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An efficient algorithm based on Gegenbauer wavelets for the solutions of variable-order fractional differential equations

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Abstract.

The article is devoted to a new computational algorithm based on the Gegenbauer wavelets (GWs) to solve the linear and nonlinear variable-order fractional differential equations. The novel operational matrices for derivatives of positive integer and variable order are derived. New piecewise functions are introduced to obtain the said operational matrices. In the proposed method, the given problem via Gegenbauer wavelets is transformed to a system of algebraic equations. The obtained solutions are endorsing the accuracy and efficiency of the suggested method and are in excellent agreement with the existing literature. The convergence and error bound analysis are presented in our study to show the credibility of the computational method and support the mathematical formulation of the algorithm. The discussed problems reconfirm the appropriateness of said algorithm and perceived that the proposed algorithm is an efficient tool to tackle the nonlinear fractional order problems of complex nature.

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Authors and Affiliations

  1. School of Mathematical Sciences, Peking University, 100871, Beijing, China

    Muhammad Usman, Muhammad Hamid & Wei Wang

  2. Department of Electrical Engineering, Bahria University, Islamabad Campus, Islamabad, Pakistan

    Rizwan Ul Haq

Authors
  1. Muhammad Usman
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  2. Muhammad Hamid
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  3. Rizwan Ul Haq
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  4. Wei Wang
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Correspondence to Rizwan Ul Haq.

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Usman, M., Hamid, M., Ul Haq, R. et al. An efficient algorithm based on Gegenbauer wavelets for the solutions of variable-order fractional differential equations. Eur. Phys. J. Plus 133, 327 (2018). https://doi.org/10.1140/epjp/i2018-12172-1

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  • DOI: https://doi.org/10.1140/epjp/i2018-12172-1

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