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Journal articles on the topic 'Improper fractional integrals'
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1
Chii-Huei, Yu. "Application of Reciprocal Substitution Method in Solving Some Improper Fractional Integrals." International Journal of Mathematics and Physical Sciences Research 11, no. 1 (2023): 1–5. https://doi.org/10.5281/zenodo.7883131.
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<strong>Abstract:</strong> In this paper, based on Jumarie type of Riemann-Liouville (R-L) fractional calculus, we use reciprocal substitution method to find two improper fractional integrals. Change of variables for fractional integral and a new multiplication of fractional analytic functions play important roles in this paper. In fact, our results are generalizations of traditional calculus results. <strong>Keywords:</strong> Jumarie type of R-L fractional calculus, reciprocal substitution method, improper fractional integrals, change of variables for fractional integral, new multiplication, fractional analytic functions. <strong>Title:</strong> Application of Reciprocal Substitution Method in Solving Some Improper Fractional Integrals <strong>Author:</strong> Chii-Huei Yu <strong>International Journal of Mathematics and Physical Sciences Research </strong> <strong>ISSN 2348-5736 (Online)</strong> <strong>Vol. 11, Issue 1, April 2023 - September 2023</strong> <strong>Page No: 1-5</strong> <strong>Research Publish Journals</strong> <strong>Website: www.researchpublish.com</strong> <strong>Published Date: 01-May-2023</strong> <strong>DOI: https://doi.org/10.5281/zenodo.7883131</strong> <strong>Paper Download Link (Source)</strong> <strong>https://www.researchpublish.com/papers/application-of-reciprocal-substitution-method-in-solving-some-improper-fractional-integrals</strong>
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Chii-Huei, Yu. "Using Interval Recurrence Formula to Solve Two Improper Fractional Integrals." International Journal of Mathematics and Physical Sciences Research 11, no. 1 (2023): 53–58. https://doi.org/10.5281/zenodo.8076853.
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<strong>Abstract:</strong> In this paper, based on Jumarie type of Riemann-Liouville (R-L) fractional calculus and a new multiplication of fractional analytic functions, we use interval recurrence formula to solve two improper fractional integrals. In fact, our result is a generalization of classical calculus result. <strong>Keywords:</strong> Jumarie type of R-L fractional calculus, new multiplication, fractional analytic functions, interval recurrence formula, improper fractional integrals. <strong>Title:</strong> Using Interval Recurrence Formula to Solve Two Improper Fractional Integrals <strong>Author:</strong> Chii-Huei Yu <strong>International Journal of Mathematics and Physical Sciences Research </strong> <strong>ISSN 2348-5736 (Online)</strong> <strong>Vol. 11, Issue 1, April 2023 - September 2023</strong> <strong>Page No: 53-58</strong> <strong>Research Publish Journals</strong> <strong>Website: www.researchpublish.com</strong> <strong>Published Date: 24-June-2023</strong> <strong>DOI: https://doi.org/10.5281/zenodo.8076853</strong> <strong>Paper Download Link (Source)</strong> <strong>https://www.researchpublish.com/papers/using-interval-recurrence-formula-to-solve-two-improper-fractional-integrals</strong>
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Chii-Huei, Yu. "Application of Differentiation under Fractional Integral Sign." International Journal of Mathematics and Physical Sciences Research 10, no. 2 (2022): 40–46. https://doi.org/10.5281/zenodo.7486538.
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<strong>Abstract:</strong> In this paper, based on Jumarie type of Riemann-Liouville (R-L) fractional calculus, we use differentiation under fractional integral sign to evaluate two improper fractional integrals. Integration by parts for fractional calculus and a new multiplication of fractional analytic functions play important roles in this article. In fact, our results are generalizations of the results in ordinary calculus. <strong>Keywords:</strong> Jumarie type of R-L fractional calculus, differentiation under fractional integral sign, improper fractional integrals, integration by parts, new multiplication, fractional analytic functions. <strong>Title:</strong> Application of Differentiation under Fractional Integral Sign <strong>Author:</strong> Chii-Huei Yu <strong>International Journal of Mathematics and Physical Sciences Research </strong> <strong>ISSN 2348-5736 (Online)</strong> <strong>Vol. 10, Issue 2, October 2022 - March 2023</strong> <strong>Page No: 40-46</strong> <strong>Research Publish Journals</strong> <strong>Website: www.researchpublish.com</strong> <strong>Published Date: 27-December-2022</strong> <strong>DOI: https://doi.org/10.5281/zenodo.7486538</strong> <strong>Paper Download Link (Source)</strong> <strong>https://www.researchpublish.com/papers/application-of-differentiation-under-fractional-integral-sign</strong>
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Kalam, B., and G. Vainikko. "About the convergence type of improper integrals defining fractional derivatives." Acta et Commentationes Universitatis Tartuensis de Mathematica 23, no. 1 (2019): 95–102. http://dx.doi.org/10.12697/acutm.2019.23.10.
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This article continues the analysis of the class of fractionally differentiable functions. We complete the main result of [4] that characterises the class of fractionally differentiable functions in terms of the pointwise convergence of certain improper integrals containing these functions. Our aim is to present an example, which shows that in order to obtain all fractionally differentiable functions, one may not replace the conditional convergence of those integrals by their absolute convergence.
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Srivastava, Rekha, Ritu Agarwal, and Sonal Jain. "A family of the incomplete hypergeometric functions and associated integral transform and fractional derivative formulas." Filomat 31, no. 1 (2017): 125–40. http://dx.doi.org/10.2298/fil1701125s.
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Recently, Srivastava et al. [Integral Transforms Spec. Funct. 23 (2012), 659-683] introduced the incomplete Pochhammer symbols that led to a natural generalization and decomposition of a class of hypergeometric and other related functions as well as to certain potentially useful closed-form representations of definite and improper integrals of various special functions of applied mathematics and mathematical physics. In the present paper, our aim is to establish several formulas involving integral transforms and fractional derivatives of this family of incomplete hypergeometric functions. As corollaries and consequences, many interesting results are shown to follow from our main results.
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Ghorbani, Hojjat, Yaghoub Mahmoudi, and Farhad Dastmalchi Saei. "Numerical Study of Fractional Mathieu Differential Equation Using Radial Basis Functions." Mathematical Modelling of Engineering Problems 7, no. 4 (2020): 568–76. http://dx.doi.org/10.18280/mmep.070409.
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In this paper, we introduce a method based on Radial Basis Functions (RBFs) for the numerical approximation of Mathieu differential equation with two fractional derivatives in the Caputo sense. For this, we suggest a numerical integration method for approximating the improper integrals with a singularity point at the right end of the integration domain, which appear in the fractional computations. We study numerically the affects of characteristic parameters and damping factor on the behavior of solution for fractional Mathieu differential equation. Some examples are presented to illustrate applicability and accuracy of the proposed method. The fractional derivatives order and the parameters of the Mathieu equation are changed to study the convergency of the numerical solutions.
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Bansal, Manish Kumar, and Devendra Kumar. "On a family of the incomplete H-functions and associated integral transforms." Journal of Applied Analysis 27, no. 1 (2021): 143–52. http://dx.doi.org/10.1515/jaa-2020-2040.
