied method is illustrated by solving these problems for several variables. One of the important variable is the resolution parameter which enables to use low degree polynomials and decrease the computational cost. Results show that the proposed method yields highly accurate solutions by using quite low degree polynomials.

Keywords

Taylor wavelets, Troesch equation, Duffing equation

References

• [1] M.A. Al-Jawary and S.G. Abd-Al-Razaq, Analytic and numerical solution for Duffing equations, Int. J. Basic Appl. Sci. 5 (2), 115-119, 2016.
• [2] N. Alias, A. Manaf, A. Ali and M. Habib, Solving Troesch’s problem by using modified nonlinear shooting method, J. Teknol. 78 (4-4), 45-52, 2011.
• [3] A. Anapalı, Ö. Yalçın and M. Gülsu, Numerical solutions of Duffing equations involving linear integral with shifted Chebyshev polynomials, AKU-J. Sci. Eng. 15, 1-11, 2015.
• [4] G.B. Arfken and H.J. Weber, Mathematical Methods for Physicists, Elsevier, London, 2005.
• [5] A. Beléndez, D.I. Méndez, E. Fernández, S. Marini and I. Pascual, An explicit approximate solution to the Duffing-harmonic oscillator by a cubication method, Phys. Lett. A. 373 (32), 2805-2809, 2009.
• [6] M. Ben-Romdhane and H. Temimi, A novel computational method for solving Troesch’s problem with high-sensitivity parameter, Int. J. Comput. Meth. Eng. Sci. Mech. 18 (4-5), 230-237, 2017.
• [7] M. Bisheh-Niasar, A. Saadatmandi and M. Akrami-Arani, A new family of high-order difference schemes for the solution of second order boundary value problems, IJMC 9 (3), 187-199, 2018.
• [8] B. Bülbül and M. Sezer, Numerical solution of Duffing equation by using an improved Taylor matrix method, J. Appl Math. 2013, (6pp), 2013.
• [9] S.H. Chang, A variational iteration method for solving Troesch’s problem, J. Comput. Appl. Math. 234 (10), 3043-3047, 2010.
• [10] S.H. Chang and I.L. Chang, A new algorithm for calculating one-dimensional differential transform of nonlinear functions, Appl. Math. Comput. 195 (2), 799-808, 2008.
• [11] E. Deeba, S.A. Khuri and S. Xiei, An algorithm for solving boundary value problems, J. Comput. Phys. 159 (2), 125-138, 2000.
• [12] E.H. Doha, D. Baleanu, A.H. Bhrawi and R.M. Hafez, A Jacobi collocation method for Troesch’s problem in plasma physics, P. Romanian Acad. A 15 (2), 130-138, 2014.
• [13] M. El-Gamel, Numerical solution of Troesch’s problem by Sinc-collocation method Appl. Math. 4 (4), 707-712, 2013.
• [14] M. El-Gamel and M. Sameeh, A Chebyshev collocation method for solving Troesch’s problem, IJMCAR 3 (2), 23-32, 2013.
• [15] X. Feng, L. Mei and G. He, An efficient algorithm for solving Troesch’s problem, Appl. Math. Comput. 189 (1), 500507, 2007.
• [16] U. Filobello-Nino, H. Vazquez-Leal, B. Benhammouda, A. Perez-Sesma and J. Cervantes-Perez, Perturbation method and Laplace-Pade approximation as a novel tool to find approximate solutions for Troesch’s problem, Nova Scientia 7 (14), 57-73, 2015.
• [17] F. Geng and M. Cui, A novel method for nonlinear two-point boundary value problems: Combination of ADM and RKM, Appl. Math. Comput. 217 (9), 4676-4681, 2011.
• [18] S. Gümgüm, Laguerre wavelet method for solving Troesch equation, BAUN J. Inst. Sci. 21 (2), 494-502, 2019.
• [19] S. Gümgüm, Taylor wavelet solution of linear and nonlinear Lane-Emden equations, Appl. Numer. Math. 158, 44-53, 2020.
• [20] S. Gümgüm, D. Ersoy-Özdek and G. Özaltun, Legendre wavelet solution of high order nonlinear ordinary delay differential equations, Turk. J. Math. 43 (3), 1339-1352, 2019.
• [21] S. Gümgüm, D. Ersoy-Özdek, G. Özaltun and N. Bildik, Legendre wavelet solution of neutral differential equations with proportional delays, J. Appl. Math. Comput. 61 (1), 389-404, 2019.
• [22] S. Gümgüm, N. Baykuş-Savaşaneril, Ö.K. Kürkçü and M. Sezer, Lucas polynomial approach for second order nonlinear differential equations, SDU J. Nat. Appl. Sci. 24 (1), 230-236, 2020.
• [23] M. Kaminski and A. Corigliano, Numerical solution of the Duffing equation with random coefficients, Meccanica 50 (7), 1841-1853, 2015.
• [24] E. Keshavarza and Y. Ordokhania, A fast numerical algorithm based on the Taylor wavelets for solving the fractional integro-differential equations with weakly singular kernels, Math. Method. Appl. Sci. 42 (13), 4427-4443, 2019.
• [25] E. Keshavarza, Y. Ordokhania and M. Razzaghi, The Taylor wavelets method for solving the initial and boundary value problems of Bratu-type equations, Appl. Numer. Math. 128, 205-216, 2018.
