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Journal articles on the topic 'Fractional calculus operator'

Author: Grafiati

Published: 4 June 2021

Last updated: 4 February 2022

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1

Patanarapeelert, Nichaphat, and Thanin Sitthiwirattham. "On Fractional Symmetric Hahn Calculus." Mathematics 7, no. 10 (September 20, 2019): 873. http://dx.doi.org/10.3390/math7100873.

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In this paper, we study fractional symmetric Hahn difference calculus. The new idea of the symmetric Hahn difference operator, the fractional symmetric Hahn integral, and the fractional symmetric Hahn operators of Riemann–Liouville and Caputo types are presented. In addition, we formulate some fundamental properties based on these fractional symmetric Hahn operators.

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2

Li, Ming, and Wei Zhao. "Essay on Fractional Riemann-Liouville Integral Operator versus Mikusinski’s." Mathematical Problems in Engineering 2013 (2013): 1–3. http://dx.doi.org/10.1155/2013/635412.

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This paper presents the representation of the fractional Riemann-Liouville integral by using the Mikusinski operators. The Mikusinski operators discussed in the paper may yet provide a new view to describe and study the fractional Riemann-Liouville integral operator. The present result may be useful for applying the Mikusinski operational calculus to the study of fractional calculus in mathematics and to the theory of filters of fractional order in engineering.

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3

Yilmazer, R., and O. Ozturk. "N-Fractional Calculus Operator Method to the Euler Equation." Issues of Analysis 25, no. 2 (December 2018): 144–52. http://dx.doi.org/10.15393/j3.art.2018.5730.

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4

Xu, Zhisong, and Mingqiu Li. "Rational Implementation of Fractional Calculus Operator Based on Quadratic Programming." Mathematical Problems in Engineering 2021 (February 12, 2021): 1–12. http://dx.doi.org/10.1155/2021/6646718.

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When fractional calculus operators and models are implemented rationally, there exist some problems such as low approximation accuracy of rational approximation function, inability to specify arbitrary approximation frequency band, or poor robustness. Based on the error criterion of the least squares method, a quadratic programming method based on the frequency-domain error is proposed. In this method, the frequency-domain error minimization between the fractional operator s ± r and its rational approximation function is transformed into a quadratic programming problem. The results show that the construction method of the optimal rational approximation function of fractional calculus operator is effective, and the optimal rational approximation function constructed can effectively approximate the fractional calculus operator and model for the specified approximation frequency band.

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5

Momenzadeh, Mohammad, and Sajedeh Norozpou. "Alternative fractional derivative operator on non-newtonian calculus and its approaches." Nexo Revista Científica 34, no. 02 (June 9, 2021): 906–15. http://dx.doi.org/10.5377/nexo.v34i02.11616.

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Nowadays, study on fractional derivative and integral operators is one of the hot topics of mathematics and lots of investigations and studies make their attentions in this field. Most of these concerns raised from the vast application of these operators in study of phenomena’s models. These operators interpreted by Newtonian calculus, however different types of calculi are existed and we introduce the fractional derivative operators focused on Bi-geometric calculus and also their fractional differential equations are studied.

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6

EL-NABULSI, RAMI AHMAD. "FRACTIONAL FIELD THEORIES FROM MULTI-DIMENSIONAL FRACTIONAL VARIATIONAL PROBLEMS." International Journal of Geometric Methods in Modern Physics 05, no. 06 (September 2008): 863–92. http://dx.doi.org/10.1142/s0219887808003119.

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Fractional calculus has recently attracted considerable attention. In particular, various fractional differential equations are used to model nonlinear wave theory that arises in many different areas of physics such as Josephson junction theory, field theory, theory of lattices, etc. Thus one may expect fractional calculus, in particular fractional differential equations, plays an important role in quantum field theories which are expected to satisfy fractional generalization of Klein–Gordon and Dirac equations. Until now, in high-energy physics and quantum field theories the derivative operator has only been used in integer steps. In this paper, we want to extend the idea of differentiation to arbitrary non-integers steps. We will address multi-dimensional fractional action-like problems of the calculus of variations where fractional field theories and fractional differential Dirac operators are constructed.

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7

Tilahun, Kelelaw, Hagos Tadessee, and D. L. Suthar. "The Extended Bessel-Maitland Function and Integral Operators Associated with Fractional Calculus." Journal of Mathematics 2020 (June 23, 2020): 1–8. http://dx.doi.org/10.1155/2020/7582063.

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The aim of this paper is to introduce a presumably and remarkably altered integral operator involving the extended generalized Bessel-Maitland function. Particular properties are considered for the extended generalized Bessel-Maitland function connected with fractional integral and differential operators. The integral operator connected with operators of the fractional calculus is also observed. We point out important links to known findings from some individual cases with our key outcomes.