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Abstract Recently, Srivastava, Saxena and Parmar [H. M. Srivastava, R. K. Saxena and R. K. Parmar, Some families of the incomplete H-functions and the incomplete H ¯ {\overline{H}} -functions and associated integral transforms and operators of fractional calculus with applications, Russ. J. Math. Phys. 25 2018, 1, 116–138] suggested incomplete H-functions (IHF) that paved the way to a natural extension and decomposition of H-function and other connected functions as well as to some important closed-form portrayals of definite and improper integrals of different kinds of special functions of physical sciences. In this article, our key aim is to present some new integral transform (Jacobi transform, Gegenbauer transform, Legendre transform and 𝖯 δ {\mathsf{P}_{\delta}} -transform) of this family of incomplete H-functions. Further, we give several interesting new and known results which are special cases our key results.
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Younis, Jihad, Bouchenak Ahmed, Mazin AlJazzazi, Rasha Al Hejaj, and Hassen Aydi. "Existence and Uniqueness Study of the Conformable Laplace Transform." Journal of Mathematics 2022 (April 26, 2022): 1–7. http://dx.doi.org/10.1155/2022/4554065.
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This paper tackles the topic of conformable Laplace transform. The authors aim at discussing its existence by exploring and providing the kind of functions that possess a conformable Laplace transform. Furthermore, the comparison theorem of conformable improper integrals is presented to further explain and justify the existence of conformable Laplace transform for some functions. The uniqueness is also established in order to determine the inverse of conformable Laplace transform for functions. Moreover, we present a table of the conformable Laplace transform of the usual functions. Finally, as an application, we use the conformable Laplace transform for solving some fractional differential equations.
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Younis, Jihad, Bouchenak Ahmed, Mazin AlJazzazi, Rasha Al Hejaj, and Hassen Aydi. "Existence and Uniqueness Study of the Conformable Laplace Transform." Journal of Mathematics 2022 (April 26, 2022): 1–7. http://dx.doi.org/10.1155/2022/4554065.
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This paper tackles the topic of conformable Laplace transform. The authors aim at discussing its existence by exploring and providing the kind of functions that possess a conformable Laplace transform. Furthermore, the comparison theorem of conformable improper integrals is presented to further explain and justify the existence of conformable Laplace transform for some functions. The uniqueness is also established in order to determine the inverse of conformable Laplace transform for functions. Moreover, we present a table of the conformable Laplace transform of the usual functions. Finally, as an application, we use the conformable Laplace transform for solving some fractional differential equations.
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Chii-Huei, Yu. "Some Type of Improper Fractional Integral." International Journal of Computer Science and Information Technology Research 10, no. 4 (2022): 53–57. https://doi.org/10.5281/zenodo.7476989.
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<strong>Abstract:</strong> In this paper, based on Jumarie type of Riemann-Liouville (R-L) fractional calculus, we evaluate some type of improper fractional integral. A new multiplication of fractional analytic functions plays an important role in this paper. The main methods we used are integration by parts for fractional calculus and fractional L’Hospital’s rule. At the same time, some examples are given to illustrate our result. In fact, our result is a generalization of the classical calculus result. <strong>Keywords:</strong> Jumarie type of R-L fractional calculus, improper fractional integral, new multiplication, fractional analytic functions, integration by parts, fractional L’Hospital’s rule. <strong>Title:</strong> Some Type of Improper Fractional Integral <strong>Author:</strong> Chii-Huei Yu <strong>International Journal of Computer Science and Information Technology Research</strong> <strong>ISSN 2348-1196 (print), ISSN 2348-120X (online)</strong> <strong>Vol. 10, Issue 4, October 2022 - December 2022</strong> <strong>Page No: 53-57</strong> <strong>Research Publish Journals</strong> <strong>Website: www.researchpublish.com</strong> <strong>Published Date: 23-December-2022</strong> <strong>DOI: https://doi.org/10.5281/zenodo.7476989</strong> <strong>Paper Download Link (Source)</strong> <strong>https://www.researchpublish.com/papers/some-type-of-improper-fractional-integral</strong>
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Chii-Huei, Yu. "Research on an Improper Fractional Integral." International Journal of Recent Research in Physics and Chemical Sciences (IJRRPCS) 10, no. 1 (2023): 37–40. https://doi.org/10.5281/zenodo.8240834.
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<strong>Abstract:</strong> In this paper, based on Jumarie type of Riemann-Liouville (R-L) fractional calculus and a new multiplication of fractional analytic functions, we study an improper fractional integral. Moreover, our result is a generalization of traditional calculus result. <strong>Keywords:</strong> Jumarie type of R-L fractional calculus, new multiplication, fractional analytic functions, improper fractional integral. <strong>Title:</strong> Research on an Improper Fractional Integral <strong>Author:</strong> Chii-Huei Yu <strong>International Journal of Recent Research in Physics and Chemical Sciences (IJRRPCS)</strong> <strong>ISSN 2350-1030</strong> <strong>Vol. 10, Issue 1, April 2023 - September 2023</strong> <strong>Page No: 37-40</strong> <strong>Paper Publications</strong> <strong>Website: www.paperpublications.org</strong> <strong>Published Date: 12-August-2023</strong> <strong>DOI: https://doi.org/10.5281/zenodo.8240834</strong> <strong>Paper Download Link (Source)</strong> <strong>https://www.paperpublications.org/upload/book/Research%20on%20an%20Improper%20Fractional%20Integral-12082023-1.pdf</strong>
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Chii-Huei, Yu. "An Improper Fractional Integral Involving Fractional Exponential Function." International Journal of Recent Research in Mathematics Computer Science and Information Technology 10, no. 1 (2023): 57–61. https://doi.org/10.5281/zenodo.8248860.
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<strong>Abstract:</strong> In this paper, based on Jumarie type of Riemann-Liouville (R-L) fractional calculus and a new multiplication of fractional analytic functions, we use integration by parts for fractional calculus to solve an improper fractional integral involving fractional exponential function. In fact, our result is a generalization of classical calculus result. <strong>Keywords:</strong> Jumarie type of R-L fractional calculus, new multiplication, fractional analytic functions, integration by parts for fractional calculus, improper fractional integral, fractional exponential function. <strong>Title:</strong> An Improper Fractional Integral Involving Fractional Exponential Function <strong>Author:</strong> Chii-Huei Yu <strong>International Journal of Recent Research in Mathematics Computer Science and Information Technology</strong> <strong>ISSN 2350-1022</strong> <strong>Vol. 10, Issue 1, April 2023 - September 2023</strong> <strong>Page No: 57-61</strong> <strong>Paper Publications</strong> <strong>Website: www.paperpublications.org</strong> <strong>Published Date: 15-August-2023</strong> <strong>DOI: https://doi.org/10.5281/zenodo.8248860</strong> <strong>Paper Download Link (Source)</strong> <strong>https://www.paperpublications.org/upload/book/An%20Improper%20Fractional%20Integral-15082023-3.pdf</strong>
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Chii-Huei, Yu. "Solution of Some Type of Improper Fractional Integral." International Journal of Interdisciplinary Research and Innovations 11, no. 1 (2023): 11–16. https://doi.org/10.5281/zenodo.7536528.