• [26] M. Khalid, F. Zaidi and M. Sultana, A numerical solution of Troesch’s problem via optimal Homotopy asymptotic method, Int. J. Comput. Appl. 140 (5), 1-5, 2016.
• [27] S.A. Khuri, A numerical algorithm for solving Troesch’s problem, Int. J. Comput. Math. 80 (4), 493-498, 2003.
• [28] S.A. Khuri and A. Sayfy, Troesch’s problem: A B-spline collocation approach, Math. Comput. Model. 54 (9-10), 1907-1918, 2011.
• [29] S. O. Korkut Uysal and G. Tanoglu, An efficient iterative algorithm for solving nonlinear oscillation problems, Filomat, 31 (9), 2713-2726, 2017.
• [30] G.R. Liu and T.Y. Wu, Numerical solution for differential equations of Duffing-type non-linearity using the generalized quadrature rule, J. Soun Vib. 237 (5), 805-817, 2000.
• [31] P.A. Lott, Periodic solutions to Duffing’s equation via the Homotopy method, PhD Thesis, The University of Southern Missisipi, 2001.
• [32] S.A. Malik, I.M. Qureshi, M. Zubair and M. Amir, Numerical solution to Troesch’s problem using hybrid heuristic computing, J. Basic. Appl. Sci. Res. 3 (7), 10-16, 2013.
• [33] V.S. Markin, A.A. Chernenko, Y.A. Chizmadehev and Y.G. Chirkov, Aspects of the theory of gas porous electrodes in Fuel Cells: Their Electrochemical Kinetics, New York, USA, 1966.
• [34] S.H. Mirmoradi, I. Hosseinpour, S. Ghanbarpour and A. Barari, Application of an approximate analytical method to nonlinear Troesch’s problem, App. Math. Sci. 3 (32), 1579-1585, 2009.
• [35] S. Momani, S. Abuasad and Z. Odibat, Variational iteration method for solving nonlinear boundary value problems, Appl. Math. Comput. 183 (2), 1351-1358, 2006.
• [36] R. Najafi and B.N. Saray, Numerical solution of the forced Duffing equations using Legendre multiwavelets, Comput. Methods Differ. Equ. 5 (1), 43-55, 2017.
• [37] A.K. Nasab, Z.P. Atabakan and A. Kılıçman, An efficient approach for solving nonlinear Troesch’s and Bratu’s Problems by wavelet analysis method, Math. Probl. Eng. 2013, (10pp), 2013.
• [38] B.V. Rathish-Kuma and M. Mehra, Wavelet multilayer Taylor Galerkin schemes for hyperbolic and parabolic problems, Appl. Math. Comput. 166 (2), 312-323, 2005.
• [39] S.M. Roberts and J.S. Shipman, On the closed form solution of Troesch’s problem, J. Comput. Phys. 21 (3), 291-304, 1976.
• [40] A. Saadatmandi and T. Abdolahi-Niasar, Numerical solution of Troesch’s problem using Christov rational functions, Comput. Methods Differ. Equ. 3 (4), 247-257, 2015.
• [41] M.R. Scott and W.H. Vandevender, A comparison of several invariant imbedding algorithms for the solution of two-point boundary-value problems, Appl. Math. Comput. 1 (3), 187-218, 1975.
• [42] S.C. Shiralashetti and S. Kumbinarasaiah, Theoretical study on continuous polynomial wavelet bases through wavelet series collocation method for nonlinear Lane-Emden type equations, Appl. Math. Comput. 315, 591-602, 2017.
• [43] R. Singh, H. Garg and V. Guleria, Haar wavelet collocation method for Lane-Emden equations with Dirichlet, Neumann and Neumann-Robin boundary conditions J. Comput. Appl. Math. 346, 150-161, 2019.
• [44] K. Tabatabaei and E. Gunerhan, Numerical solution of Duffing equation by the differential transform method, Appl. Math. Inf. Sci. Lett. 2 (1), 1-6, 2014.
• [45] H. Temimi, M. Ben-Romdhane, A.R. Ansari and G.I. Shishkin, Finite difference numerical solution of Troesch’s problem on a piecewise uniform Shishkin mesh, Calcolo 54, 225-242, 2017.
• [46] H. Temimi and H. Kürkçü, An accurate asymptotic approximation and precise numerical solution of highly sensitive Troesch’s problem, Appl. Math. Comput. 235, 253-260, 2014.
• [47] P.T. Toan, T.N. Vo and M. Razzaghi, Taylor wavelet method for fractional delay differential equations, Eng. Comput. 37, 231-240, 2019.
• [48] B.A. Troesch, A simple approach to a sensitive two-point boundary value problem, J. Comput. Phys. 21 (3), 279-290, 1976.
• [49] A.R. Vahidi, E. Babolian, G.H. Asadi-Cordshooli and F. Samiee, Restarted Adomian’s decomposition method for Duffing’s equation, Int. J. Math. Anal. 3 (15), 711-717, 2009.
• [50] E.S. Weibel, On the confinement of a plasma by magnetostatic fields, Phys. Fluids. 2 (1), 52-56, 1959.
• [51] E. Yusufoğlu, Numerical solution of Duffing equation by the Laplace decomposition algorithm Appl. Math. Comput. 177 (2), 572-580, 2006.
• [52] M. Zarebnia and M. Sajjadian, The Sinc-Galerkin method for solving Troesch’s problem, Math. Comput. Model. 56 (9-10), 218-228, 2012.