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8

Razzaghi, Mohsen. "Hybrid approximations for fractional calculus." ITM Web of Conferences 29 (2019): 01001. http://dx.doi.org/10.1051/itmconf/20192901001.

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In this paper, a numerical method for solving the fractional differential equations is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting ofblock-pulse functions and Taylor polynomials are presented. The Riemann-Liouville fractional integral operator for hybrid functions is given. This operator is then utilized to reduce the solution of the initial value problems for fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

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9

Ibrahim, Rabha W., and Hamid A. Jalab. "The Fractional Complex Step Method." Discrete Dynamics in Nature and Society 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/515973.

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It is well known that the complex step method is a tool that calculates derivatives by imposing a complex step in a strict sense. We extended the method by employing the fractional calculus differential operator in this paper. The fractional calculus can be taken in the sense of the Caputo operator, Riemann-Liouville operator, and so forth. Furthermore, we derived several approximations for computing the fractional order derivatives. Stability of the generalized fractional complex step approximations is demonstrated for an analytic test function.

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10

Faraj, Ahmad, Tariq Salim, Safaa Sadek, and Jamal Ismail. "Generalized Mittag-Leffler Function Associated with Weyl Fractional Calculus Operators." Journal of Mathematics 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/821762.

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This paper is devoted to study further properties of generalized Mittag-Leffler functionEα,β,pγ,δ,qassociated with Weyl fractional integral and differential operators. A new integral operatorℰα,β,p,w,∞γ,δ,qdepending on Weyl fractional integral operator and containingEα,β,pγ,δ,q(z)in its kernel is defined and studied, namely, its boundedness. Also, composition of Weyl fractional integral and differential operators with the new operatorℰα,β,p,w,∞γ,δ,qis established.

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11

Bas, Erdal, Resat Yilmazer, and Etibar Panakhov. "Fractional Solutions of Bessel Equation with -Method." Scientific World Journal 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/685695.

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This paper deals with the design fractional solution of Bessel equation. We obtain explicit solutions of the equation with the help of fractional calculus techniques. Using the -fractional calculus operator method, we derive the fractional solutions of the equation.

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12

Kukushkin, M. V. "ON SOME QUALITATIVE PROPERTIES OF THE OPERATOR OF FRACTIONAL DIFFERENTIATION IN KIPRIYANOV SENSE." Vestnik of Samara University. Natural Science Series 23, no. 2 (September 21, 2017): 32–43. http://dx.doi.org/10.18287/2541-7525-2017-23-2-32-43.

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In this paper we investigated the qualitative properties of the operator of fractional differentiation in Kipriyanov sense. Based on the concept of multidimensional generalization of operator of fractional differentiation in Marchaud sense we have adapted earlier known techniques of proof theorems of one-dimensional theory of fractional calculus for the operator of fractional differentiation in Kipriyanov sense. Along with the previously known definition of the fractional derivative in the direction we used a new definition of multidimensional fractional integral in the direction of allowing you to expand the domain of definition of formally adjoint operator. A number of theorems that have analogs in one-dimensional theory of fractional calculus is proved. In particular the sufficient conditions of representability of a fractional integral in the direction are received. Integral equality the result of which is the construction of the formal adjoint operator defined on the set of functions representable by the fractional integral in direction is proved.

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13

Tarasov, Vasily E. "General Fractional Calculus: Multi-Kernel Approach." Mathematics 9, no. 13 (June 26, 2021): 1501. http://dx.doi.org/10.3390/math9131501.

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For the first time, a general fractional calculus of arbitrary order was proposed by Yuri Luchko in 2021. In Luchko works, the proposed approaches to formulate this calculus are based either on the power of one Sonin kernel or the convolution of one Sonin kernel with the kernels of the integer-order integrals. To apply general fractional calculus, it is useful to have a wider range of operators, for example, by using the Laplace convolution of different types of kernels. In this paper, an extended formulation of the general fractional calculus of arbitrary order is proposed. Extension is achieved by using different types (subsets) of pairs of operator kernels in definitions general fractional integrals and derivatives. For this, the definition of the Luchko pair of kernels is somewhat broadened, which leads to the symmetry of the definition of the Luchko pair. The proposed set of kernel pairs are subsets of the Luchko set of kernel pairs. The fundamental theorems for the proposed general fractional derivatives and integrals are proved.

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14

Atici, Ferhan M., and Paul Eloe. "Discrete fractional calculus with the nabla operator." Electronic Journal of Qualitative Theory of Differential Equations, no. 3 (2009): 1–12. http://dx.doi.org/10.14232/ejqtde.2009.4.3.

