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Search Results (3)

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Keywords = fractional Cauchy–Euler equation

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25 pages, 1086 KiB  
Article
On the Existence, Uniqueness and a Numerical Approach to the Solution of Fractional Cauchy–Euler Equation
by Nazim I. Mahmudov, Suzan Cival Buranay and Mtema James Chin
Axioms 2024, 13(9), 627; https://doi.org/10.3390/axioms13090627 - 12 Sep 2024
Viewed by 1068
Abstract
In this research paper, we consider a model of the fractional Cauchy–Euler-type equation, where the fractional derivative operator is the Caputo with order 0<α<2. The problem also constitutes a class of examples of the Cauchy problem of the [...] Read more.
In this research paper, we consider a model of the fractional Cauchy–Euler-type equation, where the fractional derivative operator is the Caputo with order 0<α<2. The problem also constitutes a class of examples of the Cauchy problem of the Bagley–Torvik equation with variable coefficients. For proving the existence and uniqueness of the solution of the given problem, the contraction mapping principle is utilized. Furthermore, a numerical method and an algorithm are developed for obtaining the approximate solution. Also, convergence analyses are studied, and simulations on some test problems are given. It is shown that the proposed method and the algorithm are easy to implement on a computer and efficient in computational time and storage. Full article
(This article belongs to the Special Issue Applied Mathematics and Numerical Analysis: Theory and Applications)
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10 pages, 2242 KiB  
Article
Constructing Solutions to Multi-Term Cauchy–Euler Equations with Arbitrary Fractional Derivatives
by Pavel B. Dubovski and Jeffrey A. Slepoi
Mathematics 2024, 12(13), 1928; https://doi.org/10.3390/math12131928 - 21 Jun 2024
Viewed by 951
Abstract
We further extend the results of other researchers on existence theory to homogeneous fractional Cauchy–Euler equations [...] Read more.
We further extend the results of other researchers on existence theory to homogeneous fractional Cauchy–Euler equations ∑i=1mdixαiDαiu(x)+μu(x)=0,αi>0, with the derivatives in Caputo or Riemann–Liouville sense. Unlike the existing works, we consider multi-term equations without any restrictions on the order of fractional derivatives. The results are based on the characteristic equations which generate the solutions. Depending on the roots of the characteristic equations (real, multiple, or complex), we construct the corresponding solutions and prove their linear independence. Full article
(This article belongs to the Special Issue Analytical and Computational Methods in Differential Equations, Special Functions, Transmutations and Integral Transforms, 2nd Edition)
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20 pages, 329 KiB  
Article
Hermite-Hadamard-Fejér Type Inequalities with Generalized K-Fractional Conformable Integrals and Their Applications
by Humaira Kalsoom and Zareen A. Khan
Mathematics 2022, 10(3), 483; https://doi.org/10.3390/math10030483 - 2 Feb 2022
Cited by 6 | Viewed by 1628
Abstract
In this work, we introduce new definitions of left and right-sides generalized conformable K-fractional derivatives and integrals. We also prove new identities associated with the left and right-sides of the Hermite-Hadamard-Fejér type inequality for ϕ-preinvex functions. Moreover, we use these new [...] Read more.
In this work, we introduce new definitions of left and right-sides generalized conformable K-fractional derivatives and integrals. We also prove new identities associated with the left and right-sides of the Hermite-Hadamard-Fejér type inequality for ϕ-preinvex functions. Moreover, we use these new identities to prove some bounds for the Hermite-Hadamard-Fejér type inequality for generalized conformable K-fractional integrals regarding ϕ-preinvex functions. Finally, we also present some applications of the generalized definitions for higher moments of continuous random variables, special means, and solutions of the homogeneous linear Cauchy-Euler and homogeneous linear K-fractional differential equations to show our new approach. Full article
(This article belongs to the Special Issue Theory and Applications of Fractional Equations and Calculus)
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