Series solution for a delay differential equation arising in electrodynamics
In this study, the pantograph equation is investigated using the homotopy perturbation and variational iteration methods. The pantograph equation is a delay differential equation that arises in quite different fields of pure and applied mathematics, such as number theory, dynamical systems, probabil...
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| Vydáno v: | Communications in numerical methods in engineering Ročník 25; číslo 11; s. 1084 - 1096 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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Chichester, UK
John Wiley & Sons, Ltd
01.11.2009
Wiley |
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| ISSN: | 1069-8299, 1099-0887 |
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| Abstract | In this study, the pantograph equation is investigated using the homotopy perturbation and variational iteration methods. The pantograph equation is a delay differential equation that arises in quite different fields of pure and applied mathematics, such as number theory, dynamical systems, probability, mechanics and electrodynamics. The procedure of present methods are based on the search for a solution in the form of a series with easily computed components. Application of these techniques to this problem shows the rapid convergence of the sequence constructed by these methods to the exact solution. Moreover, this technique reduces the volume of calculations by not requiring discretization of the variables, linearization or small perturbations. Copyright © 2009 John Wiley & Sons, Ltd. |
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| AbstractList | In this study, the pantograph equation is investigated using the homotopy perturbation and variational iteration methods. The pantograph equation is a delay differential equation that arises in quite different fields of pure and applied mathematics, such as number theory, dynamical systems, probability, mechanics and electrodynamics. The procedure of present methods are based on the search for a solution in the form of a series with easily computed components. Application of these techniques to this problem shows the rapid convergence of the sequence constructed by these methods to the exact solution. Moreover, this technique reduces the volume of calculations by not requiring discretization of the variables, linearization or small perturbations. Copyright © 2009 John Wiley & Sons, Ltd. |
| Author | Koçak, Hüseyin Yıldırım, Ahmet |
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| Cites_doi | 10.1016/j.mcm.2007.09.016 10.1016/S0020-7462(98)00048-1 10.1155/2008/869614 10.1016/j.cam.2006.07.009 10.1016/j.physleta.2007.04.072 10.1088/0031-8949/78/06/065004 10.1016/S0045-7825(99)00018-3 10.1002/cnm.1200 10.2528/PIER07090403 10.1142/S0217979206033796 10.1007/s10543-005-0022-3 10.1142/S0217979208048668 10.1016/j.jsv.2007.05.021 10.1098/rspa.1971.0078 10.1016/j.camwa.2008.07.020 10.1016/j.amc.2006.01.084 10.1002/cnm.1154 10.1016/j.jmaa.2006.02.063 10.1016/j.camwa.2006.12.083 10.1088/0031-8949/75/6/007 10.1080/00207160902874653 10.1016/j.amc.2003.07.017 10.1016/S0168-9274(97)00028-7 10.1016/S0020-7462(98)00085-7 10.1515/zna-2008-10-1102 10.1142/S0217979206034819 10.1016/j.na.2008.03.012 10.1080/00207160802247646 10.1007/s002110050470 10.1088/0031-8949/75/4/031 10.1016/j.amc.2006.08.086 |
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| Keywords | Electrodynamics Probabilistic approach delay differential equations the variational iteration method Delay equation Exact solution Applied mathematics Homotopy Perturbation techniques Variational calculus Modelling Pantographs Iterative methods Linearization the homotopy perturbation method |
