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Research Article Open access CC BY 3.0

Procedure for Exact Solutions of Nonlinear Pantograph Delay Differential Equations

Brahim Benhammouda, Hector Vazquez-Leal, Luis Hernandez-Martinez

Journal of Advances in Mathematics and Computer Science · pp. 2738–2751 · Published 12 Jul 2014

10.9734/BJMCS/2014/11839

Abstract

This work presents the application of the power series method (PSM) to find solutions of nonlinear delay differential equations of pantograph type (PDDEs). Three equations are solved to show that PSM can provide analytical solutions of PDDEs in convergent series form. The nonlinear pantograph cases study are: a first order equation, a second order equation, and a second order singular equation. Additionally, we present the post-treatment of the power series solutions with the Laplace-Pad´e (LP) resummation method as a powerful technique to find exact solutions. The proposed methodology possesses a simple procedure based on a few straightforward steps and it does not depend on a perturbation parameter.

Pantograph equations Power series method Laplace transform Pad´e approximant Analytical solutions.

References (47)

  1. 1 Ordinary Differential Equations [DOI]
  2. 2 Essentials of Pade approximants
  3. 3 The dynamics of a current collection system for an electric locomotive [DOI]
  4. 4 Theory of Differential Equations. [DOI]
  5. 5 Introduction to Ordinary Differential Equations [DOI]
  6. 6 Analytic approximate solutions for steady flow over a rotating disk in porous medium with heat transfer by homotopy analysis method [DOI]
  7. 7 Solutions of non-linear oscillators by the modified differential transform method [DOI]
  8. 8 An aftertreatment technique for improving the accuracy of Adomian's decomposition method [DOI]
  9. 9 Exact solutions of some coupled nonlinear partial differential equations using the homotopy perturbation method [DOI]
  10. 10 Approximate solution of multi-pantograph equation with variable coefficients [DOI]
  11. 11 A STUDY OF NON-NEWTONIAN FLOW AND HEAT TRANSFER OVER A NON-ISOTHERMAL WEDGE USING THE HOMOTOPY ANALYSIS METHOD [DOI]
  12. 12 An approximate analytic solution of the nonlinear Riccati differential equation [DOI]
  13. 13 Nonlinear vibration of an electrostatically actuated microbeam [DOI]
  14. 14 Properties of analytic solution and numerical solution of multi-pantograph equation [DOI]
  15. 15 Variational iteration method for solving a generalized pantograph equation [DOI]
  16. 16 The Pantograph Equation in the Complex Plane [DOI]
  17. 17 A new homotopy perturbation method for solving systems of partial differential equations [DOI]
  18. 18 A Spectral Method for Pantograph-Type Delay Differential Equations and Its Convergence Analysis [DOI]
  19. 19 On the numerical solution of the model for HIV infection of CD4+ T cells [DOI]
  20. 20 The homotopy perturbation method for solving neutral functional–differential equations with proportional delays [DOI]
  21. 21 On the attainable order of collocation methods for pantograph integro-differential equations [DOI]
  22. 22 The modified homotopy perturbation method for solving strongly nonlinear oscillators [DOI]
  23. 23 Solution of the differential algebraic equations via homotopy perturbation method and their engineering applications [DOI]
  24. 24 The modified algorithm for the differential transform method to solution of Genesio systems [DOI]
  25. 25 Runge–Kutta methods for the multi-pantograph delay equation [DOI]
  26. 26 Approximate Solutions of Generalized Pantograph Equations by the Differential Transform Method [DOI]
  27. 27 A reliable aftertreatment for improving the differential transformation method and its application to nonlinear oscillators with fractional nonlinearities [DOI]
  28. 28 Vibration of electrostatically actuated microbeam by means of homotopy perturbation method [DOI]
  29. 29 On Bounded Solutions of the Balanced Generalized Pantograph Equation [DOI]
  30. 30 A new approach to the solution of the prey and predator problem and comparison of the results with the Adomian method [DOI]
  31. 31 On the Numerical Solution of Generalized Pantograph Equation
  32. 32 Application of series method with Padé and Laplace-Padé resummation methods to solve a model for the evolution of smoking habit in Spain [DOI]
  33. 33 Laplace Decomposition Approximation Solution For A System Of Multi-Pantograph Equations [DOI]
  34. 34 APPLICATION OF HOMOTOPY PERTURBATION METHOD TO SOLVE MULTIDIMENSIONAL SCHRODINGERS EQUATIONS
  35. 35 An Enhanced Modified Homotopy Perturbation Method for Solving Nonlinear Volterra and Fredholm Integro-Differential Equation
  36. 36 Approximation Algorithm for a System of Pantograph Equations [DOI]
  37. 37 Power series approximation to solutions of nonlinear systems of differential equations [DOI]
  38. 38 Analytical solutions for systems of partial differential–algebraic equations [DOI]
  39. 39 An analytic study on the multi-pantograph delay equations with variable coefficients
  40. 40 A Novel Power Series Method for Solving Second Order Partial Differential Equations
  41. 41 Method of Successive Approximations for Solving the Multi-Pantograph Delay Equations
  42. 42 Solution Of Nonlinear Second-Order Pantograph Equations Via Differential Transformation Method [DOI]
  43. 43 Some nonlinear mechanical problems solved with analytical solutions
  44. 44 Exact solutions for differential-algebraic equations [DOI]
  45. 45 A Straightforward Approach To Solve Ordinary Nonlinear Differential Systems
  46. 46 Power Serıes Solutıon of Non-Lınear Fırst Order Dıfferentıal Equation Systems
  47. 47 Advanced engineering mathematics [DOI]

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