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<strong>Abstract:</strong> In this paper, based on Jumarie type of Riemann-Liouville (R-L) fractional calculus, we find the solution of some type of improper fractional integral. Differentiation under fractional integral sign and a new multiplication of fractional analytic functions play important roles in this paper. Moreover, some examples are given to illustrate our main result. In fact, our result is a generalization of ordinary calculus result. <strong>Keywords:</strong> Jumarie type of R-L fractional calculus, improper fractional integral, differentiation under fractional integral sign, new multiplication, fractional analytic functions. <strong>Title:</strong> Solution of Some Type of Improper Fractional Integral <strong>Author:</strong> Chii-Huei Yu <strong>International Journal of Interdisciplinary Research and Innovations</strong> <strong>ISSN 2348-1218 (print), ISSN 2348-1226 (online)</strong> <strong>Vol. 11, Issue 1, January 2023 - March 2023</strong> <strong>Page No: 11-16</strong> <strong>Research Publish Journals</strong> <strong>Website: www.researchpublish.com</strong> <strong>Published Date: 14-January-2023</strong> <strong>DOI: https://doi.org/10.5281/zenodo.7536528</strong> <strong>Paper Download Link (Source)</strong> <strong>https://www.researchpublish.com/papers/solution-of-some-type-of-improper-fractional-integral</strong>
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Chii-Huei, Yu. "Methods for Solving Some Type of Improper Fractional Integral." International Journal of Engineering Research and Reviews 11, no. 2 (2023): 14–18. https://doi.org/10.5281/zenodo.7886143.
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<strong>Abstract:</strong> In this paper, we use differentiation under fractional integral sign and integration by parts for fractional calculus to find the exact solution of some type of improper fractional integral. Jumarie type of Riemann-Liouville (R-L) fractional calculus and a new multiplication of fractional analytic functions play important roles in this paper. On the other hand, some examples are provided to illustrate our result. In fact, our result is a generalization of ordinary calculus result. <strong>Keywords:</strong> Differentiation under fractional integral sign, integration by parts for fractional calculus, exact solution, improper fractional integral, Jumarie type of R-L fractional calculus, new multiplication, fractional analytic functions. <strong>Title:</strong> Methods for Solving Some Type of Improper Fractional Integral <strong>Author:</strong> Chii-Huei Yu <strong>International Journal of Engineering Research and Reviews</strong> <strong>ISSN 2348-697X (Online)</strong> <strong>Vol. 11, Issue 2, April 2023 - June 2023</strong> <strong>Page No: 14-18</strong> <strong>Research Publish Journals</strong> <strong>Website: www.researchpublish.com</strong> <strong>Published Date: 02-May-2023</strong> <strong>DOI: https://doi.org/10.5281/zenodo.7886143</strong> <strong>Paper Download Link (Source)</strong> <strong>https://www.researchpublish.com/papers/methods-for-solving-some-type-of-improper-fractional-integral</strong>
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Chii-Huei, Yu, and Yang Kuang-Wu. "Solving Some Type of Improper Fractional Integral Using Differentiation under Fractional Integral Sign and Integration by Parts for Fractional Calculus." International Journal of Engineering Research and Reviews 12, no. 4 (2024): 34–38. https://doi.org/10.5281/zenodo.13939320.
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<strong>Abstract: </strong>In this paper, based on Jumarie’s modified Riemann-Liouville (R-L) fractional calculus and a new multiplication of fractional analytic functions, we study some type of improper fractional integral. We can obtain the exact solution of this improper fractional integral by using differentiation under fractional integral sign and integration by parts for fractional calculus. In fact, our result is a generalization of classical calculus result. <strong>Keywords:</strong> Jumarie’s modified R-L fractional calculus, new multiplication, fractional analytic functions, improper fractional integral, differentiation under fractional integral sign, integration by parts for fractional calculus. <strong>Title:</strong> Solving Some Type of Improper Fractional Integral Using Differentiation under Fractional Integral Sign and Integration by Parts for Fractional Calculus <strong>Author:</strong> Chii-Huei Yu, Kuang-Wu Yang <strong>International Journal of Engineering Research and Reviews</strong> <strong>ISSN 2348-697X (Online)</strong> <strong>Vol. 12, Issue 4, October 2024 - December 2024</strong> <strong>Page No: 34-38</strong> <strong>Research Publish Journals</strong> <strong>Website: www.researchpublish.com</strong> <strong>Published Date: 16-October-2024</strong> <strong>DOI: https://doi.org/10.5281/zenodo.13939320</strong> <strong>Paper Download Link (Source)</strong> <strong>https://www.researchpublish.com/papers/solving-some-type-of-improper-fractional-integral-using-differentiation-under-fractional-integral-sign-and-integration-by-parts-for-fractional-calculus</strong>
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Butt, S. I., and H. Inam. "New fractional refinements of harmonic Hermite-Hadamard-Mercer type inequalities via support line." Filomat 38, no. 14 (2024): 5179–207. https://doi.org/10.2298/fil2414179b.
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In this research, we first provide new and refined fractional integral Mercer inequalities for harmonic convex functions by deploying the idea of line of support. Thus, these refinements allow us to develop new extensions for integral inequalities pertaining harmonic convex functions. We also provide some new fractional auxiliary equalities in Mercer sense. By employing Mercer?s harmonic convexity on them, we exhibit new fractional Mercer variants of trapezoid and midpoint type inequalities. We prove new Hermite-Hadamard (H-H) type inequalities with special functions involving fractional integral operators. For the development of these new integral inequalities, we use Power-mean, H?lder?s and improved H?lder integral inequalities. We unveiled complicated integrals into simple forms by involving hypergeometric functions. Visual illustrations demonstrate the accuracy and supremacy of the offered technique. As an application, new bounds regarding hypergeometric functions as well as special means of R (real numbers) and quadrature rule are exemplified to show the applicability and validity of the offered technique.
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Chii-Huei, Yu. "Evaluating an Improper Fractional Integral Based on Jumarie's Modified Riemann-Liouville Fractional Calculus." International Journal of Computer Science and Information Technology Research 11, no. 1 (2023): 22–27. https://doi.org/10.5281/zenodo.7551811.