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15

Ayant, F. Y. "Fractional Calculus Operator Associated with Wright's Function." International Journal of Mathematics Trends and Technology 57, no. 3 (May 25, 2018): 140–47. http://dx.doi.org/10.14445/22315373/ijmtt-v57p521.

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16

Chaurasia, V. B. L., and Vinod Gill. "New fractional calculus results involving Srivastava’s general class of multivariable polynomials and The H̅ - function." Journal of Applied Mathematics, Statistics and Informatics 11, no. 1 (May 1, 2015): 19–32. http://dx.doi.org/10.1515/jamsi-2015-0002.

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Abstract A significantly large number of earlier works on the subjects of fractional calculus give interesting account of the theory and applications of fractional calculus operators in many different areas of mathematical analysis. In the present paper, we study and develop an important result involving a fractional differential operator for the product of general multivariable polynomials, general polynomial set and two -functions. The result discussed here can be used to investigate a wide class of new and known results, hitherto scattered in the literature. For the sake of illustration, six interesting special case have also been recorded here of our main findings.

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17

Tadesse, Hagos, Haile Habenom, Anita Alaria, and Biniyam Shimelis. "Composition Formulae for the k -Fractional Calculus Operator with the S -Function." Journal of Mathematics 2021 (July 10, 2021): 1–12. http://dx.doi.org/10.1155/2021/7379820.

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In this study, the S-function is applied to Saigo’s k -fractional order integral and derivative operators involving the k -hypergeometric function in the kernel; outcomes are described in terms of the k -Wright function, which is used to represent image formulas of integral transformations such as the beta transform. Several special cases, such as the fractional calculus operator and the S -function, are also listed.

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18

Kumar, D., and F. Y. Ayant. "Fractional calculus pertaining to multivariable I-function defined by Prathima." Journal of Applied Mathematics, Statistics and Informatics 15, no. 2 (December 1, 2019): 61–73. http://dx.doi.org/10.2478/jamsi-2019-0009.

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Abstract In this paper, we study a pair of unified and extended fractional integral operator involving the multivariable I-functions and general class of multivariable polynomials. Here, we use Mellin transforms to obtain our main results. Certain properties of these operators concerning to their Mellin-transforms have been investigated. On account of the general nature of the functions involved herein, a large number of known (may be new also) fractional integral operators involved simpler functions can be obtained. We will also quote the particular case of the multivariable H-function.

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19

Tarasov, Vasily E. "General Fractional Dynamics." Mathematics 9, no. 13 (June 22, 2021): 1464. http://dx.doi.org/10.3390/math9131464.

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General fractional dynamics (GFDynamics) can be viewed as an interdisciplinary science, in which the nonlocal properties of linear and nonlinear dynamical systems are studied by using general fractional calculus, equations with general fractional integrals (GFI) and derivatives (GFD), or general nonlocal mappings with discrete time. GFDynamics implies research and obtaining results concerning the general form of nonlocality, which can be described by general-form operator kernels and not by its particular implementations and representations. In this paper, the concept of “general nonlocal mappings” is proposed; these are the exact solutions of equations with GFI and GFD at discrete points. In these mappings, the nonlocality is determined by the operator kernels that belong to the Sonin and Luchko sets of kernel pairs. These types of kernels are used in general fractional integrals and derivatives for the initial equations. Using general fractional calculus, we considered fractional systems with general nonlocality in time, which are described by equations with general fractional operators and periodic kicks. Equations with GFI and GFD of arbitrary order were also used to derive general nonlocal mappings. The exact solutions for these general fractional differential and integral equations with kicks were obtained. These exact solutions with discrete timepoints were used to derive general nonlocal mappings without approximations. Some examples of nonlocality in time are described.

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20

Ali, R. S., S. Mubeen, I. Nayab, Serkan Araci, G. Rahman, and K. S. Nisar. "Some Fractional Operators with the Generalized Bessel–Maitland Function." Discrete Dynamics in Nature and Society 2020 (July 24, 2020): 1–15. http://dx.doi.org/10.1155/2020/1378457.

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In this paper, we aim to determine some results of the generalized Bessel–Maitland function in the field of fractional calculus. Here, some relations of the generalized Bessel–Maitland functions and the Mittag-Leffler functions are considered. We develop Saigo and Riemann–Liouville fractional integral operators by using the generalized Bessel–Maitland function, and results can be seen in the form of Fox–Wright functions. We establish a new operator Zν,η,ρ,γ,w,a+μ,ξ,m,σϕ and its inverse operator Dν,η,ρ,γ,w,a+μ,ξ,m,σϕ, involving the generalized Bessel–Maitland function as its kernel, and also discuss its convergence and boundedness. Moreover, the Riemann–Liouville operator and the integral transform (Laplace) of the new operator have been developed.

