Numerical solutions of fractional order integro-differential equations by using artificial neural networks and power series method

Mohd Noor; Musim Malik; Mohammad Sajid; Mohd Noor; Musim Malik; Mohammad Sajid

doi:10.3934/math.2026313

AIMS Mathematics

2026, Volume 11, Issue 3: 7610-7632. doi: 10.3934/math.2026313

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Numerical solutions of fractional order integro-differential equations by using artificial neural networks and power series method

1.
School of Mathematical & Statistical Sciences, Indian Institute of Technology, Mandi, 175005, Himachal Pradesh, India
2.
Department of Mechanical Engineering, College of Engineering, Qassim University, Saudi Arabia

Received: 24 January 2026 Revised: 14 March 2026 Accepted: 18 March 2026 Published: 23 March 2026
MSC : 68T07, 45D05, 45E10

This study presents a neural network-based framework for finding approximate series solutions to Abel's integral equation (AIE) and fractional order Volterra-type integro-differential equations (FOVIDEs). The suggested method changes the problem by writing the solution as a power series that has been truncated. This makes the original differential equation a problem of estimating parameters. Then, a special neural network is trained to find the coefficients of the series with good accuracy. Numerical tests show that the proposed method works, maintains accuracy, and converges. This shows that it can be used as a strong computational tool for solving fractional order systems.
- Abel's integral equation,
- artificial neural networks,
- fractional order Volterra integro-differential equations,
- power series
Citation: Mohd Noor, Musim Malik, Mohammad Sajid. Numerical solutions of fractional order integro-differential equations by using artificial neural networks and power series method[J]. AIMS Mathematics, 2026, 11(3): 7610-7632. doi: 10.3934/math.2026313

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Abstract

This study presents a neural network-based framework for finding approximate series solutions to Abel's integral equation (AIE) and fractional order Volterra-type integro-differential equations (FOVIDEs). The suggested method changes the problem by writing the solution as a power series that has been truncated. This makes the original differential equation a problem of estimating parameters. Then, a special neural network is trained to find the coefficients of the series with good accuracy. Numerical tests show that the proposed method works, maintains accuracy, and converges. This shows that it can be used as a strong computational tool for solving fractional order systems.

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