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<strong>Abstract:</strong> In this paper, based on Jumarie’s modified Riemann-Liouville (R-L) fractional calculus, we find the solution of some type of improper fractional integral. Change of variable for fractional calculus, integration by parts for fractional calculus, fractional L’Hospital’s rule, and a new multiplication of fractional analytic functions play important roles in this article. In fact, our result is a generalization of the traditional calculus result. <strong>Keywords:</strong> Jumarie’s modified R-L fractional calculus, improper fractional integral, change of variable, integration by parts, fractional L’Hospital’s rule, new multiplication, fractional analytic functions. <strong>Title:</strong> Evaluating an Improper Fractional Integral Based on Jumarie’s Modified Riemann-Liouville Fractional Calculus <strong>Author:</strong> Chii-Huei Yu <strong>International Journal of Computer Science and Information Technology Research</strong> <strong>ISSN 2348-1196 (print), ISSN 2348-120X (online)</strong> <strong>Vol. 11, Issue 1, January 2023 - March 2023</strong> <strong>Page No: 22-27</strong> <strong>Research Publish Journals</strong> <strong>Website: www.researchpublish.com</strong> <strong>Published Date: 19-January-2023</strong> <strong>DOI: https://doi.org/10.5281/zenodo.7551811</strong> <strong>Paper Download Link (Source)</strong> <strong>https://www.researchpublish.com/papers/evaluating-an-improper-fractional-integral-based-on-jumaries-modified-riemann-liouville-fractional-calculus</strong>
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Munir, Arslan, Ather Qayyum, Siti Supadi, Hüseyin Budak, and Irza Faiz. "A study of improved error bounds for Simpson type inequality via fractional integral operator." Filomat 38, no. 10 (2024): 3415–27. https://doi.org/10.2298/fil2410415m.
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Fractional integral operators have been studied extensively in the last few decades, and many different types of fractional integral operators have been introduced by various mathematicians. In 1967 Michele Caputo introduced Caputo fractional derivatives, which defined one of these fractional operators, the Caputo Fabrizio fractional integral operator. The main aim of this article is to established the new integral equalities related to Caputo-Fabrizio fractional integral operator. By incorporating this identity and convexity theory to obtained a novel class of Simpson type inequality. In this paper, we present a novel generalization of Simpson type inequality via s-convex and quasi-convex functions. Then, utilizing this identity the bounds of classical Simpson type inequality is improved. Finally, we discussed some applications to Simps,on s quadrature rule.
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Botmart, Thongchai, Soubhagya Kumar Sahoo, Bibhakar Kodamasingh, Muhammad Amer Latif, Fahd Jarad, and Artion Kashuri. "Certain midpoint-type Fejér and Hermite-Hadamard inclusions involving fractional integrals with an exponential function in kernel." AIMS Mathematics 8, no. 3 (2022): 5616–38. http://dx.doi.org/10.3934/math.2023283.
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<abstract><p>In this paper, using positive symmetric functions, we offer two new important identities of fractional integral form for convex and harmonically convex functions. We then prove new variants of the Hermite-Hadamard-Fejér type inequalities for convex as well as harmonically convex functions via fractional integrals involving an exponential kernel. Moreover, we also present improved versions of midpoint type Hermite-Hadamard inequality. Graphical representations are given to validate the accuracy of the main results. Finally, applications associated with matrices, q-digamma functions and modifed Bessel functions are also discussed.</p></abstract>
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Ibrahimov, V., G. Shafieva, K. Rahimova, A. Quliyeva, and I. Qurbanov. "ON SOME WAYS FOR CALCULATION DEFINITE INTEGRALS." Slovak international scientific journal, no. 79 (January 10, 2024): 27–32. https://doi.org/10.5281/zenodo.10480866.
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One of the classical problems is the calculation of the definite integrals. The specialists encounter very often the calculation of definite integrals. Traditionally, to calculate some definite integrals were used integral sums as the rectangular method, trapezoid and Simpson methods etc. By using that all methods have their own advantages and disadvantages. Currently, experts are trying to construct more accurate methods with the improved properties. Therefore, construction of methods with the improved properties is always relevant. Following what was said, here is a suggestion for a new way for calculation of definite integral. For this aim offered here to use the some relationship between the both ODEs and definite integral. And suggested to use the intersection of definite integrals with the initial-value problem for the Ordinary Differential Equations. In the first time this idea was used by Euler in the result of which appeared Euler's explicit method. Here, this idea has been generalized and has constructed more general methods for the calculation of definite integrals. And the advantages of these methods were also shown. The advantages of the suggested methods here have been illustrated by using some simple examples.
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Shevchenko, Georgiy, and Lauri Viitasaari. "Adapted integral representations of random variables." International Journal of Modern Physics: Conference Series 36 (January 2015): 1560004. http://dx.doi.org/10.1142/s2010194515600046.
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We study integral representations of random variables with respect to general Hölder continuous processes and with respect to two particular cases; fractional Brownian motion and mixed fractional Brownian motion. We prove that an arbitrary random variable can be represented as an improper integral, and that the stochastic integral can have any distribution. If in addition the random variable is a final value of an adapted Hölder continuous process, then it can be represented as a proper integral. It is also shown that in the particular case of mixed fractional Brownian motion, any adapted random variable can be represented as a proper integral.
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Hyder, Abd-Allah, Mohamed A. Barakat, Doaa Rizk, Rasool Shah, and Kamsing Nonlaopon. "Study of HIV model via recent improved fractional differential and integral operators." AIMS Mathematics 8, no. 1 (2023): 1656–71. http://dx.doi.org/10.3934/math.2023084.
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<abstract><p>In this article, a new fractional mathematical model is presented to investigate the contagion of the human immunodeficiency virus (HIV). This model is constructed via recent improved fractional differential and integral operators. Other operators like Caputo, Riemann-Liouville, Katugampola, Jarad and Hadamard are being extended and generalized by these improved fractional differential and integral operators. Banach's and Leray-Schauder nonlinear alternative fixed point theorems are utilized to examine the existence and uniqueness results of the proposed fractional HIV model. Moreover, different kinds of Ulam stability for the fractional HIV model are established. It is simple to recognize that the extracted results can be reduced to some results acquired in multiple works of literature.</p></abstract>
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Ali, Muhammad Aamir, Wei Liu, Shigeru Furuichi, and Michal Fečkan. "Improved Hermite–Hadamard Inequality Bounds for Riemann–Liouville Fractional Integrals via Jensen’s Inequality." Fractal and Fractional 8, no. 9 (2024): 547. http://dx.doi.org/10.3390/fractalfract8090547.
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This paper derives the sharp bounds for Hermite–Hadamard inequalities in the context of Riemann–Liouville fractional integrals. A generalization of Jensen’s inequality called the Jensen–Mercer inequality is used for general points to find the new and refined bounds of fractional Hermite–Hadamard inequalities. The existing Hermite–Hadamard inequalities in classical or fractional calculus have been proved for convex functions, typically involving only two points as in Jensen’s inequality. By applying the general points in Jensen–Mercer inequalities, we extend the scope of the existing results, which were previously proved for two points in the Jensen’s inequality or the Jensen–Mercer inequality. The use of left and right Riemann–Liouville fractional integrals in inequalities is challenging because of the general values involved in the Jensen–Mercer inequality, which we overcame by considering different cases. The use of the Jensen–Mercer inequality for general points to prove the refined bounds is a very interesting finding of this work, because it simultaneously generalizes many existing results in fractional and classical calculus. The application of these new results is demonstrated through error analysis of numerical integration formulas. To show the validity and significance of the findings, various numerical examples are tested. The numerical examples clearly demonstrate the significance of this new approach, as using more points in the Jensen–Mercer inequality leads to sharper bounds.