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21

Razzaghi, M. "A numerical scheme for problems in fractional calculus." ITM Web of Conferences 20 (2018): 02001. http://dx.doi.org/10.1051/itmconf/20182002001.

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In this paper, a new numerical method for solving the fractional differential equations with boundary value problems is presented. The method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The Riemann-Liouville fractional integral operator for hybrid functions is given. This operator is then utilized to reduce the solution of the boundary value problems for fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

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22

Mohammed, Pshtiwan Othman, Thabet Abdeljawad, and Faraidun Kadir Hamasalh. "On Discrete Delta Caputo–Fabrizio Fractional Operators and Monotonicity Analysis." Fractal and Fractional 5, no. 3 (September 9, 2021): 116. http://dx.doi.org/10.3390/fractalfract5030116.

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The discrete delta Caputo-Fabrizio fractional differences and sums are proposed to distinguish their monotonicity analysis from the sense of Riemann and Caputo operators on the time scale Z. Moreover, the action of Q− operator and discrete delta Laplace transform method are also reported. Furthermore, a relationship between the discrete delta Caputo-Fabrizio-Caputo and Caputo-Fabrizio-Riemann fractional differences is also studied in detail. To better understand the dynamic behavior of the obtained monotonicity results, the fractional difference mean value theorem is derived. The idea used in this article is readily applicable to obtain monotonicity analysis of other discrete fractional operators in discrete fractional calculus.

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23

Oliveira, Daniela S., and Edmundo Capelas de Oliveira. "On a Caputo-type fractional derivative." Advances in Pure and Applied Mathematics 10, no. 2 (April 1, 2019): 81–91. http://dx.doi.org/10.1515/apam-2017-0068.

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Abstract In this paper, we present a new differential operator of arbitrary order defined by means of a Caputo-type modification of the generalized fractional derivative recently proposed by Katugampola. The generalized fractional derivative, when convenient limits are considered, recovers the Riemann–Liouville and the Hadamard derivatives of arbitrary order. Our differential operator recovers as limiting cases the arbitrary order derivatives proposed by Caputo and by Caputo–Hadamard. Some properties are presented as well as the relation between this differential operator of arbitrary order and the Katugampola generalized fractional operator. As an application we prove the fundamental theorem of fractional calculus associated with our operator.

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24

Yilmazer, Resat, and Karmina Ali. "On discrete fractional solutions of the hydrogen atom type equations." Thermal Science 23, Suppl. 6 (2019): 1935–41. http://dx.doi.org/10.2298/tsci190311354y.

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Discrete fractional calculus deals with sums and differences of arbitrary orders. In this study, we acquire new discrete fractional solutions of hydrogen atom type equations by using discrete fractional nabla operator ??(0 < ? < 1). This operator is applied homogeneous and non-homogeneous hydrogen atom type equations. So, we obtain many particular solutions of these equations.

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25

WOON, S. C. "ANALYTIC CONTINUATION OF OPERATORS APPLICATIONS: FROM NUMBER THEORY AND GROUP THEORY TO QUANTUM FIELD AND STRING THEORIES." Reviews in Mathematical Physics 11, no. 04 (April 1999): 463–501. http://dx.doi.org/10.1142/s0129055x99000179.

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We are used to thinking of an operator acting once, twice, and so on. However, an operator can be analytically continued to the operator raised to a complex power. Applications include (s,r) diagrams and an extension of Fractional Calculus where commutativity of fractional derivatives is preserved, generating integrals and non-standard derivations of theorems in Number Theory, non-integer power series and breaking of Leibniz and Chain rules, pseudo-groups and symmetry deforming models in particle physics and cosmology, non-local effect in analytically continued matrix representations and its connection with noncommutative geometry, particle-physics-like scatterings of zeros of analytically continued Bernoulli polynomials, and analytic continuation of operators in QM, QFT and Strings.

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26

Kukushkin, Maksim V. "Riemann–Liouville Operator in Weighted Lp Spaces via the Jacobi Series Expansion." Axioms 8, no. 2 (June 23, 2019): 75. http://dx.doi.org/10.3390/axioms8020075.

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In this paper, we use the orthogonal system of the Jacobi polynomials as a tool to study the Riemann–Liouville fractional integral and derivative operators on a compact of the real axis. This approach has some advantages and allows us to complete the previously known results of the fractional calculus theory by means of reformulating them in a new quality. The proved theorem on the fractional integral operator action is formulated in terms of the Jacobi series coefficients and is of particular interest. We obtain a sufficient condition for a representation of a function by the fractional integral in terms of the Jacobi series coefficients. We consider several modifications of the Jacobi polynomials, which gives us the opportunity to study the invariant property of the Riemann–Liouville operator. In this direction, we have shown that the fractional integral operator acting in the weighted spaces of Lebesgue square integrable functions has a sequence of the included invariant subspaces.

