## Abstract

The fractional-order Bagley–Torvik equation has many applications in the field of life science and engineering. In this paper, we develop a new scheme based on the existing finite element method for the numerical solution of the Bagley–Torvik equation of order (0, 2). We adopt the formulation of the equation in a simple and generalized way. The existence and uniqueness of the solution and its error estimations are derived based on the technique we derived. A series of numerical examples are provided to demonstrate the accuracy, efficiency, and simplicity of the method. The results are depicted graphically and in a table to compare the exact and approximate solutions obtained by following the numerical methods available in the literature. The numerical experiment shows that using a small number of quadratic functions, the accuracy of our numerical technique is better than the existing methods. Since the Bagley–Torvik equation represents the general form of fractional-order boundary value problems, the numerical technique indicates the identical path to solve the similar type of the fractional-order boundary value problems.

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## Acknowledgements

The authors are grateful to the reviewers for their constructive suggestions to develop the quality of the manuscript significantly. The author M. Kamrujjaman research was partially supported by the University Grant Commission (UGC), for year 2019-2020, Bangladesh.

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Ali, H., Kamrujjaman, M. & Shirin, A. Numerical solution of a fractional-order Bagley–Torvik equation by quadratic finite element method.
*J. Appl. Math. Comput.* **66**, 351–367 (2021). https://doi.org/10.1007/s12190-020-01440-6

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DOI: https://doi.org/10.1007/s12190-020-01440-6

### Keywords

- Bagley–Torvik equation
- Fractional-order BVPs
- Finite element method
- Quadratic shape function
- Error estimation