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Lozynskyy, Andriy, Andriy Chaban, Tomasz Perzyński, Andrzej Szafraniec, and Lidiia Kasha. "Application of Fractional-Order Calculus to Improve the Mathematical Model of a Two-Mass System with a Long Shaft." Energies 14, no. 7 (2021): 1854. http://dx.doi.org/10.3390/en14071854.
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Based on the general theory of fractional order derivatives and integrals, application of the Caputo–Fabrizio operator is analyzed to improve a mathematical model of a two-mass system with a long shaft and concentrated parameters. Thus, the real transmission of complex electric drives, which consist of long shafts with a sufficient degree of adequacy, is presented as a two-mass system. Such a system is described by ordinary fractional order differential equations. In addition, it is well known that an elastic mechanical wave, propagating along a drive transmission with a long stiff shaft, creates a retardation effect on distribution of the time–space angular velocity, the rotation angle of the shaft, and its elastic moment. The approach proposed in the current work helps to take in account the moving elastic wave along the shaft of electric drive mechanism. On this basis, it is demonstrated that the use of the fractional order integrator in the model for the elastic moment enables it to reproduce real transient processes in the joint coordinates of the system. It also provides an accuracy equivalent to the model with distributed parameters. The distance between the traditional model and the model in which the fractional integral is used for the elastic moment modelling in a two-mass system, with a long shaft, is analyzed.
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Gao, Feng, and Chunmei Chi. "Improvement on Conformable Fractional Derivative and Its Applications in Fractional Differential Equations." Journal of Function Spaces 2020 (August 1, 2020): 1–10. http://dx.doi.org/10.1155/2020/5852414.
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In this paper, we made improvement on the conformable fractional derivative. Compared to the original one, the improved conformable fractional derivative can be a better replacement of the classical Riemann-Liouville and Caputo fractional derivative in terms of physical meaning. We also gave the definition of the corresponding fractional integral and illustrated the applications of the improved conformable derivative to fractional differential equations by some examples.
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Zheng, Shiqi, Xiaoqi Tang, and Bao Song. "Fractional order controllers tuning strategy for permanent magnet synchronous motor servo drive system based on genetic algorithm–wavelet neural network hybrid method." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 228, no. 17 (2014): 3218–39. http://dx.doi.org/10.1177/0954406214525603.
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In this paper, a novel tuning strategy for the fractional order proportional integral and fractional order [proportional integral] controllers is proposed for the permanent magnet synchronous motor servo drive system. The tuning strategy is based on a genetic algorithm–wavelet neural network hybrid method. Firstly, the initial values of the control parameters of the fractional order controllers are selected according to a new global tuning rule, which is based on the genetic algorithm and considers both the time- and frequency-domain specifications. Secondly, the wavelet neural network is utilized to update the control parameters based on the selected initial values in an online manner which improves the capability of handling parameter variations and time-varying operating conditions. Furthermore, to improve the computational efficiency, a recursive least squares algorithm, which provides information to the wavelet neural network, is used to identify the permanent magnet synchronous motor drive system. Finally, experimental results on the permanent magnet synchronous motor drive system show both of the two proposed fractional order controllers work efficiently, with improved performance comparing with their traditional counterpart.
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Sánchez-Rivero, Maibeth, Manuel A. Duarte-Mermoud, Juan Carlos Travieso-Torres, Marcos E. Orchard, and Gustavo Ceballos-Benavides. "Analysis of Fractional Order-Adaptive Systems Represented by Error Model 1 Using a Fractional-Order Gradient Approach." Mathematics 12, no. 20 (2024): 3212. http://dx.doi.org/10.3390/math12203212.
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In adaptive control, error models use system output error and adaptive laws to update controller parameters for control or identification tasks. Fractional-order calculus, involving non-integer-order derivatives and integrals, is increasingly important for modeling, estimation, and control due to its ability to generalize classical methods and offer improved robustness to disturbances. This paper addresses the gap in the literature where fractional-order gradient methods have not yet been extensively applied in identification and adaptive control schemes. We introduce a fractional-order error model with fractional-order gradient (FOEM1-FG), which integrates fractional gradient operators based on the Caputo fractional derivative. By using theoretical analysis and simulations, we confirm that FOEM1-FG maintains stability and ensures bounded output errors across a variety of input signals. Notably, the fractional gradient’s performance improves as the order, β, increases with β>1, leading to faster convergence. Compared to existing integer-order methods, the proposed approach provides a more flexible and efficient solution in adaptive identification and control schemes. Our results show that FOEM1-FG offers superior stability and convergence characteristics, contributing new insights to the field of fractional calculus in adaptive systems.
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Hinze, Matthias, André Schmidt, and Remco I. Leine. "Numerical solution of fractional-order ordinary differential equations using the reformulated infinite state representation." Fractional Calculus and Applied Analysis 22, no. 5 (2019): 1321–50. http://dx.doi.org/10.1515/fca-2019-0070.
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Abstract In this paper, we propose a novel approach for the numerical solution of fractional-order ordinary differential equations. The method is based on the infinite state representation of the Caputo fractional differential operator, in which the entire history of the state of the system is considered for correct initialization. The infinite state representation contains an improper integral with respect to frequency, expressing the history dependence of the fractional derivative. The integral generally has a weakly singular kernel, which may lead to problems in numerical computations. A reformulation of the integral generates a kernel that decays to zero at both ends of the integration interval leading to better convergence properties of the related numerical scheme. We compare our method to other schemes by considering several benchmark problems.
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Huang, Tingsheng, Chunyang Wang, and Xuelian Liu. "Depth Image Denoising Algorithm Based on Fractional Calculus." Electronics 11, no. 12 (2022): 1910. http://dx.doi.org/10.3390/electronics11121910.
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Depth images are often accompanied by unavoidable and unpredictable noise. Depth image denoising algorithms mainly attempt to fill hole data and optimise edges. In this paper, we study in detail the problem of effectively filtering the data of depth images under noise interference. The classical filtering algorithm tends to blur edge and texture information, whereas the fractional integral operator can retain more edge and texture information. In this paper, the Grünwald–Letnikov-type fractional integral denoising operator is introduced into the depth image denoising process, and the convolution template of this operator is studied and improved upon to build a fractional integral denoising model and algorithm for depth image denoising. Depth images from the Redwood dataset were used to add noise, and the mask constructed by the fractional integral denoising operator was used to denoise the images by convolution. The experimental results show that the fractional integration order with the best denoising effect was −0.4 ≤ ν ≤ −0.3 and that the peak signal-to-noise ratio was improved by +3 to +6 dB. Under the same environment, median filter denoising had −15 to −30 dB distortion. The filtered depth image was converted to a point cloud image, from which the denoising effect was subjectively evaluated. Overall, the results prove that the fractional integral denoising operator can effectively handle noise in depth images while preserving their edge and texture information and thus has an excellent denoising effect.
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Wei, Wei, H. M. Srivastava, Yunyi Zhang, Lei Wang, Peiyi Shen, and Jing Zhang. "A Local Fractional Integral Inequality on Fractal Space Analogous to Anderson’s Inequality." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/797561.