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27

Khalighi, Moein, Leila Eftekhari, Soleiman Hosseinpour, and Leo Lahti. "Three-Species Lotka-Volterra Model with Respect to Caputo and Caputo-Fabrizio Fractional Operators." Symmetry 13, no. 3 (February 25, 2021): 368. http://dx.doi.org/10.3390/sym13030368.

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In this paper, we apply the concept of fractional calculus to study three-dimensional Lotka-Volterra differential equations. We incorporate the Caputo-Fabrizio fractional derivative into this model and investigate the existence of a solution. We discuss the uniqueness of the solution and determine under what conditions the model offers a unique solution. We prove the stability of the nonlinear model and analyse the properties, considering the non-singular kernel of the Caputo-Fabrizio operator. We compare the stability conditions of this system with respect to the Caputo-Fabrizio operator and the Caputo fractional derivative. In addition, we derive a new numerical method based on the Adams-Bashforth scheme. We show that the type of differential operators and the value of orders significantly influence the stability of the Lotka-Volterra system and numerical results demonstrate that different fractional operator derivatives of the nonlinear population model lead to different dynamical behaviors.

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28

Ozturk, Okkes. "Discrete fractional solutions of radial Schrödinger equation for Makarov potential." Infinite Dimensional Analysis, Quantum Probability and Related Topics 20, no. 03 (September 2017): 1750019. http://dx.doi.org/10.1142/s0219025717500199.

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The fractional calculus that is a theory of integral and derivative with arbitrary order is an important subject of applied mathematics. This theory has extensive usage fields in science and engineering. Discrete mathematics, the study of finite structures, is one of the fastest growing areas in mathematics and optimization. Recently, many considerable scientific works were published on fractional calculus and discrete fractional calculus (DFC). The purpose of this paper is to obtain discrete fractional solutions of radial Schrödinger equation for Makarov potential by means of nabla DFC operator. Moreover, we introduce hypergeometric forms of these solutions.

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29

Acar, Nihan, and Ferhan Atici. "Exponential functions of discrete fractional calculus." Applicable Analysis and Discrete Mathematics 7, no. 2 (2013): 343–53. http://dx.doi.org/10.2298/aadm130828020a.

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In this paper, exponential functions of discrete fractional calculus with the nabla operator are studied. We begin with proving some properties of exponential functions along with some relations to the discrete Mittag-Leffler functions. We then study sequential linear difference equations of fractional order with constant coefficients. A corresponding characteristic equation is defined and considered in two cases where characteristic real roots are same or distinct. We define a generalized Casoratian for a set of discrete functions. As a consequence, for solutions of sequential linear difference equations, their nonzero Casoratian ensures their linear independence.

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30

Al-Masaeed, Mohamed, Eqab M. Rabei, Ahmed Al-Jamel, and Dumitru Baleanu. "Quantization of fractional harmonic oscillator using creation and annihilation operators." Open Physics 19, no. 1 (January 1, 2021): 395–401. http://dx.doi.org/10.1515/phys-2021-0035.

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Abstract In this article, the Hamiltonian for the conformable harmonic oscillator used in the previous study [Chung WS, Zare S, Hassanabadi H, Maghsoodi E. The effect of fractional calculus on the formation of quantum-mechanical operators. Math Method Appl Sci. 2020;43(11):6950–67.] is written in terms of fractional operators that we called α \alpha -creation and α \alpha -annihilation operators. It is found that these operators have the following influence on the energy states. For a given order α \alpha , the α \alpha -creation operator promotes the state while the α \alpha -annihilation operator demotes the state. The system is then quantized using these creation and annihilation operators and the energy eigenvalues and eigenfunctions are obtained. The eigenfunctions are expressed in terms of the conformable Hermite functions. The results for the traditional quantum harmonic oscillator are found to be recovered by setting α = 1 \alpha =1 .

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31

George Maria Selvam, A., and S. Britto Jacob. "STABILITY OF NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS IN THE FRAME OF ATANGANA-BALEANU OPERATOR." Advances in Mathematics: Scientific Journal 10, no. 5 (May 4, 2021): 2319–33. http://dx.doi.org/10.37418/amsj.10.5.3.

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Theory of fractional calculus with singular and non-singular kernels is pioneering and has garnered significant interest recently. Fair amount of literature on the qualitative properties of fractional differential and integral equations involving different types of operators is available. This manuscript aims to analyze the stability of a class of nonlinear fractional differential equation in terms of Atangana-Baleanu-Caputo operator. Sufficient conditions for the existence and uniqueness of solutions are obtained by employing classical fixed point theorems and Banach contraction principle. Also adequate conditions for Hyers-Ulam stability are established. To substantiate our analytic results, an example is provided with numerical simulation.

