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The Green Function for a Class of Caputo Fractional Differential Equations with a Convection Term

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Abstract

A class of two-point boundary value problem whose highest-order term is a Caputo fractional derivative of order α ∈ (1, 2] with a convection term is considered. Its boundary conditions are of Robin type including the Dirichlet boundary conditions as a special case. An explicit formula for the associated Green function is obtained in terms of two-parameter Mittag-Leffler functions. This work improves Meng and Stynes’ work [9] in three aspects. Firstly, the Green function is constructed by the use of the Laplace transform. This method may be more straightforward and easy to be generalized to solving other problems. Secondly, the monotonicity of a function is proven using monotone definition rather than using its derivative. Thirdly, a direct proof of the necessity of the positive property of Green function is given so that we can avoid the difficulty of construction any counter-example.

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Authors and Affiliations

  1. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, PR China

    Zhanbing Bai & Sujing Sun

  2. School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, 221116, PR China

    Zengji Du

  3. School of Engineering, University of California, Merced, CA, 95343, USA

    YangQuan Chen

Authors
  1. Zhanbing Bai
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  2. Sujing Sun
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  3. Zengji Du
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  4. YangQuan Chen
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Correspondence to Zhanbing Bai.

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Bai, Z., Sun, S., Du, Z. et al. The Green Function for a Class of Caputo Fractional Differential Equations with a Convection Term. Fract Calc Appl Anal 23, 787–798 (2020). https://doi.org/10.1515/fca-2020-0039

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  • DOI: https://doi.org/10.1515/fca-2020-0039

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