An existence result with numerical solution of nonlinear fractional integral equations

Kazemi, Manochehr; Deep, Amar; Nieto, Juan J.

doi:10.1002/mma.9128

Cited by 9 publications

(16 citation statements)

References 39 publications

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“…For its numerical solution, they also suggested an iterative approach based on the midpoint rule. We shall demonstrate the falsity of Theorem 5.1 in [12] in this note, among others. We further show that, generally speaking, the recursive connection described in [12] is untrue since sentences in the recursion sequence cannot be calculated from previously unknown phrases; the method can only be used in a specific situation, and the midpoint rule is not good at all to find numerically a fixed point of equation ( 2) in fractional cases (0 < τ < 1); instead, using Jacobi's quadrature rule works well.…”

Section: Introductionmentioning

confidence: 63%

“…The goal of article [12] is to examine in order to determine fixed points of equation ( 2) and a subjected iterative method to solve numerically it if the conditions are considered appropriate for all continuous functions in (3) (refer to [12, conditions (1)-(3)]). Consequently, Kazemi et al [12] demonstrated the existence of a fixed point for T using Petryshyn's theorem. For its numerical solution, they also suggested an iterative approach based on the midpoint rule.…”

Section: Introductionmentioning

confidence: 99%

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Note on "An existence result with numerical solution of nonlinear fractional integral equations"

Golshan,

Khandani,

Ezzat

2024

Preprint

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The recent iterative numerical method in [Kazemi, Manochehr, Amar Deep, and Juan Nieto, "An existence result with numerical solution of nonlinear fractional integral equations," Mathematical Methods in the Applied Sciences ( 2023)] requires some corrections, as we point out in this note. We give a counterexample to one of their main statements (Theorem 5.1). The iterative method used is untrue, and we will correct it.With an example, it can be seen that using the midpoint rule will not be suitable and accurate in fractional cases (0 < τ < 1); instead, using Jacobi's quadrature rule can work well. Also, to show the validity of our numerical method, a non-linear example is considered.MSC: 60H20, 47H10.

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Section: Introductionmentioning

confidence: 63%

Section: Introductionmentioning

confidence: 99%

Note on "An existence result with numerical solution of nonlinear fractional integral equations"

Golshan,

Khandani,

Ezzat

2024

Preprint

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“…Notably, the research underscored the Newton-Raphson method's robustness, as it consistently converged efficiently with minimal iteration counts across various benchmark problems. This underscored the method's superiority in terms of efficiency and reliability, especially when dealing with complex nonlinear equations [21].…”

Section: Related Workmentioning

confidence: 99%

AI-Enhanced Performance Evaluation of Python, MATLAB, and Scilab for Solving Nonlinear Systems of Equations: A Comparative Study Using the Broyden Method

Azure,

Wiredu,

Musah

et al. 2023

AJCM

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This research extensively evaluates three leading mathematical software packages: Python, MATLAB, and Scilab, in the context of solving nonlinear systems of equations with five unknown variables. The study's core objectives include comparing software performance using standardized benchmarks, employing key performance metrics for quantitative assessment, and examining the influence of varying hardware specifications on software efficiency across HP ProBook, HP EliteBook, Dell Inspiron, and Dell Latitude laptops. Results from this investigation reveal insights into the capabilities of these software tools in diverse computing environments. On the HP ProBook, Python consistently outperforms MATLAB in terms of computational time. Python also exhibits a lower robustness index for problems 3 and 5 but matches or surpasses MATLAB for problem 1, for some initial guess values. In contrast, on the HP EliteBook, MATLAB consistently exhibits shorter computational times than Python across all benchmark problems. However, Python maintains a lower robustness index for most problems, except for problem 3, where MATLAB performs better. A notable challenge is Python's failure to converge for problem 4 with certain initial guess values, while MATLAB succeeds in producing results. Analysis on the Dell Inspiron reveals a split in strengths. Python demonstrates superior computational efficiency for some problems, while MATLAB excels in handling others. This pattern extends to the robustness index, with Python showing lower values for some problems, and MATLAB achieving the lowest indices for other problems. In conclusion, this research offers valuable insights into the comparative performance of Python, MATLAB, and Scilab in solving nonlinear systems of equations. It underscores the importance of considering both software and hardware specifica-How to cite this paper: Azure, I., Wiredu,

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“…Example 2.6. Kazemi et al [18] used the following conditions to check the fixed point existence solution of fractional integral equation z = T z, where…”

Section: Case Studymentioning

confidence: 99%

“…Thus, (II)-(III) hold and equation ( 13) has a fixed point solution in B ρ (E). Also, from (II) it is needed to add conditions [18] and compare with (4)).…”

Section: Case Studymentioning

confidence: 99%

An existence result for implicit functional equations

Golshan

2023

Preprint

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In this article, we attempt to provide a more general method based on Petryshyn’s fixed-point theorem to ensure the existence of solutions to implicit functional equations. These implicit functional equations include fractional, non-fractional, (fractional) stochastic integral equations, etc., and any combination of them in C( I). Some results regarding the existence of fixed points in implicit functional integral equations will be reviewed in the literature. We show that this general result unifies and improves many of the main results in the literature. To illustrate that our approach is more general than other methods, we present some concrete examples. Also, we apply our method to create new functional equations in practice and check the existence of solutions.

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An existence result with numerical solution of nonlinear fractional integral equations

Cited by 9 publications

References 39 publications

Note on "An existence result with numerical solution of nonlinear fractional integral equations"

Note on "An existence result with numerical solution of nonlinear fractional integral equations"

AI-Enhanced Performance Evaluation of Python, MATLAB, and Scilab for Solving Nonlinear Systems of Equations: A Comparative Study Using the Broyden Method

An existence result for implicit functional equations

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