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A Lagrange spectral collocation method for weakly singular fuzzy fractional Volterra integro-differential equations

  • Fuzzy systems and their mathematics
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Abstract

A linear fractional-order weakly singular fuzzy Volterra integro-differential equation has been examined. In this case, the Caputo fractional-order derivative has been considered. A new type of spectral collocation method based on the Lagrange interpolation basis polynomial has been studied and modified for the equation. In the spectral collocation technique, it is necessary to choose collocation points to find the numerical solution of the equation. We have chosen the collocation points based on the Chebyshev extreme points or Gauss–Lobatto–Chebyshev points of order N. We have used the fractional Gauss–Jacobi quadrature method to approximate the fractional integral terms of the proposed equation. Also, the integral operators have been approximated by the Gauss quadrature rule. A theorem has been given to demonstrate that there exists a unique solution for the proposed equation. In addition, Banach’s fixed point principle has been applied in the proof of the existence and uniqueness theorem. The convergence analysis of the proposed numerical technique is given in the form of some lemmas and theorems. Some numerical experiments have been performed to verify the proposed method. Five different kinds of errors have been computed and compared to do the error analysis. Also, these kinds of error analysis have been examined by analyzing the result in the form of different graphs and tables. The numerical results of the proposed technique have been compared with an existing method, Adomian decomposition method (ADM).

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Funding

For the financial assistance of this research work, Sandip Moi thanks the Council of Scientific and Industrial Research (CSIR), Government of India (File No.- 08/003(0135)/2019-EMR-I).

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, B. Garden, Howrah, 711103, India

    Sandip Moi & Smita Pal Sarkar

  2. Discipline of Mathematics, School of Sciences, Indira Gandhi National Open University, New Delhi, 110068, India

    Suvankar Biswas

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  1. Sandip Moi
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  2. Suvankar Biswas
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Correspondence to Suvankar Biswas.

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Moi, S., Biswas, S. & Sarkar, S.P. A Lagrange spectral collocation method for weakly singular fuzzy fractional Volterra integro-differential equations. Soft Comput 27, 4483–4499 (2023). https://doi.org/10.1007/s00500-023-07829-2

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