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32

Li, Qi, Muhammad Shoaib Saleem, Peiyu Yan, Muhammad Sajid Zahoor, and Muhammad Imran. "On Strongly Convex Functions via Caputo–Fabrizio-Type Fractional Integral and Some Applications." Journal of Mathematics 2021 (April 2, 2021): 1–10. http://dx.doi.org/10.1155/2021/6625597.

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The theory of convex functions plays an important role in the study of optimization problems. The fractional calculus has been found the best to model physical and engineering processes. The aim of this paper is to study some properties of strongly convex functions via the Caputo–Fabrizio fractional integral operator. In this paper, we present Hermite–Hadamard-type inequalities for strongly convex functions via the Caputo–Fabrizio fractional integral operator. Some new inequalities of strongly convex functions involving the Caputo–Fabrizio fractional integral operator are also presented. Moreover, we present some applications of the proposed inequalities to special means.

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33

EL-NABULSI, AHMAD RAMI. "THE FRACTIONAL CALCULUS OF VARIATIONS FROM EXTENDED ERDÉLYI-KOBER OPERATOR." International Journal of Modern Physics B 23, no. 16 (June 30, 2009): 3349–61. http://dx.doi.org/10.1142/s0217979209052923.

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Fractional calculus of variations (FCV) has recently attracted considerable attention as it is deeply related to the fractional quantization procedure. In this work, the FCV from extended Erdélyi-Kober fractional integral is constructed. Our main goal is to exhibit a general treatment for dissipative systems, in particular the harmonic oscillator (HO) that has time-dependent mass and time-dependent frequency. The general linear equation of damped Erdélyi-Kober harmonic oscillator is constructed from which a time-dependent mass generalized law was derived exhibiting different types of behavior. This relatively new time-dependent mass law permits us to point out several possible cases simultaneously in contrast to many models discussed in the literature and without making use of any types of fractional derivatives. Some results on Hamiltonian part, namely Hamilton equations for the damped HO were obtained and discussed in detail.

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34

Tarasov, Vasily E. "Generalized Memory: Fractional Calculus Approach." Fractal and Fractional 2, no. 4 (September 24, 2018): 23. http://dx.doi.org/10.3390/fractalfract2040023.

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The memory means an existence of output (response, endogenous variable) at the present time that depends on the history of the change of the input (impact, exogenous variable) on a finite (or infinite) time interval. The memory can be described by the function that is called the memory function, which is a kernel of the integro-differential operator. The main purpose of the paper is to answer the question of the possibility of using the fractional calculus, when the memory function does not have a power-law form. Using the generalized Taylor series in the Trujillo-Rivero-Bonilla (TRB) form for the memory function, we represent the integro-differential equations with memory functions by fractional integral and differential equations with derivatives and integrals of non-integer orders. This allows us to describe general economic dynamics with memory by the methods of fractional calculus. We prove that equation of the generalized accelerator with the TRB memory function can be represented by as a composition of actions of the accelerator with simplest power-law memory and the multi-parametric power-law multiplier. As an example of application of the suggested approach, we consider a generalization of the Harrod-Domar growth model with continuous time.

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35

Mohammed, Aabed, Maslina Darus, and Daniel Breaz. "Fractional Calculus for Certain Integral Operator Involving Logarithmic Coefficients." Journal of Mathematics and Statistics 5, no. 2 (February 1, 2009): 118–22. http://dx.doi.org/10.3844/jms2.2009.118.122.

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36

Mohammed. "Fractional Calculus for Certain Integral Operator Involving Logarithmic Coefficients." Journal of Mathematics and Statistics 5, no. 2 (February 1, 2009): 118–22. http://dx.doi.org/10.3844/jmssp.2009.118.122.

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37

Singh, Yudhveer, and Ravi Shanker Dubey. "Fractional calculus operator with generalize k-Mittag-Leffler function." Journal of Interdisciplinary Mathematics 23, no. 2 (February 17, 2020): 545–53. http://dx.doi.org/10.1080/09720502.2020.1731971.

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38

Yilmazer, Resat. "Discrete fractional solution of a nonhomogeneous non-Fuchsian differential equations." Thermal Science 23, Suppl. 1 (2019): 121–27. http://dx.doi.org/10.2298/tsci180917336y.

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In this article, we also present new fractional solutions of the non-homogeneous and homogeneous non-Fuchsian differential equation by using nabla-discrete fractional calculus operator ??(0 < ? < 1). So, we acquire new solution of these equation in the discrete fractional form via a newly developed method.