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Anderson's inequality (Anderson, 1958) as well as its improved version given by Fink (2003) is known to provide interesting examples of integral inequalities. In this paper, we establish local fractional integral analogue of Anderson's inequality on fractal space under some suitable conditions. Moreover, we also show that the local fractional integral inequality on fractal space, which we have proved in this paper, is a new generalization of the classical Anderson's inequality.
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Gao, Dongdong, and Jianli Li. "Existence Results for Impulsive Fractional Differential Inclusions with Two Different Caputo Fractional Derivatives." Discrete Dynamics in Nature and Society 2019 (February 18, 2019): 1–9. http://dx.doi.org/10.1155/2019/1323176.
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In this paper, we study the impulsive fractional differential inclusions with two different Caputo fractional derivatives and nonlinear integral boundary value conditions. Under certain assumptions, new criteria to guarantee the impulsive fractional impulsive fractional differential inclusion has at least one solution are established by using Bohnenblust-Karlin’s fixed point theorem. Also, some previous results will be significantly improved.
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Yu, Bo, Yi-Fei Pu, and Qiu-Yan He. "Fractional-Order Dual-Slope Integral Fast Analog-to-Digital Converter with High Sensitivity." Journal of Circuits, Systems and Computers 29, no. 05 (2019): 2050083. http://dx.doi.org/10.1142/s0218126620500838.
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The dual-slope integral analog-to-digital converter is widely used in low-speed, high-precision measurement owing to its high precision and strong resistance on crosstalk interference. To meet the requirements of higher accuracy and faster measurement, the integral sensitivity and conversion speed of the dual-slope integral analog-to-digital converter must be improved. Therefore, based on fractional-order calculus, we propose a fractional-order dual-slope integral analog-to-digital converter. First, constant-current charging curves were provided to explain the source of the idea of the fractional-order dual-slope integral analog-to-digital converter. Then, the working principle of the fractional-order dual-slope integral analog-to-digital converter is described in detail. The calculation formula of analog-to-digital conversion is derived and analyzed. Moreover, the relationship of the voltage-measurement error with the operation-order error of the fractor and the reference voltage error is theoretically derived. Furthermore, we theoretically analyze the resistance of the proposed analog-to-digital converter to crosstalk interference, as well as the requirements for the first fractional integral time when crosstalk interference is suppressed. Specifically, we prove that the proposed analog-to-digital converter has a higher sensitivity and conversion speed than the classical converter, and we provide a quantitative calculation formula.
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Wang, Dawei, Hongbo Zou, and Jili Tao. "A new design of fractional-order dynamic matrix control with proportional–integral–derivative-type structure." Measurement and Control 52, no. 5-6 (2019): 567–76. http://dx.doi.org/10.1177/0020294019843939.
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Most dynamic systems in practice are of fractional order and often the models using fractional-order equations can grasp their intrinsic properties with more accuracy compared with conventional differential equations. In this paper, a fractional-order modeling-based proportional–integral–derivative-type dynamic matrix control is developed and tested on a typical industrial heating furnace system with fractional-order dynamics. The Oustaloup approximation method is first adopted to obtain the model approximation of the processes, which paves the way for the application of integer order dynamic matrix control to the fractional-order systems. Meanwhile, a set of proportional–integral–derivative-type operators are introduced in the cost function to further optimize the dynamic matrix control in terms of tracking and disturbance-rejection performance. The resulting controller bears both the merits of the dynamic matrix control and the proportional–integral–derivative, and thus improved control performance is obtained. In addition, an industrial heating furnace process system is given to test the performance of the proposed method in comparison with traditional integer order model-based dynamic matrix control, and results show that the proposed method gives improved system performance.
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Zhang, Haiyang, Yi Zhao, Lianglin Xiong, Junzhou Dai, and Yi Zhang. "New Event-Triggered Synchronization Criteria for Fractional-Order Complex-Valued Neural Networks with Additive Time-Varying Delays." Fractal and Fractional 8, no. 10 (2024): 569. http://dx.doi.org/10.3390/fractalfract8100569.
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This paper explores the synchronization control issue for a class of fractional-order Complex-valued Neural Networks (FOCVNNs) with additive time-varying delays (TVDs) utilizing a sampled-data-based event-triggered mechanism (SDBETM). First, an innovative free-matrix-based fractional-order integral inequality (FMBFOII) and an improved fractional-order complex-valued integral inequality (FOCVII) are proposed, which are less conservative than the existing classical fractional-order integral inequality (FOII). Secondly, an SDBETM is inducted to conserve network resources. In addition, a novel Lyapunov–Krasovskii functional (LKF) enriched with additional information regarding the fractional-order derivative, additive TVDs, and triggering instants is constructed. Then, through the integration of the innovative FOCVII, LKF, SDBETM, and other analytical methodologies, we deduce two criteria in the form of linear matrix inequalities (LMIs) to ensure the synchronization of the master–slave FOCVNNs. Finally, numerical simulations are illustrated to confirm the validity of the proposed results.
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Ouyang, Fan, Hongyan Liu, and Yanying Ma. "An Improved Numerical Scheme for 2D Nonlinear Time-Dependent Partial Integro-Differential Equations with Multi-Term Fractional Integral Items." Fractal and Fractional 9, no. 3 (2025): 167. https://doi.org/10.3390/fractalfract9030167.
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This paper is dedicated to investigating a highly accurate numerical solution for a class of 2D nonlinear time-dependent partial integro-differential equations with multi-term fractional integral items. These integrals are weakly singular with respect to time, which are handled using the product integration rule on graded meshes to compensate for the influence generated by the initial weak singular nature of the exact solution. The temporal derivative is approximated by a generalized Crank–Nicolson difference scheme, while the nonlinear term is approximated by a linearized method. Furthermore, the stability and convergence of the derived time semi-discretization scheme are strictly proved by revising the finite discrete parameters. Meanwhile, the differential matrices of the spatial high-order derivatives based on barycentric rational interpolation are utilized to obtain the fully discrete scheme. Finally, the effectiveness and reliability of the proposed method are validated by means of several numerical experiments.
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Han, Rubing, Shuonan Wu, and Hao Zhou. "A Monotone Discretization for the Fractional Obstacle Problem and Its Improved Policy Iteration." Fractal and Fractional 7, no. 8 (2023): 634. http://dx.doi.org/10.3390/fractalfract7080634.
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In recent years, the fractional Laplacian has attracted the attention of many researchers, the corresponding fractional obstacle problems have been widely applied in mathematical finance, particle systems, and elastic theory. Furthermore, the monotonicity of the numerical scheme is beneficial for numerical stability. The purpose of this work is to introduce a monotone discretization method for addressing obstacle problems involving the integral fractional Laplacian with homogeneous Dirichlet boundary conditions over bounded Lipschitz domains. Through successful monotone discretization of the fractional Laplacian, the monotonicity is preserved for the fractional obstacle problem and the uniform boundedness, existence, and uniqueness of the numerical solutions of the fractional obstacle problems are proved. A policy iteration is adopted to solve the discrete nonlinear problems, and the convergence after finite iterations can be proved through the monotonicity of the scheme. Our improved policy iteration, adapted to solution regularity, demonstrates superior performance by modifying discretization across different regions. Numerical examples underscore the efficacy of the proposed method.