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| References | He JH. A coupling method of a homotopy technique and a perturbation technique for non-linear problems. International Journal of Non-Linear Mechanics 2000; 35:37-43. Yıldırım A. The homotopy perturbation method for approximate solution of the modified KdV equation. Zeitschrift für Naturforschung A, A Journal of Physical Sciences, 2008; 63a:621. Yıldırım A. An algorithm for solving the fractional nonlinear Schrödinger equation by means of the homotopy perturbation method. International Journal of Nonlinear Sciences and Numerical Simulation 2009; (10):445-451. Yıldırım A. Applying He's variational iteration method for solving differential-difference equation. Mathematical Problems in Engineering 2008; 2008:1-7. Article ID 869614. Yıldırım A. He's homotopy perturbation method for solving the space- and time-fractional telegraph equations. International Journal of Computer Mathematics 2009; DOI: 10.1080/00207160902874653. Yıldırım A. Variational iteration method for modified Camassa-Holm and Degasperis-Procesi equations. Communications in Numerical Methods in Engineering 2008; DOI: 10.1002/cnm.1154. Shakeri F, Dehghan M. Inverse problem of diffusion equation by He's homotopy perturbation method. Physica Scripta 2007; 75:551. Ishiwata E, Muroya Y. Rational approximation method for delay differential equations with proportional delay. Applied Mathematics and Computation 2007; 187:741-747. Fan Z, Liu MZ, Cao W. Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations. Journal of Mathematical Analysis and Applications 2007; 325:1142-1159. Yıldırım A. He's homotopy perturbation method for nonlinear differential-difference equations. International Journal of Computer Mathematics 2008; DOI: 10.1080/00207160802247646. He JH, Wu XH. Variational iteration method: new development and applications. Computers and Mathematics with Applications 2007; 54:881-894. Baker CTH, Bukwwar E. Continuous 2-methods for the stochastic pantograph equation. Electronic Transactions on Numerical Analysis 2000; 11:131-151. Yıldırım A, Öziş T. Solutions of singular IVPs of Lane-Emden type by the variational iteration method. Nonlinear Analysis Series A: Theory, Methods and Applications 2008; DOI: 10.1016/j.na.2008.03.012. Liu MZ, Li DS. Properties of analytic solution and numerical solution of multi-pantograph equation. Applied Mathematics and Computation 2004; 155:853-871. Dehghan M, Shakeri F. Solution of an integro-differential equation arising in oscillating magnetic fields using He's homotopy perturbation method. PIER 2008; 78:361. Zhao JJ, Xu Y, Wang HX, Liu MZ. Stability of a class of Runge-Kutta methods for a family of pantograph equations of neutral type. Applied Mathematics and Computation 2006; 181:1170-1181. Öziş T, Yıldırım A. A study of nonlinear oscillators with u1/3 force by He's variational iteration method. Journal of Sound and Vibration 2007; 306:372-376. He JH. New interpretation of homotopy perturbation method. International Journal of Modern Physics B 2006; 20:2561. He JH. Variational iteration method-a kind of non-linear analytical technique: some examples. International Journal of Non-Linear Mechanics 1999; 34:699-708. He JH. Variational iteration method-some recent results and new interpretations. Journal of Computational and Applied Mathematics 2007; 207:3-17. Yıldırım A. Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method. Computers and Mathematics with Applications 2008; 56:3175-3180. Li DS, Liu MZ. Exact solution properties of a multi-pantograph delay differential equation. Journal of Harbin Institute of Technology 2000; 32:1-3. Dehghan M, Shakeri F. Solution of a partial differential equation subject to temperature overspecification by He's homotopy perturbation method. Physica Scripta 2007; 75:778. He JH. Some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physics B 2006; 20:1141. Liu MZ, Yang ZW, Hu GD. Asymptotical stability of numerical methods with constant stepsize for pantograph equations. BIT Numerical Mathematics 2005; 45:743-759. He JH. Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering 1999; 178:257-262. Zhang C, Sun G. Boundedness and asymptotic stability of multistep methods for generalized pantograph equations. Journal of Computational Mathematics 2004; 22:447-456. Koto T. Stability of Runge-Kutta methods for the generalized pantograph equation. Numerische Mathematik 1999; 84:233-247. Yıldırım A. Application of the homotopy perturbation method for the Fokker-Planck equation. Communications in Numerical Methods in Engineering 2008; DOI: 10.1002/cnm.1200. He JH. An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering. International Journal of Modern Physics B 2008; 22:3487. Ockendon JR, Taylor AB. The dynamics of a current collection system for an electric locomotive. Proceedings of the Royal Society of London, Series A 1971; 322:447-468. Dehghan M, Shakeri F. The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics. Physica Scripta 2008; 78:065004. Yıldırım A, Öziş T. Solutions of singular IVPs of Lane-Emden type by homotopy perturbation method. Physics Letters A 2007; 369:70. Liu Y. Numerical investigation of the pantograph equation. Applied Numerical Mathematics 1997; 24:309-317. Tamasan A. Differentiability with respect to lag for nonlinear pantograph equations. Pure and Applied Mathematics 1998; 9:215-220. He JH. Recent development of the homotopy perturbation method. Topological Methods in Nonlinear Analysis 2008; 31:205. Shakeri F, Dehghan M. Solution of the delay differential equations via homotopy perturbation method. Mathematical and Computer Modelling 2008; 48:486. 2004; 22 2007; 207 2007; 306 2007; 325 2007; 369 2008; 63a 2007; 187 1997; 24 2009 2008 2008; 78 2008; 56 1971; 322 1999; 84 2008; 31 2007; 75 2007; 54 2008; 2008 2005; 45 2004; 155 2006; 20 2000; 35 2000; 32 2000; 11 1999; 34 2008; 48 2006; 181 2008; 22 1999; 178 1998; 9 Baker CTH (e_1_2_1_7_2) 2000; 11 Tamasan A (e_1_2_1_6_2) 1998; 9 e_1_2_1_22_2 Yıldırım A (e_1_2_1_34_2) 2009 e_1_2_1_23_2 e_1_2_1_20_2 e_1_2_1_21_2 e_1_2_1_26_2 e_1_2_1_27_2 e_1_2_1_24_2 e_1_2_1_25_2 He JH (e_1_2_1_36_2) 2008; 31 e_1_2_1_28_2 e_1_2_1_29_2 Li DS (e_1_2_1_8_2) 2000; 32 e_1_2_1_30_2 e_1_2_1_4_2 e_1_2_1_5_2 e_1_2_1_2_2 Zhang C (e_1_2_1_10_2) 2004; 22 e_1_2_1_11_2 e_1_2_1_3_2 e_1_2_1_12_2 e_1_2_1_33_2 e_1_2_1_32_2 e_1_2_1_31_2 e_1_2_1_15_2 e_1_2_1_38_2 e_1_2_1_16_2 e_1_2_1_37_2 e_1_2_1_13_2 e_1_2_1_14_2 e_1_2_1_35_2 e_1_2_1_19_2 e_1_2_1_17_2 e_1_2_1_9_2 e_1_2_1_18_2 e_1_2_1_39_2 |
| References_xml | – reference: Li DS, Liu MZ. Exact solution properties of a multi-pantograph delay differential equation. Journal of Harbin Institute of Technology 2000; 32:1-3. – reference: Yıldırım A. An algorithm for solving the fractional nonlinear Schrödinger equation by means of the homotopy perturbation method. International Journal of Nonlinear Sciences and Numerical Simulation 2009; (10):445-451. – reference: Yıldırım A, Öziş T. Solutions of singular IVPs of Lane-Emden type by homotopy perturbation method. Physics Letters A 2007; 369:70. – reference: Yıldırım A. The homotopy perturbation method for approximate solution of the modified KdV equation. Zeitschrift für Naturforschung A, A Journal of Physical Sciences, 2008; 63a:621. – reference: He JH. A coupling method of a homotopy technique and a perturbation technique for non-linear problems. International Journal of Non-Linear Mechanics 2000; 35:37-43. – reference: Shakeri F, Dehghan M. Inverse problem of diffusion equation by He's homotopy perturbation method. Physica Scripta 2007; 75:551. – reference: Yıldırım A. Application of the homotopy perturbation method for the Fokker-Planck equation. Communications in Numerical Methods in Engineering 2008; DOI: 10.1002/cnm.1200. – reference: Shakeri F, Dehghan M. Solution of the delay differential equations via homotopy perturbation method. Mathematical and Computer Modelling 2008; 48:486. – reference: Dehghan M, Shakeri F. Solution of an integro-differential equation arising in oscillating magnetic fields using He's homotopy perturbation method. PIER 2008; 78:361. – reference: Dehghan M, Shakeri F. Solution of a partial differential equation subject to temperature overspecification by He's homotopy perturbation method. Physica Scripta 2007; 75:778. – reference: Fan Z, Liu MZ, Cao W. Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations. Journal of Mathematical Analysis and Applications 2007; 325:1142-1159. – reference: Tamasan A. Differentiability with respect to lag for nonlinear pantograph equations. Pure and Applied Mathematics 1998; 9:215-220. – reference: Baker CTH, Bukwwar E. Continuous 2-methods for the stochastic pantograph equation. Electronic Transactions on Numerical Analysis 2000; 11:131-151. – reference: He JH. Variational iteration method-some recent results and new interpretations. Journal of Computational and Applied Mathematics 2007; 207:3-17. – reference: Yıldırım A. Variational iteration method for modified Camassa-Holm and Degasperis-Procesi equations. Communications in Numerical Methods in Engineering 2008; DOI: 10.1002/cnm.1154. – reference: Yıldırım A. Applying He's variational iteration method for solving differential-difference equation. Mathematical Problems in Engineering 2008; 2008:1-7. Article ID 869614. – reference: Liu MZ, Li DS. Properties of analytic solution and numerical solution of multi-pantograph equation. Applied Mathematics and Computation 2004; 155:853-871. – reference: He JH. An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering. International Journal of Modern Physics B 2008; 22:3487. – reference: Dehghan M, Shakeri F. The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics. Physica Scripta 2008; 78:065004. – reference: Ockendon JR, Taylor AB. The dynamics of a current collection system for an electric locomotive. Proceedings of the Royal Society of London, Series A 1971; 322:447-468. – reference: Zhang C, Sun G. Boundedness and asymptotic stability of multistep methods for generalized pantograph equations. Journal of Computational Mathematics 2004; 22:447-456. – reference: Liu Y. Numerical investigation of the pantograph equation. Applied Numerical Mathematics 1997; 24:309-317. – reference: He JH. New interpretation of homotopy perturbation method. International Journal of Modern Physics B 2006; 20:2561. – reference: He JH. Variational iteration method-a kind of non-linear analytical technique: some examples. International Journal of Non-Linear Mechanics 1999; 34:699-708. – reference: Liu MZ, Yang ZW, Hu GD. Asymptotical stability of numerical methods with constant stepsize for pantograph equations. BIT Numerical Mathematics 2005; 45:743-759. – reference: He JH. Recent development of the homotopy perturbation method. Topological Methods in Nonlinear Analysis 2008; 31:205. – reference: He JH. Some asymptotic methods for strongly nonlinear equations. International Journal of Modern Physics B 2006; 20:1141. – reference: Ishiwata E, Muroya Y. Rational approximation method for delay differential equations with proportional delay. Applied Mathematics and Computation 2007; 187:741-747. – reference: Yıldırım A, Öziş T. Solutions of singular IVPs of Lane-Emden type by the variational iteration method. Nonlinear Analysis Series A: Theory, Methods and Applications 2008; DOI: 10.1016/j.na.2008.03.012. – reference: Yıldırım A. He's homotopy perturbation method for nonlinear differential-difference equations. International Journal of Computer Mathematics 2008; DOI: 10.1080/00207160802247646. – reference: Öziş T, Yıldırım A. A study of nonlinear oscillators with u1/3 force by He's variational iteration method. Journal of Sound and Vibration 2007; 306:372-376. – reference: Yıldırım A. Solution of BVPs for fourth-order integro-differential equations by using homotopy perturbation method. Computers and Mathematics with Applications 2008; 56:3175-3180. – reference: He JH, Wu XH. Variational iteration method: new development and applications. Computers and Mathematics with Applications 2007; 54:881-894. – reference: Koto T. Stability of Runge-Kutta methods for the generalized pantograph equation. Numerische Mathematik 1999; 84:233-247. – reference: Yıldırım A. He's homotopy perturbation method for solving the space- and time-fractional telegraph equations. International Journal of Computer Mathematics 2009; DOI: 10.1080/00207160902874653. – reference: He JH. Homotopy perturbation technique. Computer Methods in Applied Mechanics and Engineering 1999; 178:257-262. – reference: Zhao JJ, Xu Y, Wang HX, Liu MZ. Stability of a class of Runge-Kutta methods for a family of pantograph equations of neutral type. Applied Mathematics and Computation 2006; 181:1170-1181. – volume: 181 start-page: 1170 year: 2006 end-page: 1181 article-title: Stability of a class of Runge–Kutta methods for a family of pantograph equations of neutral type