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A novel method to approximate fractional differential equations based on the theory of functional connections

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Abstract

In this paper, we propose a new method of using the theory of functional connections (TFC) to approximate the solution of fractional differential equations. For functions with one constraint at one point, several constraints at one point, distinct points, and relative constraints, the theoretical approach of the suggested method is investigated. The choice of the basis function is described, and the issue of using monomials is discussed. For the first time in the literature, the suggested method is used to solve fractional differential initial value problems, boundary value problems, and higher-order problems. Wherever the exact solution exists, the numerical results are compared. The numerical findings for the fractional order scenario are compared with the predictor-corrector method and polynomial least squares method. The error plot for the integer order case with the exact solution is also provided. The proposed approach is also used to solve a corneal shape model.

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Funding

The first author received the financial support of UGC NFOBC Ph.D. Fellowship (Ref. 202122-TN13000109).

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Authors and Affiliations

  1. Department of Mathematics, National Institute of Technology Puducherry, Karaikal, 609609, India

    Sivalingam S M & V. Govindaraj

  2. Institute for the Future of Knowledge, University of Johannesburg, PO Box 524, Auckland Park, Johannesburg, 2006, South Africa

    Pushpendra Kumar

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  1. Sivalingam S M
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  2. Pushpendra Kumar
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  3. V. Govindaraj
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Contributions

Sivalingam S. M.: conceptualization, visualization, software, resources, formal analysis, investigation, writing—-original draft. P. Kumar: investigation, formal analysis, resources, visualization, writing—-review & editing. V. Govindaraj: supervision, formal analysis, writing—-review & editing.

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Correspondence to Pushpendra Kumar.

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S M, S., Kumar, P. & Govindaraj, V. A novel method to approximate fractional differential equations based on the theory of functional connections. Numer Algor 95, 527–549 (2024). https://doi.org/10.1007/s11075-023-01580-3

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  • DOI: https://doi.org/10.1007/s11075-023-01580-3

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