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Mahmoud, Ahmed G., Mohamed A. El-Beltagy, and Ahmed M. Zobaa. "Novel Fractional Order Differential and Integral Models for Wind Turbine Power–Velocity Characteristics." Fractal and Fractional 8, no. 11 (2024): 656. http://dx.doi.org/10.3390/fractalfract8110656.
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This work presents an improved modelling approach for wind turbine power curves (WTPCs) using fractional differential equations (FDE). Nine novel FDE-based models are presented for mathematically modelling commercial wind turbine modules’ power–velocity (P-V) characteristics. These models utilize Weibull and Gamma probability density functions to estimate the capacity factor (CF), where accuracy is measured using relative error (RE). Comparative analysis is performed for the WTPC mathematical models with a varying order of differentiation (α) from 0.5 to 1.5, utilizing the manufacturer data for 36 wind turbines with capacities ranging from 150 to 3400 kW. The shortcomings of conventional mathematical models in various meteorological scenarios can be overcome by applying the Riemann–Liouville fractional integral instead of the classical integer-order integrals. By altering the sequence of differentiation and comparing accuracy, the suggested model uses fractional derivatives to increase flexibility. By contrasting the model output with actual data obtained from the wind turbine datasheet and the historical data of a specific location, the models are validated. Their accuracy is assessed using the correlation coefficient (R) and the Mean Absolute Percentage Error (MAPE). The results demonstrate that the exponential model at α=0.9 gives the best accuracy of WTPCs, while the original linear model was the least accurate.
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Sahoo, Soubhagya Kumar, Fahd Jarad, Bibhakar Kodamasingh, and Artion Kashuri. "Hermite-Hadamard type inclusions via generalized Atangana-Baleanu fractional operator with application." AIMS Mathematics 7, no. 7 (2022): 12303–21. http://dx.doi.org/10.3934/math.2022683.
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<abstract><p>Defining new fractional operators and employing them to establish well-known integral inequalities has been the recent trend in the theory of mathematical inequalities. To take a step forward, we present novel versions of Hermite-Hadamard type inequalities for a new fractional operator, which generalizes some well-known fractional integral operators. Moreover, a midpoint type fractional integral identity is derived for differentiable mappings, whose absolute value of the first-order derivatives are convex functions. Moreover, considering this identity as an auxiliary result, several improved inequalities are derived using some fundamental inequalities such as Hölder-İşcan, Jensen and Young inequality. Also, if we take the parameter $ \rho = 1 $ in most of the results, we derive new results for Atangana-Baleanu equivalence. One example related to matrices is also given as an application.</p></abstract>
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Gao, Jin, and Hui Li. "Tuning Parameters of the Fractional Order PID-LQR Controller for Semi-Active Suspension." Electronics 12, no. 19 (2023): 4115. http://dx.doi.org/10.3390/electronics12194115.
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In order to further improve the control effect of proportion integral differential (PID) control and linear quadratic regulator (LQC) control, and improve vehicle ride comfort and enhance body stability, the 7 DOF semi-active suspension model was established, and the fractional order PIλDμ-LQR controller was designed by combining fractional order PIλDμ control theory and LQR control theory. The semi-active suspension model in this paper is more complex, and there are many parameters in the controller. The optimal weighting coefficient of 12 vehicle smoothness evaluation indicators and parameters Kp, Ki, Kd, λ and μ in the controller were founded by NSGA-II algorithm. After optimization, the optimized parameters were brought into the controller for random pavement simulation. Compared to the traditional passive suspension, fractional order PIλDμ individual control and LQR separate control, the simulation results show that the effect of fractional order PIλDμ-LQR control is very significant. The evaluation index of vehicle smoothness has been significantly improved, and the use of fractional order PIλDμ-LQR control has significantly improved the working performance of the suspension and improved the smoothness of the vehicle. At the same time, the adjusting force output of the actuator is very balanced, which inhibits the roll of the body and improves the anti-roll performance. After simulation, the excellent performance of the designed fractional PIλDμ-LQR controller was verified, and the introduced NSGA-II algorithm played an important role in the controller parameter tuning work, which shows that the fractional order PIλDμ-LQR controller and NSGA-II algorithm cooperate with each other to achieve good control effects.
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Al-Dhaifallah, Mujahed. "Construction and Evaluation of a Control Mechanism for Fuzzy Fractional-Order PID." Applied Sciences 12, no. 14 (2022): 6832. http://dx.doi.org/10.3390/app12146832.
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In this research, a control mechanism for fuzzy fractional-order proportional integral derivatives was suggested (FFOPID). The fractional calculus application has been used in different fields of engineering and science and showed to be improved in the past few years. However, there are few studies on the implementation of the fuzzy fractional-order controller for control in real time. Therefore, for an experimental pressure control model, a fractional order PID controller with intelligent fuzzy tuning was constructed and its results were calculated through simulation. To highlight proposed control scheme advantages, the performances of the controller were inspected under load disturbances and variations in set-point conditions. Furthermore, with classical PID control schemes and fractional order proportional integral derivative (FOPID), a comparative study was made. It is revealed from the results that the suggested control scheme outclasses other categories of the control schemes.
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Lozynskyy, Andriy, Jacek Kozyra, Andriy Kutsyk, et al. "AVR Fractional-Order Controller Based on Caputo–Fabrizio Fractional Derivatives and Integral Operators." Energies 17, no. 23 (2024): 5913. http://dx.doi.org/10.3390/en17235913.
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The application of a fractional-order controller (FOC) using the Caputo–Fabrizio representation in the automatic voltage regulation (AVR) system of a synchronous generator is shown in this paper. The mathematical model of the system is created and the adequacy of the model is confirmed. The efficiency of the proposed regulator in different operating regimes is demonstrated. In particular, the proposed controller improves voltage regulation in a wide range of changes in the coordinates that characterize the power system operation mode, and it increases the system’s robustness to both uncertainties and nonlinearities that often occur in power systems. The synthesized fractional-order regulator provides higher response and control accuracy compared to traditional regulators used in automatic voltage regulation (AVR) systems.
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Qayyum, Mubashir, Efaza Ahmad, Muhammad Bilal Riaz, and Jan Awrejcewicz. "Improved Soliton Solutions of Generalized Fifth Order Time-Fractional KdV Models: Laplace Transform with Homotopy Perturbation Algorithm." Universe 8, no. 11 (2022): 563. http://dx.doi.org/10.3390/universe8110563.
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The main purpose of this research is to propose a new methodology to observe a class of time-fractional generalized fifth-order Korteweg–de Vries equations. Laplace transform along with a homotopy perturbation algorithm is utilized for the solution and analysis purpose in the current study. This extended technique provides improved and convergent series solutions through symbolic computation. The proposed methodology is applied to time-fractional Sawada–Kotera, Ito, Lax’s, and Kaup–Kupershmidt models, which are induced from a generalized fifth-order KdV equation. For validity purposes, obtained and existing results at integral orders are compared. Convergence analysis was also performed by computing solutions and errors at different values in a fractional domain. Dynamic behavior of the fractional parameter is also studied graphically. Simulations affirm the dominance of the proposed algorithm in terms of accuracy and fewer computations as compared to other available schemes for fractional KdVs. Hence, the projected algorithm can be utilized for more advanced fractional models in physics and engineering.
