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A general form of the generalized Taylor’s formula with some applications

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  • El-Ajou, Ahmad
  • Abu Arqub, Omar
  • Al-Smadi, Mohammed

Abstract

In this article, a new general form of fractional power series is introduced in the sense of the Caputo fractional derivative. Using this approach some results of the classical power series are circulated and proved to this fractional power series, whilst a new general form of the generalized Taylor’s formula is also obtained. Some applications including fractional power series solutions for higher-order linear fractional differential equations subject to given nonhomogeneous initial conditions are provided and analyzed to guarantee and to confirm the performance of the proposed results. The results reveal that the new fractional expansion is very effective, straightforward, and powerful for formulating the exact solutions in the form of a rapidly convergent series with easily computable components.

Suggested Citation

  • El-Ajou, Ahmad & Abu Arqub, Omar & Al-Smadi, Mohammed, 2015. "A general form of the generalized Taylor’s formula with some applications," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 851-859.
    • Handle: RePEc:eee:apmaco:v:256:y:2015:i:c:p:851-859
    DOI: 10.1016/j.amc.2015.01.034
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    References listed on IDEAS

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    1. Tarasov, Vasily E. & Zaslavsky, George M., 2005. "Fractional Ginzburg–Landau equation for fractal media," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 354(C), pages 249-261.
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    Cited by:

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    2. Aylin Bayrak, Mine & Demir, Ali, 2018. "A new approach for space-time fractional partial differential equations by residual power series method," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 215-230.
    3. Ali, Khalid K. & Wazwaz, Abdul-Majid & Maneea, M., 2024. "Efficient solutions for fractional Tsunami shallow-water mathematical model: A comparative study via semi analytical techniques," Chaos, Solitons & Fractals, Elsevier, vol. 178(C).
    4. Mohammed Shqair & Ahmad El-Ajou & Mazen Nairat, 2019. "Analytical Solution for Multi-Energy Groups of Neutron Diffusion Equations by a Residual Power Series Method," Mathematics, MDPI, vol. 7(7), pages 1-20, July.
    5. Shadimetov, Kh.M. & Hayotov, A.R. & Nuraliev, F.A., 2016. "Optimal quadrature formulas of Euler–Maclaurin type," Applied Mathematics and Computation, Elsevier, vol. 276(C), pages 340-355.
    6. Jaradat, I. & Al-Dolat, M. & Al-Zoubi, K. & Alquran, M., 2018. "Theory and applications of a more general form for fractional power series expansion," Chaos, Solitons & Fractals, Elsevier, vol. 108(C), pages 107-110.
    7. Arqub, Omar Abu & Maayah, Banan, 2019. "Fitted fractional reproducing kernel algorithm for the numerical solutions of ABC – Fractional Volterra integro-differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 126(C), pages 394-402.

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