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Numerical solutions of the Bagley–Torvik equation by using generalized functions with fractional powers of Laguerre polynomials

  • Şuayip Yüzbaşı EMAIL logo and Gamze Yıldırım
Published/Copyright: May 3, 2022
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International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 24 Issue 3

Abstract

In this study, a collocation approach is presented to solve Bagley–Torvik equation, which is a class of fractional differential equations. As most fractional differential equations do not have exact analytical solutions, it is needed numerical methods. This study is important because it presents a numerical method for fractional differential equations. The main purpose of this method is to obtain the approximate solution based on Laguerre polynomials of the Bagley–Torvik equation. To date, a collocation method based on the Laguerre polynomials has not been studied for the solutions of the Bagley–Torvik equation. This reveals the novelty of the study. The approximate solution is sought in form of the fractional powers of the Laguerre polynomials. By using the Caputo derivative, the matrix relation is created for term with fractional derivative in the equation. Similarly, the matrix relation of second derivative is computed in equation. Then, by using these matrix relations and the collocation points, the Bagley–Torvik problem is converted into a system of the linear algebraic equations. The solution of this system gives the coefficients of the assumed solution. Secondly, an error estimation method is given with the help of the residual function and also the Laguerre polynomial solution is improved by using the estimated error function. Then, the method is applied to four examples and the obtained numerical results are shown in tables and graphs. Also, the comparisons are made with other methods in the literature and so the presented method gives better results than other methods.

Keywords: Bagley–Torvik equation; Caputo fractional derivative; collocation method; fractional differential equations; initial-boundary conditions; Laguerre polynomials
AMS: 34B05; 34K37; 65G99; 65L60; 65L80
Corresponding author: Şuayip Yüzbaşı, Department of Mathematics, Faculty of Science, Akdeniz University, TR 07058, Antalya, Türkiye, E-mail:

Acknowledgements

Authors would like to thank the reviewers for their constructive comments and suggestions to improve the paper.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

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Received: 2021-03-20
Accepted: 2022-04-07
Published Online: 2022-05-03

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. One-dimensional optimal system and similarity transformations for the 3 + 1 Kudryashov–Sinelshchikov equation
  4. Hopf bifurcations in a network of FitzHugh–Nagumo biological neurons
  5. Gravity modulation effect on weakly nonlinear thermal convection in a fluid layer bounded by rigid boundaries
  6. A new numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ equation
  7. Chebyshev spectral method for solving a class of local and nonlocal elliptic boundary value problems
  8. Adaptive extragradient methods for solving variational inequalities in real Hilbert spaces
  9. Elastic trend filtering
  10. A new approach to representations of homothetic motions in Lorentz space
  11. Numerical simulation of variable-order fractional differential equation of nonlinear Lane–Emden type appearing in astrophysics
  12. Stability analysis and numerical simulations of the fractional COVID-19 pandemic model
  13. Numerical solutions of the Bagley–Torvik equation by using generalized functions with fractional powers of Laguerre polynomials
  14. N-fold Darboux transformation and exact solutions for the nonlocal Fokas–Lenells equation on the vanishing and plane wave backgrounds
  15. Closed form soliton solutions to the space-time fractional foam drainage equation and coupled mKdV evolution equations
  16. Master-slave synchronization in the Duffing-van der Pol and Φ6 Duffing oscillators
  17. Singularity analysis and analytic solutions for the Benney–Gjevik equations
  18. Trajectory controllability of nonlinear fractional Langevin systems
  19. Similarity transformations for modified shallow water equations with density dependence on the average temperature
  20. Non-equidistant partition predictor–corrector method for fractional differential equations with exponential memory
  21. Dynamical analysis of fractional-order of IVGTT glucose–insulin interaction
  22. Dynamic response of Mathieu–Duffing oscillator with Caputo derivative
  23. Non-Newtonian fluid flow having fluid–particle interaction through a porous zone in a channel with permeable walls
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https://doi.org/10.1515/ijnsns-2021-0120
Keywords for this article
Bagley–Torvik equation; Caputo fractional derivative; collocation method; fractional differential equations; initial-boundary conditions; Laguerre polynomials

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. One-dimensional optimal system and similarity transformations for the 3 + 1 Kudryashov–Sinelshchikov equation
  4. Hopf bifurcations in a network of FitzHugh–Nagumo biological neurons
  5. Gravity modulation effect on weakly nonlinear thermal convection in a fluid layer bounded by rigid boundaries
  6. A new numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ equation
  7. Chebyshev spectral method for solving a class of local and nonlocal elliptic boundary value problems
  8. Adaptive extragradient methods for solving variational inequalities in real Hilbert spaces
  9. Elastic trend filtering
  10. A new approach to representations of homothetic motions in Lorentz space
  11. Numerical simulation of variable-order fractional differential equation of nonlinear Lane–Emden type appearing in astrophysics
  12. Stability analysis and numerical simulations of the fractional COVID-19 pandemic model
  13. Numerical solutions of the Bagley–Torvik equation by using generalized functions with fractional powers of Laguerre polynomials
  14. N-fold Darboux transformation and exact solutions for the nonlocal Fokas–Lenells equation on the vanishing and plane wave backgrounds
  15. Closed form soliton solutions to the space-time fractional foam drainage equation and coupled mKdV evolution equations
  16. Master-slave synchronization in the Duffing-van der Pol and Φ6 Duffing oscillators
  17. Singularity analysis and analytic solutions for the Benney–Gjevik equations
  18. Trajectory controllability of nonlinear fractional Langevin systems
  19. Similarity transformations for modified shallow water equations with density dependence on the average temperature
  20. Non-equidistant partition predictor–corrector method for fractional differential equations with exponential memory
  21. Dynamical analysis of fractional-order of IVGTT glucose–insulin interaction
  22. Dynamic response of Mathieu–Duffing oscillator with Caputo derivative
  23. Non-Newtonian fluid flow having fluid–particle interaction through a porous zone in a channel with permeable walls
  24. A class of computationally efficient Newton-like methods with frozen inverse operator for nonlinear systems
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