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Dergiler / Hacettepe Journal of Mathematics and Statistics / 2021 / Cilt: 50 - Sayı: 1

Numerical investigation of dynamic Euler-Bernoulli equation via 3-Scale Haar wavelet collocation method

Ömer Oruç
Alaattin Esen
Fatih Bulut
Sayfa
159–179
Yayın Tarihi
2022-10-18
DOI
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⚡ TL;DR(1 Cümlede Hızlı Çıkarım)

In the applications of the proposed method, we achieve high accuracy even with small number of collocation points.

📌 3 Maddede Bu Makale

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Amaç & Hipotez:In this study, we analyze the performance of a numerical scheme based on 3-scale Haar wavelets for dynamic Euler-Bernoulli equation, which is a fourth order time dependent partial differential equation.
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Temel Bulgular & Veri:For its solution, we first rewrite the fourth order time dependent partial differential equation as a system of partial differential equations by introducing a new variable, and then use finite difference approximations to discretize in time, as well as 3-scale Haar wavelets to discretize in space.
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Sonuç & Katkı:In the applications of the proposed method, we achieve high accuracy even with small number of collocation points.
🔬Tür:Araştırma Makalesi
📍Kapsam:Genişletilmiş Akademik Örneklem
⚙️Değişkenler:3-Scale Haar wavelets · Vibrating beam · Dynamic Euler-Bernoulli equation

Abstract

In this study, we analyze the performance of a numerical scheme based on 3-scale Haar wavelets for dynamic Euler-Bernoulli equation, which is a fourth order time dependent partial differential equation. This type of equations governs the behaviour of a vibrating beam and have many applications in elasticity. For its solution, we first rewrite the fourth order time dependent partial differential equation as a system of partial differential equations by introducing a new variable, and then use finite difference approximations to discretize in time, as well as 3-scale Haar wavelets to discretize in space. By doing so, we obtain a system of algebraic equations whose solution gives wavelet coefficients for constructing the numerical solution of the partial differential equation. To test the accuracy and reliability of the numerical scheme based on 3-scale Haar wavelets, we apply it to five test problems including variable and constant coefficient, as well as homogeneous and non-homogeneous partial differential equations. The obtained results are compared wherever possible with those from previous studies. Numerical results are tabulated and depicted graphically. In the applications of the proposed method, we achieve high accuracy even with small number of collocation points.

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Bu Makaleye Atıf Yap

Oruç, Ö., Esen, A., Bulut, F. (2022). Numerical investigation of dynamic Euler-Bernoulli equation via 3-Scale Haar wavelet collocation method. Hacettepe Journal of Mathematics and Statistics, Cilt: 50 - Sayı: 1, 159–179. https://ajindex.com/articles/529536

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Kaynakça (17)

  1. C. Andrade and S. McKee, High accuracy A.D.I. methods for fourth-order parabolic equations with variable coefficients, J. Comput. Appl. Math. 3 (1), 11–14, 1977
  2. T. Aziz, A. Khan and J. Rashidinia, Spline methods for the solution of fourth-order parabolic partial differential equations, Appl. Math. Comput. 167, 153–166, 2005
  3. H.T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Sys- tems, Birkhauser, Boston, 1989
  4. H. Caglar and N. Caglar, Fifth-degree B-spline solution for a fourth-order parabolic partial differential equations, Appl. Math. Comput. 201, 597–603, 2008
  5. C. Chen and C.H. Hsiao, Haar wavelet method for solving lumped and distributed parameter systems, IEE Proc. Control Theory Appl. 144, 87–94, 1997
  6. C. Chen and C.H. Hsiao, Wavelet approach to optimising dynamic systems, IEE Proc. Control Theory Appl. 146, 213–219, 1997
  7. L. Collatz, Hermitian methods for initial value problems in partial differential equa- tions, in: J.J.H. Miller (Ed.), Topics in Numerical Analysis, Academic Press, New York, 41–61, 1973
  8. S.D. Conte, A stable implicit finite difference approximation to a fourth order parabolic equation, J. Assoc. Comput. Mech. 4, 18–23, 1957
  9. S.H. Crandall, Numerical treatment of a fourth order partial differential equations, J. Assoc. Comput. Mech. 1, 111–118, 1954
  10. A. Danaee, Arshad Khan, Islam Khan, Tariq Aziz and D.J. Evans, Hopscotch proce- dure for a fourth-order parabolic partial differential equation, Math. Comput. Simulat. XXIV, 326–329, 1982
  11. D.J. Evans, A stable explicit method for the finite difference solution of a fourth order parabolic partial differential equation, Comput. J. 8, 280–287, 1965
  12. D.J. Evans and W.S.Yousif, A note on solving the fourth order parabolic equation by the age method, Int. J. Comput. Math. 40, 93–97, 1991
  13. G. Fairweather and A.R. Gourlay, Some stable difference approximations to a fourth order parabolic partial differential equation, Math. Comput. 21, 1–11, 1967
  14. H. Haddadpour, An exact solution for variable coefficients fourth-order wave equation using the Adomian method, Math. Comput. Model. 44, 144–1152, 2006
  15. C.H. Hsiao, Haar wavelet direct method for solving variational problems, Math. Com- put. Simul. 64, 569–585, 2004
  16. C.H. Hsiao and W.J. Wang, State analysis of time-varying singular nonlinear systems via Haar wavelets Math. Comput. Simul. 51, 91–100, 1999
  17. C.H. Hsiao and W.J. Wang, State analysis of t