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A numerical approximation for generalized fractional Sturm–Liouville problem with application

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  • Goel, Eti
  • Pandey, Rajesh K.
  • Yadav, S.
  • Agrawal, Om P.

Abstract

In this paper, we present a numerical scheme for the generalized fractional Sturm–Liouville problem (GFSLP) with mixed boundary conditions. The GFSLP is defined in terms of the B-operator consisting of an integral operator with a kernel and a differential operator. One of the main features of the B-operator is that for different kernels, it leads to different Sturm–Liouville Problems (SLPs), and thus the same formulation can be used to discuss different SLPs. We prove the well-posedness of the proposed GFSLP. Further, the approximated eigenvalues of the GFSLP are obtained for two different kernels namely a modified power kernel and the Prabhakar kernel in the B-operator. We obtain real eigenvalues and corresponding orthogonal eigenfunctions. Theoretical and numerical convergence orders of eigenvalues and eigenvectors are also discussed. Further, the numerically obtained eigenvalues and eigenfunctions are used to construct an approximate solution of the one-dimensional fractional diffusion equation defined in a bounded domain.

Suggested Citation

  • Goel, Eti & Pandey, Rajesh K. & Yadav, S. & Agrawal, Om P., 2023. "A numerical approximation for generalized fractional Sturm–Liouville problem with application," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 207(C), pages 417-436.
    • Handle: RePEc:eee:matcom:v:207:y:2023:i:c:p:417-436
    DOI: 10.1016/j.matcom.2023.01.003
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    References listed on IDEAS

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    1. Kashfi Sadabad, Mahnaz & Jodayree Akbarfam, Aliasghar, 2021. "An efficient numerical method for estimating eigenvalues and eigenfunctions of fractional Sturm–Liouville problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 185(C), pages 547-569.
    2. Al-Mdallal, Qasem M., 2009. "An efficient method for solving fractional Sturm–Liouville problems," Chaos, Solitons & Fractals, Elsevier, vol. 40(1), pages 183-189.
    3. Metzler, Ralf & Klafter, Joseph, 2000. "Boundary value problems for fractional diffusion equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 278(1), pages 107-125.
    4. Tatiana Odzijewicz & Agnieszka B. Malinowska & Delfim F. M. Torres, 2012. "Fractional Calculus of Variations in Terms of a Generalized Fractional Integral with Applications to Physics," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-24, May.
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