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Behr, Nicolas, Giuseppe Dattoli, and Ambra Lattanzi. "Operator Ordering and Solution of Pseudo-Evolutionary Equations." Axioms 8, no. 1 (March 16, 2019): 35. http://dx.doi.org/10.3390/axioms8010035.

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The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary differential equation setting. A combination of techniques, involving procedures of umbral and of operational nature, has been demonstrated to be a very promising tool in order to approach within a unifying context non-canonical evolution problems. This article covers the extension of this approach to the solution of pseudo-evolutionary equations. We will comment on the explicit formulation of the necessary techniques, which are based on certain time- and operator ordering tools. We will in particular demonstrate how Volterra-Neumann expansions, Feynman-Dyson series and other popular tools can be profitably extended to obtain solutions for fractional differential equations. We apply the method to several examples, in which fractional calculus and a certain umbral image calculus play a role of central importance.

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Rashid, Saima, Aasma Khalid, Gauhar Rahman, Kottakkaran Sooppy Nisar, and Yu-Ming Chu. "On New Modifications Governed by Quantum Hahn’s Integral Operator Pertaining to Fractional Calculus." Journal of Function Spaces 2020 (July 14, 2020): 1–12. http://dx.doi.org/10.1155/2020/8262860.

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In the article, we present several generalizations for the generalized Čebyšev type inequality in the frame of quantum fractional Hahn’s integral operator by using the quantum shift operator σΨqς=qς+1−qσς∈l1,l2,σ=l1+ω/1−q,0<q<1,ω≥0. As applications, we provide some associated variants to illustrate the efficiency of quantum Hahn’s integral operator and compare our obtained results and proposed technique with the previously known results and existing technique. Our ideas and approaches may lead to new directions in fractional quantum calculus theory.

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El-Ashwah, R. M., M. K. Aouf, A. A. M. Hassan, and A. H. Hassan. "A New Class of Analytic Functions Defined by Using Salagean Operator." International Journal of Analysis 2013 (February 5, 2013): 1–10. http://dx.doi.org/10.1155/2013/153128.

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We derive some results for a new class of analytic functions defined by using Salagean operator. We give some properties of functions in this class and obtain numerous sharp results including for example, coefficient estimates, distortion theorem, radii of star-likeness, convexity, close-to-convexity, extreme points, integral means inequalities, and partial sums of functions belonging to this class. Finally, we give an application involving certain fractional calculus operators that are also considered.

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Yilmazer, Resat, and Okkes Ozturk. "On nabla discrete fractional calculus operator for a modified Bessel equation." Thermal Science 22, Suppl. 1 (2018): 203–9. http://dx.doi.org/10.2298/tsci170614287y.

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In thermal sciences, it is possible to encounter topics such as Bessel beams, Bessel functions or Bessel equations. In this work, we also present new discrete fractional solutions of the modified Bessel differential equation by means of the nabla-discrete fractional calculus operator. We consider homogeneous and non-homogeneous modified Bessel differential equation. So, we acquire four new solutions of these equations in the discrete fractional forms via a newly developed method

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Alarifi, Najla M., and Rabha W. Ibrahim. "Analytic Normalized Solutions of 2D Fractional Saint-Venant Equations of a Complex Variable." Journal of Function Spaces 2021 (September 10, 2021): 1–11. http://dx.doi.org/10.1155/2021/4797955.

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Saint-Venant equations describe the flow below a pressure surface in a fluid. We aim to generalize this class of equations using fractional calculus of a complex variable. We deal with a fractional integral operator type Prabhakar operator in the open unit disk. We formulate the extended operator in a linear convolution operator with a normalized function to study some important geometric behaviors. A class of integral inequalities is investigated involving special functions. The upper bound of the suggested operator is computed by using the Fox-Wright function, for a class of convex functions and univalent functions. Moreover, as an application, we determine the upper bound of the generalized fractional 2-dimensional Saint-Venant equations (2D-SVE) of diffusive wave including the difference of bed slope.

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Nolte, Bodo, Sigmar Kempfle, and Ingo Schäfer. "Does a Real Material Behave Fractionally? Applications of Fractional Differential Operators to the Damped Structure Borne Sound in Viscoelastic Solids." Journal of Computational Acoustics 11, no. 03 (September 2003): 451–89. http://dx.doi.org/10.1142/s0218396x03002024.

