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A computational approximation for the solution of retarded functional differential equations and their applications to science and engineering

  • * Corresponding author: Burcu Gürbüz

    * Corresponding author: Burcu Gürbüz
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  • Delay differential equations are of great importance in science, engineering, medicine and biological models. These type of models include time delay phenomena which is helpful for characterising the real-world applications in machine learning, mechanics, economics, electrodynamics and so on. Besides, special classes of functional differential equations have been investigated in many researches. In this study, a numerical investigation of retarded type of these models together with initial conditions are introduced. The technique is based on a polynomial approach along with collocation points which maintains an approximated solutions to the problem. Besides, an error analysis of the approximate solutions is given. Accuracy of the method is shown by the results. Consequently, illustrative examples are considered and detailed analysis of the problem is acquired. Consequently, the future outlook is discussed in conclusion.

    Mathematics Subject Classification: 34K40, 33C45, 40C05, 65L60, 65G50.


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  • Figure 1.  The pantograph of a tram (İstiklal Street, İstanbul) [24]

    Figure 2.  A representation of a pantograph and its trolley wire system [13]

    Figure 3.  $ L_n(t) $ and $ H_n(t) $ values for $ n = 0, 1, 2, 3 $ and $ t\in[0, 2] $

    Figure 4.  Exact solution comparison with some approximate solutions for $ N = 6 $ and $ 10 $

    Figure 5.  Runge-Kutta method (RKM) solution for $ u_{1}(t) $ of Example 2

    Figure 6.  Laguerre collocation method (LCM) solution for $ u_{1}(t) $, $ N = 3 $ of Example 2

    Figure 7.  Runge-Kutta method (RKM) solution for $ u_{2}(t) $ of Example 2

    Figure 8.  Laguerre collocation method (LCM) solution for $ u_{2}(t) $, $ N = 3 $ of Example 2

    Figure 9.  Comparison between RKM and LCM solutions for $ N = 3, 4 $ and $ u_{1}(t) $ of Example 2

    Figure 10.  Comparison between RKM and LCM solutions for $ N = 3, 4 $ and $ u_{2}(t) $ of Example 2

    Table 1.  $ L_\infty $, $ L_2 $ and $ RMS $ errors for $ N = 3 $

    $ t $ $ L_2 $-Error $ L_\infty $-Error $ RMS $-Error
    1 0.7560E-05 0.5247E-04 0.1000E-06
    2 0.1164E-05 0.3791E-04 0.1502E-05
    3 0.1550E-04 0.5467E-03 0.6855E-04
    4 0.8259E-03 0.7795E-03 0.1752E-03
    5 0.4643E-04 0.5467E-02 0.2916E-05
     | Show Table
    DownLoad: CSV

    Table 2.  $ L_2 $ errors of LCM and HCM for $ N = 3 $ and $ N = 4 $ of Example 2

    $ t $ LCM, $ N=3 $ LCM, $ N=4 $ HCM, $ N=3 $ HCM, $ N=4 $
    0.0 0.6250E-02 0.1593E-03 0.7081E-02 0.2301E-03
    1.2 0.4202E-03 0.2490E-04 0.5223E-02 0.5100E-03
    2.3 0.2664E-03 0.2507E-04 0.5708E-02 0.6290E-04
    4.5 0.8259E-02 0.4510E-03 0.4430E-01 0.5291E-03
    5.0 0.5531E-02 0.7410E-03 0.3548E-01 0.8302E-03
     | Show Table
    DownLoad: CSV

    Table 3.  CPU comparisons of Example 2

    $ N $ LCM
    3 1.140
    4 1.258
     | Show Table
    DownLoad: CSV
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