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Zhou, Zhibiao, Wei Xiao, and Yongshun Liang. "Partially Explore the Differences and Similarities between Riemann-Liouville Integral and Mellin Transform." Fractal and Fractional 6, no. 11 (2022): 638. http://dx.doi.org/10.3390/fractalfract6110638.
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At present many researchers devote themselves to studying the relationship between continuous fractal functions and their fractional integral. But little attention is paid to the relationship between Mellin transform and fractional integral. This paper aims to partially explore the differences and similarities between Riemann-Liouville integral and Mellin transform, then a 1-dimensional continuous and unbounded variational function defined on the closed interval [0,1] needs to be constructed. Through describing the image of the constructed function and its transformed function and proving the relevant properties, we obtain that Box dimension of its Riemann–Liouville integral of arbitrary order and its Mellin transformed function are also one. The smoothness of its Riemann–Liouville integral can only be improved, and its Mellin transformed function is differentiable.
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Chang, Chih-Wen, Sania Qureshi, Ioannis K. Argyros, Khair Muhammad Saraz, and Evren Hincal. "A Modified Fractional Newton’s Solver." Axioms 13, no. 10 (2024): 689. http://dx.doi.org/10.3390/axioms13100689.
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Fractional calculus extends the conventional concepts of derivatives and integrals to non-integer orders, providing a robust mathematical framework for modeling complex systems characterized by memory and hereditary properties. This study enhances the convergence rate of the Caputo-based Newton’s solver for solving one-dimensional nonlinear equations. By modifying the order to 1+η, we provide a thorough analysis of the convergence order and present numerical simulations that demonstrate the improved efficiency of the proposed modified fractional Newton’s solver. The numerical simulations indicate significant advancements over traditional and existing fractional Newton-type approaches.
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Gao, Zhe. "A Tuning Method via Borges Derivative of a Neural Network-Based Discrete-Time Fractional-Order PID Controller with Hausdorff Difference and Hausdorff Sum." Fractal and Fractional 5, no. 1 (2021): 23. http://dx.doi.org/10.3390/fractalfract5010023.
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In this paper, the fractal derivative is introduced into a neural network-based discrete-time fractional-order PID controller in two areas, namely, in the controller’s structure and in the parameter optimization algorithm. The first use of the fractal derivative is to reconstruct the fractional-order PID controller by using the Hausdorff difference and Hausdorff sum derived from the Hausdorff derivative and Hausdorff integral. It can avoid the derivation of the Gamma function for the order updating to realize the parameter and order tuning based on neural networks. The other use is the optimization of order and parameters by using Borges derivative. Borges derivative is a kind of fractal derivative as a local fractional-order derivative. The chain rule of composite function is consistent with the integral-order derivative. It is suitable for updating the parameters and the order of the fractional-order PID controller based on neural networks. This paper improves the neural network-based PID controller in two aspects, which accelerates the response speed and improves the control accuracy. Two illustrative examples are given to verify the effectiveness of the proposed neural network-based discrete-time fractional-order PID control scheme with fractal derivatives.
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Şanlı, Zeynep, Tuncay Köroğlu, and Mehmet Kunt. "Improved Hermite Hadamard type inequalities for harmonically convex functions via Katugampola fractional integrals." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 68, no. 2 (2019): 1556–75. http://dx.doi.org/10.31801/cfsuasmas.413019.
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Xin, Baogui, and Yuting Li. "0-1 Test for Chaos in a Fractional Order Financial System with Investment Incentive." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/876298.
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A new integer-order chaotic financial system is extended by introducing a simple investment incentive into a three-dimensional chaotic financial system. A four-dimensional fractional-order chaotic financial system is presented by bringing fractional calculus into the new integer-order financial system. By using weighted integral thought, the fractional order derivative's economics meaning is given. The 0-1 test algorithm and the improved Adams-Bashforth-Moulton predictor-corrector scheme are employed to detect numerically the chaos in the proposed fractional order financial system.
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Kasinathan, Dhanalakshmi, Ramkumar Kasinathan, Ravikumar Kasinathan, and Dimplekumar Chalishajar. "Exponential stability for higher-order impulsive fractional neutral stochastic integro-delay differential equations with mixed brownian motions and non-local conditions." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 15, no. 1 (2025): 101. https://doi.org/10.36922/ijocta.1524.
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This paper investigates the exponential stability of second-order fractional neutral stochastic integral-delay differential equations (FNSIDDEs) with impulses driven by mixed fractional Brownian motions (fBm). Existence and uniqueness conditions ensure that FNSIDDEs are acquired by formulating a Banach fixed point theorem (BFPT). Novel sufficient conditions have to prove pth moment exponential stability of FNSIDDEs via fBm employing the impulsive-integral inequality. The current study expands and improves on previous findings. Additionally, an example is presented to illustrate the efficiency of the obtained theoretical results.
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Ma, Zhiqiang, Zheng H. Zhu, and Guanghui Sun. "Fractional-order sliding mode control for deployment of tethered spacecraft system." Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 233, no. 13 (2019): 4721–34. http://dx.doi.org/10.1177/0954410019830030.
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This paper proposes a fractional-order integral sliding mode control with the order 0 < ν < 1 to stabilize the deployment of tethered spacecraft system with only tension regulation. The work in this paper is partially based on integer-order nonlinear sliding mode controller and improves its performance with fractional-order calculus. The proposed scheme makes use of integral sliding surface to obtain smaller convergence regions of state errors, and the fractional derivative is synthesized to enhance the flexibility of controller design by fining parameters for better dynamic and steady-state performance. Fractional-order observers help to eliminate external disturbances while the adaptive law is presented to remove the adverse effect in stability analyses, and fractional-order uniform ultimate boundedness is proved to guarantee the existence of the proposed sliding surface. According to theoretical analyses, the fractional order will indeed affect the dynamic and steady-state performance of control system, and the proposed method will be verified in numerical simulations compared with the nonlinear sliding mode counterpart.
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Abdulkader, Rasheed. "Controller Design based on Fractional Calculus for AUV Yaw Control." Engineering, Technology & Applied Science Research 13, no. 2 (2023): 10432–38. http://dx.doi.org/10.48084/etasr.5687.
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This research presents a fractional order integral controller strategy, which improves the steering angle for Autonomous Underwater Vehicles (AUVs). The AUV mathematical modeling is presented. A Fractional Order Proportional Integral (FOPI) control scheme is implemented to ensure the yaw angle stability of the AUV steering under system uncertainty. The FOPI controller is validated with MATLAB/Simulink and is compared to the conventional Integer Order PI (IOPI) controller to track the yaw angle of the structure. The simulation results show that the proposed FOPI controller outperforms the IOPI controller and improves the AUV system steering and the overall transient response while ensuring the system's stability with and without external disturbances such as underwater current and different loading conditions.