A COMPARATIVE STUDY OF A CLASS OF LINEAR AND NONLINEAR PANTOGRAPH DIFFERENTIAL EQUATIONS VIA DIFFERENT ORTHOGONAL POLYNOMIAL WAVELETS
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Abstract
We propose a wavelet approach on different orthogonal polynomials for solving linear and nonlinear pantograph equations with stretch kind. The pantograph differential equation is a unique proportional delay functional differential equation class. It has been used to deal with numerous physics, mathematics, and engineering applications, such as quantum mechanics, control systems, electrodynamics, and number theory. This scheme is based on constructing the operational matrix for integration via different wavelets with their collocation nodes. This study aims to examine the numerical dynamics of the pantograph equation under stretch kind through different orthogonal polynomial wavelets. Illustrative examples are presented to highlight the flexibility of this scheme, and comparisons are made between the mentioned scheme and other existing schemes using tables and graphs. These numerical results correctly predict the applicability and effectiveness of the mentioned scheme.
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References
Alomari, A.K., Noorani, M.S., & Nazar, R. (2009). Solution of delay differential equation by means of homotopy analysis method. Acta Applicandae Mathematicae, 108(2), 395-412.
Anakira, N.R., Alomari, A.K., & Hashim, I. (2013). Optimal homotopy asymptotic method for solving delay differential equations. Mathematical Problems in Engineering, Article ID 498902.
Asma, Rahman, G.u., & Gómez-Aguilar, J.F. et al. (2022). Study of Multi-Term Pantograph Differential Equations of Arbitrary Order. Few-Body Syst 63, 71. https://doi.org/10.1007/s00601-022-01770-0
Azodi, H.D., & Yaghouti, M.R. (2018). A new method based on fourth kind Chebyshev wavelets to a fractional order model of HIV infection of CD4^+T cells. Computational Methods for Differential Equations, 6(3), 353-371.
Bahmanpour, M., Tavassoli Kajani M., & Maleki, M. (2018). A Muntz wavelets collocation method for solving fractional differential equations. Comp. Appl Math, 37, 5514-5526.
Bahsi, M.M., & Cevik, M. (2015). Numerical Solution of Pantograph-Type Delay Differential Equations Using Perturbation-Iteration Algorithms. Journal of Applied Mathematics, Article ID 139821, 10 pages. http://dx.doi.org/10.1155/2015/139821.
Baker, C.T.H., Paul, C.A.H., & Wille, D.R. (1995). Issues in the numerical solution of evolutionary delay differential equations. Advances in Computational Mathematics, 3, 171-196.
Bellen, A., & Zennaro, M. (2003). Numerical Methods for Delay Differential Equations. Numerical Mathematics and Scientific Computations Series, Oxford University Press, Oxford.
Daubechies, I. (1988). Orthonormal bases of compactly supported wavelets. Comm. Pure Appl Math, 41, 909-996.
Drfel, G., & Iserles, A. (1997). The pantograph equation in the complex plane. J Math Anal Appl, 213, 117-132.
Driver, RD (1977). Ordinary and Delay Differential Equations. Applied Mathematics Series, Springer, New York.
Fox, L., Mayers, D.F., Ockendon, J.R., & Tayler, A.B. (1971). On a functional differential equation. IMA Journal of Applied Mathematics, 8(3), 271-307.
Hafshejani, M.S., Vanani, S.K., & Hafshejani, J.S. (2011). Numerical solution of Delay Differential Equations Using Legendre Wavelet Method. World Applied Sciences Journal, 13, 27-33.
Jafari, H., Mahmoudi, M. & Noori Skandari, M.H. (2021). A new numerical method to solve pantograph delay differential equations with convergence analysis. Adv Differ Equ, 129. https://doi.org/10.1186/s13662-021-03293-0
Javadi, S., Babolian, E., & Taheri, Z. (2016). Solving generalized pantograph equations by shifted orthonormal Bernstein polynomials. Journal of Computational and Applied Mathematics, 303, 1-14.
Kajani, M.T., HadiVencheh, A., & Ghasemi, M. (2009). The Chebyshev wavelets operational matrix of integration and product operation matrix. International Journal of Computer Mathematics, 86(7), 1118-1125.
KarimiVanani, S., & Aminataei, A. (2010). On the numerical solution of delay differential equations using multiquadric approximation scheme. J Functional Differential Equations, 17, 391-399.
Mallat, S. (2008). A wavelet tour of signal processing: the sparse way, Academic press, 3rd edition.
Muhammad, A.I., Muhammad, S., Syed, T.M.D., & Muhammad, R. (2017). Modified wavelets-based algorithm for nonlinear delay differential equations of fractional order. Advances in Mechanical Engineering, 9, 1-8. DOI: 10.1177/1687814017696223.
Ockendon, J.R., & Tayler, A.B. (1971). The dynamics of a current collection system for an electric locomotive. Proc Roy Sot Lond A, 322, 447-468.
Rahimkhani, P., Ordokhani, Y., & Babolian, E. (2016). Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet. Journal of Computational and Applied Mathematics, 309, 493-510.
