Exact and Numerical Treatment of a Special Kind of the Pantograph Model via Laplace Technique

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DOI: https://doi.org/10.28924/2291-8639-20-2022-71
Published: Dec 26, 2022
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Weam G. Alharbi, Exact and Numerical Treatment of a Special Kind of the Pantograph Model via Laplace Technique, Int. J. Anal. Appl., 20 (2022), 71.

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Weam G. Alharbi

Abstract

The Pantograph is a device of practical application in electric trains, by which the current is collected. The mathematical problem of this device is generally given by the delay differential equation φ’(t) = αy(t) + βφ(γt), where α and β are real constants and γ is a proportional delay parameter. In the literature, a special attention has been given to the particular case γ = −1. The objective of this paper is to extend the application of the Laplace transform (LT) combined with the Adomian decomposition method (ADM) to analyze the above model at such particular case of γ. The solution will be determined in exact form which agrees with the corresponding results in the literature. Various properties of the obtained exact solution are discussed in detail. Moreover, it will be declared that for sufficiently small values of α compared to β there exists an accurate approximate solution. The accuracy of the approximate solution is numerically validated. In addition, some numerical results are conducted for the behavior of the present solution at selected values of α and β.

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References

  1. H.I. Andrews, Third Paper: Calculating the Behaviour of an Overhead Catenary System for Railway Electrification, Proc. Inst. Mech. Eng. 179 (1964), 809-846. https://doi.org/10.1243/PIME_PROC_1964_179_050_02.
  2. M.R. Abbott, Numerical Method for Calculating the Dynamic Behaviour of a Trolley Wire Overhead Contact System for Electric Railways, Comput. J. 13 (1970), 363-368. https://doi.org/10.1093/comjnl/13.4.363.
  3. G. Gilbert, H.E.H. Davtcs, Pantograph Motion on a Nearly Uniform Railway Overhead Line, Proc. Inst. Electr. Eng. 113 (1966), 485-492. https://doi.org/10.1049/piee.1966.0078.
  4. P.M. Caine, P.R. Scott, Single-Wire Railway Overhead System, Proc. Inst. Electr. Eng. 116 (1969), 1217-1221. https://doi.org/10.1049/piee.1969.0226.
  5. J.R. Ockendon, A.B. Tayler, The Dynamics of a Current Collection System for an Electric Locomotive, Proc. R. Soc. Lond. A. Math. Phys. Eng. Sci. 322 (1971), 447-468. https://doi.org/10.1098/rspa.1971.0078.
  6. L. Fox, D. Mayers, J.R. Ockendon, A.B. Tayler, On a Functional Differential Equation, IMA J. Appl. Math. 8 (1971), 271-307. https://doi.org/10.1093/imamat/8.3.271.
  7. T. Kato, J.B. McLeod, The Functional-Differential Equation y 0 (x) = ay(λx) + by(x), Bull. Amer. Math. Soc. 77 (1971), 891-935.
  8. A. Iserles, On the Generalized Pantograph Functional-Differential Equation, Eur. J. Appl. Math. 4 (1993), 1–38. https://doi.org/10.1017/s0956792500000966.
  9. V.A. Ambartsumian, On the Fluctuation of the Brightness of the Milky Way, Dokl. Akad. Nauk SSSR, 44 (1994), 223-226.
  10. J. Patade, S. Bhalekar, On Analytical Solution of Ambartsumian Equation, Natl. Acad. Sci. Lett. 40 (2017), 291–293. https://doi.org/10.1007/s40009-017-0565-2.
  11. F.M. Alharbi, A. Ebaid, New Analytic Solution for Ambartsumian Equation, J. Math. Syst. Sci. 8 (2018), 182-186. https://doi.org/10.17265/2159-5291/2018.07.002.
  12. H. Bakodah, A. Ebaid, Exact Solution of Ambartsumian Delay Differential Equation and Comparison with DaftardarGejji and Jafari Approximate Method, Mathematics. 6 (2018), 331. https://doi.org/10.3390/math6120331.
  13. N.O. Alatawi, A. Ebaid, Solving a Delay Differential Equation by Two Direct Approaches, J. Math. Syst. Sci. 9 (2019), 54-56. https://doi.org/10.17265/2159-5291/2019.02.003.
  14. A. Ebaid, A. Al-Enazi, B.Z. Albalawi, M.D. Aljoufi, Accurate Approximate Solution of Ambartsumian Delay Differential Equation via Decomposition Method, Math. Comput. Appl. 24 (2019), 7. https://doi.org/10.3390/mca24010007.
