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Comparative analysis of Riemann-Liouville, Caputo-Fabrizio, and Atangana-Baleanu integrals

Shaikh Asif Ali, Qureshi Sania

Journal of Applied Mathematics and Computational Mechanics

2022

Vol. 21, nr 1

91--101

This study analyzes the most commonly used operators of the Riemann-Liouville, the Caputo-Fabrizio, and the Atangana-Baleanu integral operators. Firstly, a numerical scheme for the Riemann-Liouville fractional integral has been discussed. Then, two numerical techniques have been suggested for the remaining two operators. The experimental order of convergence for the schemes is further supported by the computations of absolute relative error at the final nodal point over the integration interval [0, T ]. Comparative analysis of the integrals reveals that the Riemann-Liouville fractional integral yields the most minor errors and the most significant experimental order of convergence in the majority of functions, particularly when the fractional-order parameter α → 0. It is worth noting that the Atangana-Baleanu has proved to be an essential operator for solving many dynamical systems that a single RL operator cannot handle. All of the three integral operators coincide with each other for α = 1. Mathematica 11.3 for an Intel(R) Core(TM) i3-4500U procesor running on 1.70 GHz is used to carry out all the necessary computations.

A computational method for time fractional partial integro-differential equations

Mojahedfar M., Tari Abolfazl, Shahmorad S.

Journal of Applied Analysis

2020

Vol. 26, nr 2

315--323

In this paper, a class of time fractional partial integro-differential equations (FPIDEs) with initial conditions is studied. Some operational matrices are used to reduce a FPIDE problem to a system of algebraic equations with special properties. The resulted system is solved to give an approximate solution to the problem. Error estimation is also discussed for the approximate solution. Finally, some numerical examples are given to show the accuracy of the proposed method.

Characterizations of compact operators on ℓp-type fractional sets of sequences

Özger Faruk

Demonstratio Mathematica

2019

Vol. 52, nr 1

105--115

Among the sets of sequences studied, difference sets of sequences are probably the most common type of sets. This paper considers some ℓp-type fractional difference sets via the gamma function. Although, we characterize compactness conditions on those sets using the main key of Hausdorff measure of noncompactness, we can only obtain sufficient conditions when the final space is ℓ∞. However, we use some recent results to exactly characterize the classes of compact matrix operators when the final space is the set of bounded sequences.