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Qualitative, Approximate and Numerical Approaches for the Solution of Nonlinear Differential Equations

In: Applications of Nonlinear Analysis

Author

Listed:
  • Eugenia N. Petropoulou

    (University of Patras)

  • Michail A. Xenos

    (University of Ioannina)

Abstract

The differential equations that describe many realistic problems are nonlinear and most of these cannot be solved explicitly using standard analytic techniques. In such cases, qualitative, approximate or numerical techniques are employed, in order to obtain as much information as possible. The aim of the present chapter, is to give a description of the general ideas governing these techniques together with their advantages and limitations. This is achieved by implementing various methods to an initial value problem for a specific nonlinear ordinary differential equation, which combines both van der Pol and Duffing equations. This equation is solved using (a) the fourth order Runge-Kutta, the standard finite differences and the finite elements methods, (b) a nonstandard discretization technique based on functional analysis, (c) classical perturbation techniques and (d) the homotopy analysis method. Moreover, various results are given regarding the dynamic properties of its solution. Finally, this problem is connected with a Green function and this connection is again used for its numerical solution.

Suggested Citation

  • Eugenia N. Petropoulou & Michail A. Xenos, 2018. "Qualitative, Approximate and Numerical Approaches for the Solution of Nonlinear Differential Equations," Springer Optimization and Its Applications, in: Themistocles M. Rassias (ed.), Applications of Nonlinear Analysis, pages 611-664, Springer.
    • Handle: RePEc:spr:spochp:978-3-319-89815-5_22
    DOI: 10.1007/978-3-319-89815-5_22
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