Rayal, A., Tamta, S., Rawat, S., & Kashif, M., (2022). Numerical view of Lucas-Lehmer polynomials with its characteristics Uttaranchal Journal of Applied and Life Sciences, Uttaranchal University, 3(1), 66-75.
Rayal, A., & Verma, S.R. (2020a). An approximate wavelets solution to the class of variational problems with fractional order. Journal of Applied Mathematics and Computing, 65, 735-769.
Rayal, A., & Verma, S.R. (2020b). Numerical analysis of pantograph differential equation of the stretched type associated with fractal-fractional derivatives via fractional order Legendre wavelets. Chaos, Solitons Fractals, 139(1), 110076.
Rayal, A., & Verma, S.R. (2020c). Numerical study of variational problems of moving or fixed boundary conditions by Muntz wavelets. Journal of Vibration and Control, 28, 1-16.
Rayal, A., & Verma, S.R. (2022). Two-dimensional Gegenbauer wavelets for the numerical solution of tempered fractional model of the nonlinear Klein-Gordon equation. Applied Numerical Mathematics, 174, 191-220.
Rayal, A., (2023a). An effective Taylor wavelets basis for the evaluation of numerical differentiations. Palestine Journal of Mathematics, 12(1), 551-568.
Rayal, A., Joshi, B.P., Pandey, M., & Torres, D.F.M. (2023b). Numerical Investigation of the Fractional Oscillation Equations under the Context of Variable Order Caputo Fractional Derivative via Fractional Order Bernstein Wavelets. Mathematics, 11(11), 2503; https://doi.org/10.3390/math11112503
Rayal, A., Anand, M., Chauhan, K., & Prinsa (2023c). An Overview of Mamadu-Njoseh wavelets and its properties for numerical computations. Uttaranchal Journal of Applied and Life Sciences, Uttaranchal University, 4(1), 1-8.
Rayal, A. (2023d) Muntz Wavelets Solution for the Polytropic Lane–Emden Differential Equation Involved with Conformable Type Fractional Derivative. Int. J. Appl. Comput. Math 9, 50; https://doi.org/10.1007/s40819-023-01528-0
Saadatmandi, A., & Dehghan, M. (2009). Variational iteration method for solving a generalized pantograph equation. Computers and Mathematics with Applications, 58, 2190-2196.
Saeed, U., & Rehman, M.U. (2014). Hermite Wavelet Method for Fractional Delay Differential Equations. Journal of Difference Equations, Article ID 359093, 1-8.
Sedaghat, S., Ordokhani, Y., & Dehghan, M. (2012). Numerical solution of the delay differential equations of pantograph type via Chebyshev polynomials. Communications in Nonlinear Science and Numerical Simulation, 17, 4815-4830.
Sezer, M., & Akyuz-Dascioglu, A. (2007). A Taylor method for numerical solution of generalized pantograph equations with linear functional argument. Journal of Computational and Applied Mathematics, 200, 217-225.
Shiralashetti, S.C., Kumbinarasaiah, S., Mundewadi, R.A., & Hoogar, B.S. (2016). Series solutions of pantograph equations using wavelets. Open Journal of Applied & Theoretical Mathematics, 2(4), 505-518.
Tayler, A.B. (1986). Mathematical Models in Applied Mathematics. Clarendon Pres, Oxford, 40-53.
Tohidi, E., Bhrawy, A.H., & Erfani, K. (2013). A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation. Applied Mathematical Modelling, 37, 4283-4294.
Tural-Polat, S.N. (2019). Third-kind Chebyshev Wavelet Method for the Solution of Fractional Order Riccati Differential Equations. Journal of Circuits, Systems and Computers, 28(14), 1950247.
Wang, L.P., Chen, Y.M., Liu, D.Y., & Boutat, D. (2019). Numerical algorithm to solve generalized fractional pantograph equations with variable coefficients based on shifted Chebyshev polynomials. International Journal of Computer Mathematics, 96, 2487-2510.
Yalcinbas, S., Aynigul, M., & Sezer, M. (2011). A collocation method using Hermite polynomials for approximate solution of pantograph equations. Journal of the Franklin Institute, 348, 1128-1139.
Yalcinbas, S., Sorkun, H.H., & Sezer, M. (2015). A numerical method for solutions of pantograph type differential equations with variable coefficients using Bernstein polynomials. New Trends in Mathematical Sciences, 3(4), 179-195.
Yang, C., (2018). Modified Chebyshev collocation method for pantograph-type differential equations. Appl Numer Math, 134, 132-144.
Yusufoglu, E. (2010). An efficient algorithm for solving generalized pantograph equations with linear functional argument. Applied Mathematics and Computation, 217, 3591-3595.
Yuzbas, S., Gok, E., & Sezer, M. (2014). Laguerre matrix method with the residual error estimation for solutions of a class of delay differential equations. Mathematical Methods in the Applied Sciences, 37(4), 453-463.
Zhu, L., & Wang, Y. (2013). Second Chebyshev wavelet operational matrix of integration and its application in the calculus of variations. International Journal of Computer Mathematics, 90(11), 2338-2352.