Solvability of a nonlinear integro-differential equation with fractional order using the Bernoulli matrix approach

Raniyah E. Alsulaiman; Mohamed A. Abdou; Eslam M. Youssef; Mai Taha; Raniyah E. Alsulaiman; Mohamed A. Abdou; Eslam M. Youssef; Mai Taha

doi:10.3934/math.2023377

AIMS Mathematics

2023, Volume 8, Issue 3: 7515-7534. doi: 10.3934/math.2023377

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Solvability of a nonlinear integro-differential equation with fractional order using the Bernoulli matrix approach

1.
Department of Mathematics, College of Science, Jouf University, Sakaka 2014, Saudi Arabia
2.
Department of Mathematics, Faculty of Education, Alexandria University, Alexandria 21256, Egypt

Received: 01 November 2022 Revised: 12 December 2022 Accepted: 19 December 2022 Published: 16 January 2023
MSC : 26A33, 34A08, 34A12, 37C25, 45G15

Under some suitable conditions, we study the existence and uniqueness of a solution to a new modification of a nonlinear fractional integro-differential equation (NFIDEq) in dual Banach space C_E (E, [0, T]), which simulates several phenomena in mathematical physics, quantum mechanics, and other domains. The desired conclusions are demonstrated with the use of fixed-point theorems after applying the theory of fractional calculus. The validation of the provided strategy has been done by utilizing the Bernoulli matrix approach (BMA) method as a numerical method. The major motivation for selecting the BMA approach is that it combines Bernoulli polynomial approximation with Caputo fractional derivatives and numerical integral transformation to reduce the NFIDEq to an algebraic system and then derive the numerical solution; additionally, the convergence analysis indicated that the proposed strategy has more precision than other numerical methods. Finally, as a verification of the theoretical work, we apply two examples with numerical results by using [Matlab R2022b], illustrating the comparisons between the exact solutions and numerical solutions, as well as the absolute error in each case is computed.
- fractional integro-differential equations,
- Caputo fractional derivative,
- fixed point theorem,
- Gauss quadrature formula,
- Bernoulli matrix approach
Citation: Raniyah E. Alsulaiman, Mohamed A. Abdou, Eslam M. Youssef, Mai Taha. Solvability of a nonlinear integro-differential equation with fractional order using the Bernoulli matrix approach[J]. AIMS Mathematics, 2023, 8(3): 7515-7534. doi: 10.3934/math.2023377

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Abstract

Under some suitable conditions, we study the existence and uniqueness of a solution to a new modification of a nonlinear fractional integro-differential equation (NFIDEq) in dual Banach space C_E (E, [0, T]), which simulates several phenomena in mathematical physics, quantum mechanics, and other domains. The desired conclusions are demonstrated with the use of fixed-point theorems after applying the theory of fractional calculus. The validation of the provided strategy has been done by utilizing the Bernoulli matrix approach (BMA) method as a numerical method. The major motivation for selecting the BMA approach is that it combines Bernoulli polynomial approximation with Caputo fractional derivatives and numerical integral transformation to reduce the NFIDEq to an algebraic system and then derive the numerical solution; additionally, the convergence analysis indicated that the proposed strategy has more precision than other numerical methods. Finally, as a verification of the theoretical work, we apply two examples with numerical results by using [Matlab R2022b], illustrating the comparisons between the exact solutions and numerical solutions, as well as the absolute error in each case is computed.

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