  15. A.A. Alatawi, M. Aljoufi, F.M. Alharbi, A. Ebaid, Investigation of the Surface Brightness Model in the Milky Way via Homotopy Perturbation Method, J. Appl. Math. Phys. 8 (2020), 434–442. https://doi.org/10.4236/jamp.2020.83033.
  16. E. A. Algehyne, E. R. El-Zahar, F. M. Alharbi, A. Ebaid, Development of Analytical Solution for a Generalized Ambartsumian Equation, AIMS Math. 5 (2020), 249–258. https://doi.org/10.3934/math.2020016.
  17. S.M. Khaled, E.R. El-Zahar, A. Ebaid, Solution of Ambartsumian Delay Differential Equation with Conformable Derivative, Mathematics. 7 (2019), 425. https://doi.org/10.3390/math7050425.
  18. D. Kumar, J. Singh, D. Baleanu, S. Rathore, Analysis of a Fractional Model of the Ambartsumian Equation, Eur. Phys. J. Plus. 133 (2018), 259. https://doi.org/10.1140/epjp/i2018-12081-3.
  19. A. Ebaid, C. Cattani, A.S. Al Juhani, E.R. El-Zahar, A novel exact solution for the fractional Ambartsumian equation, Adv. Differ. Equ. 2021 (2021), 88. https://doi.org/10.1186/s13662-021-03235-w.
  20. A. Ebaid, H.K. Al-Jeaid, On the Exact Solution of the Functional Differential Equation y 0 (t) = ay(t) + by(−t), Adv. Differ. Equ. Control Processes, 26 (2022), 39-49. https://doi.org/10.17654/0974324322003.
  21. G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academi Publishers, Boston, 1994.
  22. A.M. Wazwaz, Adomian Decomposition Method for a Reliable Treatment of the Bratu-Type Equations, Appl. Math. Comput. 166 (2005), 652–663. https://doi.org/10.1016/j.amc.2004.06.059.
  23. A. Ebaid, Approximate Analytical Solution of a Nonlinear Boundary Value Problem and its Application in Fluid Mechanics, Z. Naturforsch. A. 66 (2011), 423–426. https://doi.org/10.1515/zna-2011-6-707.
  24. J.S. Duan, R. Rach, A New Modification of the Adomian Decomposition Method for Solving Boundary Value Problems for Higher Order Nonlinear Differential Equations, Appl. Math. Comput. 218 (2011), 4090–4118. https://doi.org/10.1016/j.amc.2011.09.037.
  25. A. Ebaid, A New Analytical and Numerical Treatment for Singular Two-Point Boundary Value Problems via the Adomian Decomposition Method, J. Comput. Appl. Math. 235 (2011), 1914–1924. https://doi.org/10.1016/j.cam.2010.09.007.
  26. E.H. Aly, A. Ebaid, R. Rach, Advances in the Adomian Decomposition Method for Solving Two-Point Nonlinear Boundary Value Problems With Neumann Boundary Conditions, Computers Math. Appl. 63 (2012), 1056–1065. https://doi.org/10.1016/j.camwa.2011.12.010.
  27. C. Chun, A. Ebaid, M. Lee, E. Aly, An Approach for Solving Singular Two Point Boundary Value Problems: Analytical and Numerical Treatment, ANZIAM J. 52 (2012), 21-43. https://doi.org/10.21914/anziamj.v53i0.4582.
  28. A. Ebaid, M.D. Aljoufi, A.M. Wazwaz, An Advanced Study on the Solution of Nanofluid Flow Problems via Adomian’s Method, Appl. Math. Lett. 46 (2015), 117–122. https://doi.org/10.1016/j.aml.2015.02.017.
  29. J. Diblík, M. Kúdelcíková, Two Classes of Positive Solutions of First Order Functional Differential Equations of Delayed Type, Nonlinear Anal.: Theory Methods Appl. 75 (2012), 4807-4820. https://doi.org/10.1016/j.na.2012.03.030.
  30. A. Ebaid, N. Al-Armani, A New Approach for a Class of the Blasius Problem via a Transformation and Adomian’s Method, Abstr. Appl. Anal. 2013 (2013), 753049. https://doi.org/10.1155/2013/753049.
  31. A. Ebaid, Analytical Solutions for the Mathematical Model Describing the Formation of Liver Zones via Adomian’s Method, Comput. Math. Methods Med. 2013 (2013), 547954. https://doi.org/10.1155/2013/547954.
  32. A. Alshaery, A. Ebaid, Accurate Analytical Periodic Solution of the Elliptical Kepler Equation Using the Adomian Decomposition Method, Acta Astronautica. 140 (2017), 27–33. https://doi.org/10.1016/j.actaastro.2017.07.034.
  33. H.O. Bakodah, A. Ebaid, The Adomian Decomposition Method for the Slip Flow and Heat Transfer of Nanofluids Over a Stretching/shrinking Sheet, Rom. Rep. Phys. 70 (2018), 115.