publication-title: Applied Mathematics and Computation – volume: 9 start-page: 215 year: 1998 end-page: 220 article-title: Differentiability with respect to lag for nonlinear pantograph equations publication-title: Pure and Applied Mathematics – volume: 20 start-page: 1141 year: 2006 article-title: Some asymptotic methods for strongly nonlinear equations publication-title: International Journal of Modern Physics B – volume: 22 start-page: 3487 year: 2008 article-title: An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering publication-title: International Journal of Modern Physics B – volume: 22 start-page: 447 year: 2004 end-page: 456 article-title: Boundedness and asymptotic stability of multistep methods for generalized pantograph equations publication-title: Journal of Computational Mathematics – volume: 78 start-page: 361 year: 2008 article-title: Solution of an integro‐differential equation arising in oscillating magnetic fields using He's homotopy perturbation method publication-title: PIER – volume: 187 start-page: 741 year: 2007 end-page: 747 article-title: Rational approximation method for delay differential equations with proportional delay publication-title: Applied Mathematics and Computation – volume: 56 start-page: 3175 year: 2008 end-page: 3180 article-title: Solution of BVPs for fourth‐order integro‐differential equations by using homotopy perturbation method publication-title: Computers and Mathematics with Applications – year: 2008 article-title: He's homotopy perturbation method for nonlinear differential‐difference equations publication-title: International Journal of Computer Mathematics – volume: 45 start-page: 743 year: 2005 end-page: 759 article-title: Asymptotical stability of numerical methods with constant stepsize for pantograph equations publication-title: BIT Numerical Mathematics – volume: 325 start-page: 1142 year: 2007 end-page: 1159 article-title: Existence and uniqueness of the solutions and convergence of semi‐implicit Euler methods for stochastic pantograph equations publication-title: Journal of Mathematical Analysis and Applications – volume: 63a start-page: 621 year: 2008 article-title: The homotopy perturbation method for approximate solution of the modified KdV equation publication-title: Zeitschrift für Naturforschung A, A Journal of Physical Sciences – volume: 84 start-page: 233 year: 1999 end-page: 247 article-title: Stability of Runge–Kutta methods for the generalized pantograph equation publication-title: Numerische Mathematik – volume: 11 start-page: 131 year: 2000 end-page: 151 article-title: Continuous 2‐methods for the stochastic pantograph equation publication-title: Electronic Transactions on Numerical Analysis – volume: 31 start-page: 205 year: 2008 article-title: Recent development of the homotopy perturbation method publication-title: Topological Methods in Nonlinear Analysis – year: 2009 article-title: He's homotopy perturbation method for solving the space‐ and time‐fractional telegraph equations publication-title: International Journal of Computer Mathematics – volume: 178 start-page: 257 year: 1999 end-page: 262 article-title: Homotopy perturbation technique publication-title: Computer Methods in Applied Mechanics and Engineering – year: 2008 article-title: Application of the homotopy perturbation method for the Fokker–Planck equation publication-title: Communications in Numerical Methods in Engineering – volume: 34 start-page: 699 year: 1999 end-page: 708 article-title: Variational iteration method—a kind of non‐linear analytical technique: some examples publication-title: International Journal of Non‐Linear Mechanics – volume: 207 start-page: 3 year: 2007 end-page: 17 article-title: 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| SubjectTerms | Computational techniques delay differential equations Exact sciences and technology Fundamental areas of phenomenology (including applications) Mathematical methods in physics Physics the homotopy perturbation method the variational iteration method |
| Title | Series solution for a delay differential equation arising in electrodynamics |
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