Abstract
Fractional calculus has become a basic tool for modeling phenomena involving memory. However, due to the non-local nature of fractional derivatives, the computations involved in solving a fractional differential equations (FDEs) are tedious and time consuming. Develo** numerical and analytical methods for solving nonlinear FDEs has been a subject of intense research at present. In the present article, we review some of the existing numerical methods for solving FDEs and some new methods developed by our group recently. We also perform their comparative study.
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References
Oldham, K., Spanier, J.: The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order. Elsevier, Amsterdam (1974)
Daftardar-Gejji, V.: An introduction to fractional calculus. In: Daftardar-Gejji, V. (ed.) Fractional Calculus: Theory and Applications. Narosa, Delhi (2014)
Yang, Q., Chen, D., Zhao, T., Chen, Y.: Fractional calculus in image processing: a review. Fract. Calc. Appl. Anal. 19(5), 1222–1249 (2016)
Samko, S.G., Kilbas, A.A., Mariuchev, O.I.: Fractional Integrals and Derivatives. Taylor and Francis, Milton Park (1993)
Daftardar-Gejji, V., Babakhani, A.: Analysis of a system of fractional differential equations. J. Math. Anal. Appl. 293(2), 511–522 (2004)
Daftardar-Gejji, V., Jafari, H.: Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives. J. Math. Anal. Appl. 328, 1026–1033 (2007)
Adomian, G.: Solving Frontier Problems in Physics: The Decomposition Method. Kluwer Academic, Boston (1994)
Daftardar-Gejji, V., Jafari, H.: An iterative method for solving nonlinear functional equations differential equations. J. Math. Anal. Appl. 316(2), 321–354 (2006)
Daftardar-Gejji, V., Sukale, Y., Bhalekar, S.: A new predictor-corrector method for fractional differential equations. Appl. Math. Comput. 244, 158–182 (2014)
Jhinga, A., Daftardar-Gejji, V.: A new finite difference predictor-corrector method for fractional differential equations. Appl. Math. Comput. 336, 418–432 (2018)
Bhalekar, S., Daftardar-Gejji, V.: Convergence of the new iterative method. Int. J. Differ. Equ. (2011)
Diethelm, K., Ford, N., Freed, A.: Detailed error analysis for a fractional Adams method. Numer. Algorithms 36(1), 31–52 (2004)
Sukale, Y.: PhD thesis, Savitribai Phule Pune University (2016)
Li, C., Zheng, F.: Numerical methods for Fractional Calculus. CRC Press, New York (2015)
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Daftardar-Gejji, V. (2019). Numerics of Fractional Differential Equations. In: Daftardar-Gejji, V. (eds) Fractional Calculus and Fractional Differential Equations. Trends in Mathematics. Birkhäuser, Singapore. https://doi.org/10.1007/978-981-13-9227-6_1
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DOI: https://doi.org/10.1007/978-981-13-9227-6_1
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