  34. W. Li, Y. Pang, Application of Adomian Decomposition Method to Nonlinear Systems, Adv. Differ. Equ. 2020 (2020), 67. https://doi.org/10.1186/s13662-020-2529-y.
  35. A. Ebaid, H.K. Al-Jeaid, H. Al-Aly, Notes on the Perturbation Solutions of the Boundary Layer Flow of Nanofluids Past a Stretching Sheet, Appl. Math. Sci. 7 (2013), 6077–6085. https://doi.org/10.12988/ams.2013.36277.
  36. A. Ebaid, S.M. Khaled, An Exact Solution for a Boundary Value Problem with Application in Fluid Mechanics and Comparison with the Regular Perturbation Solution, Abstr. Appl. Anal. 2014 (2014), 172590. https://doi.org/10.1155/2014/172590.
  37. A. Ebaid, Remarks on the Homotopy Perturbation Method for the Peristaltic Flow of Jeffrey Fluid With NanoParticles in an Asymmetric Channel, Computers Math. Appl. 68 (2014), 77–85. https://doi.org/10.1016/j.camwa.2014.05.008.
  38. Z. Ayati, J. Biazar, On the Convergence of Homotopy Perturbation Method, J. Egypt. Math. Soc. 23 (2015), 424–428. https://doi.org/10.1016/j.joems.2014.06.015.
  39. A. Ebaid, A.F. Aljohani, E.H. Aly, Homotopy Perturbation Method for Peristaltic Motion of Gold-Blood Nanofluid With Heat Source, Int. J. Numer. Methods Heat Fluid Flow, 30 (2020), 3121-3138, https://doi.org/10.1108/HFF-11-2018-0655.
  40. A. Ebaid, Analysis of Projectile Motion in View of Fractional Calculus, Appl. Math. Model. 35 (2011), 1231–1239. https://doi.org/10.1016/j.apm.2010.08.010.
  41. S.M. Khaled, A. Ebaid, F. Al Mutairi, The Exact Endoscopic Effect on the Peristaltic Flow of a Nanofluid, J. Appl. Math. 2014 (2014), 367526. https://doi.org/10.1155/2014/367526.
  42. A. Ebaid, M.A. Al Sharif, Application of Laplace Transform for the Exact Effect of a Magnetic Field on Heat Transfer of Carbon Nanotubes-Suspended Nanofluids, Z. Naturforsch. A. 70 (2015), 471–475. https://doi.org/10.1515/zna-2015-0125.
  43. A. Ebaid, A.-M. Wazwaz, E. Alali, B.S. Masaedeh, Hypergeometric Series Solution to a Class of Second-Order Boundary Value Problems via Laplace Transform with Applications to Nanofluids, Commun. Theor. Phys. 67 (2017), 231. https://doi.org/10.1088/0253-6102/67/3/231.
  44. H. Saleh, E. Alali, A. Ebaid, Medical Applications for the Flow of Carbon-Nanotubes Suspended Nanofluids in the Presence of Convective Condition Using Laplace Transform, J. Assoc. Arab Univ. Basic Appl. Sci. 24 (2017), 206–212. https://doi.org/10.1016/j.jaubas.2016.12.001.
  45. A. Ebaid, E. Alali, H.S. Ali, The Exact Solution of a Class of Boundary Value Problems With Polynomial Coefficients and Its Applications on Nanofluids, J. Assoc. Arab Univ. Basic Appl. Sci. 24 (2017), 156–159. https://doi.org/10.1016/j.jaubas.2017.07.003.
  46. S.M. Khaled, The Exact Effects of Radiation and Joule Heating on Magnetohydrodynamic Marangoni Convection Over a Flat Surface, Therm Sci. 22 (2018), 63–72. https://doi.org/10.2298/tsci151005050k.
  47. H. Ali, E. Alali, A. Ebaid, F. Alharbi, Analytic Solution of a Class of Singular Second-Order Boundary Value Problems with Applications, Mathematics. 7 (2019), 172. https://doi.org/10.3390/math7020172.
  48. A. Ebaid, E.R. El-Zahar, A.F. Aljohani, B. Salah, M. Krid, J.T. Machado, Analysis of the Two-Dimensional Fractional Projectile Motion in View of the Experimental Data, Nonlinear Dyn. 97 (2019), 1711–1720. https://doi.org/10.1007/s11071-019-05099-y.
  49. A. Ebaid, W. Alharbi, M.D. Aljoufi, E.R. El-Zahar, The Exact Solution of the Falling Body Problem in ThreeDimensions: Comparative Study, Mathematics. 8 (2020), 1726. https://doi.org/10.3390/math8101726.
  50. A. Ebaid, H.K. Al-Jeaid, The Mittag–Leffler Functions for a Class of First-Order Fractional Initial Value Problems: Dual Solution via Riemann–Liouville Fractional Derivative, Fractal Fract. 6 (2022), 85. https://doi.org/10.3390/fractalfract6020085.
  51. A.F. Aljohani, A. Ebaid, E.A. Algehyne, Y.M. Mahrous, C. Cattani, H.K. Al-Jeaid, The Mittag-Leffler Function for Re-Evaluating the Chlorine Transport Model: Comparative Analysis, Fractal Fract. 6 (2022), 125. https://doi.org/10.3390/fractalfract6030125.
  52. M.R. Spiegel, Laplace transforms, McGraw-Hill. Inc., New York, 1965.