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Time dependent analysis of the dynamic damped behavior of continua are mathematically modelled by partial differential equations. One obtains uniqueness, existence and stability (well posed problems) by the implementation of the correct initial boundary conditions. However, by taking memory effects into consideration, any change in the past of the system changes the future dynamic behavior. Classical damping descriptions fail when describing the behavior of many materials, like teflon. This is because in classical theory the operators are local ones. The implementation of fractional time derivatives into the partial differential equations is an alternative technique to overcome these problems. Thereby the time derivative operator is a global one, memory effects in structure borne sound can be calculated. In this paper the theory of fractional time derivative operators and their application in continuum mechanics is presented. The main result when using this method for damping behavior is that a global operator is needed which takes the whole history into account. We call this theory the functional calculus method instead of the well-known fractional calculus with the use of initial conditions. In order to show the efficiency of this method the calculated impulse response of a viscoelastic rod is compared with measurements. It is shown that the damping behavior is described much better than by other models with comparably few parameters. Moreover, it is the only one that works in a wide frequency range and can describe the dispersion of the resonance frequencies. The implementation of this damping description in a Boundary Element Code is an application of dynamics of 3D continua in the frequency domain.

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Stamov, Gani, and Ivanka Stamova. "Impulsive Fractional Differential Inclusions and Almost Periodic Waves." Mathematics 9, no. 12 (June 18, 2021): 1413. http://dx.doi.org/10.3390/math9121413.

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In the present paper, the concept of almost periodic waves is introduced to discontinuous impulsive fractional inclusions involving Caputo fractional derivative. New results on the existence and uniqueness are established by using the theory of operator semigroups, Hausdorff measure of noncompactness, fixed point theorems and fractional calculus techniques. Applications to a class of fractional-order impulsive gene regulatory network (GRN) models are proposed to illustrate the results.

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Rashid, Saima, Aasma Khalid, Omar Bazighifan, and Georgia Irina Oros. "New Modifications of Integral Inequalities via ℘-Convexity Pertaining to Fractional Calculus and Their Applications." Mathematics 9, no. 15 (July 26, 2021): 1753. http://dx.doi.org/10.3390/math9151753.

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Integral inequalities for ℘-convex functions are established by using a generalised fractional integral operator based on Raina’s function. Hermite–Hadamard type inequality is presented for ℘-convex functions via generalised fractional integral operator. A novel parameterized auxiliary identity involving generalised fractional integral is proposed for differentiable mappings. By using auxiliary identity, we derive several Ostrowski type inequalities for functions whose absolute values are ℘-convex mappings. It is presented that the obtained outcomes exhibit classical convex and harmonically convex functions which have been verified using Riemann–Liouville fractional integral. Several generalisations and special cases are carried out to verify the robustness and efficiency of the suggested scheme in matrices and Fox–Wright generalised hypergeometric functions.

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Atici, Ferhan, and Meltem Uyanik. "Analysis of discrete fractional operators." Applicable Analysis and Discrete Mathematics 9, no. 1 (2015): 139–49. http://dx.doi.org/10.2298/aadm150218007a.

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In this paper, we introduce two new monotonicity concepts for a nonnegative or nonpositive valued function defined on a discrete domain. We give examples to illustrate connections between these new monotonicity concepts and the traditional ones. We then prove some monotonicity criteria based on the sign of the fractional difference operator of a function f, ??f with 0 < ? < 1. As an application, we state and prove the mean value theorem on discrete fractional calculus.

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HAMEED, Mustafa Ibrahim. "Certain Subclass of Univalent Functions Involving Fractional Q-Calculus Operator." JOURNAL OF ADVANCES IN MATHEMATICS 13, no. 4 (November 10, 2017): 7370–78. http://dx.doi.org/10.24297/jam.v13i4.6442.

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The main object of the present paper is to introduce certain subclass of univalent function associated with the concept of differential subordination. We studied some geometric properties like coefficient inequality and nieghbourhood property, the Hadamard product properties and integral operator mean inequality.

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Li, Ming, and Wei Zhao. "Solving Abel’s Type Integral Equation with Mikusinski’s Operator of Fractional Order." Advances in Mathematical Physics 2013 (2013): 1–4. http://dx.doi.org/10.1155/2013/806984.

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This paper gives a novel explanation of the integral equation of Abel’s type from the point of view of Mikusinski’s operational calculus. The concept of the inverse of Mikusinski’s operator of fractional order is introduced for constructing a representation of the solution to the integral equation of Abel’s type. The proof of the existence of the inverse of the fractional Mikusinski operator is presented, providing an alternative method of treating the integral equation of Abel’s type.

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Ma, Xiao, Xiao-Bao Shu, and Jianzhong Mao. "Existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay." Stochastics and Dynamics 20, no. 01 (June 26, 2019): 2050003. http://dx.doi.org/10.1142/s0219493720500033.

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In this paper, we investigate the existence of almost periodic solutions for fractional impulsive neutral stochastic differential equations with infinite delay in Hilbert space. The main conclusion is obtained by using fractional calculus, operator semigroup and fixed point theorem. In the end, we give an example to illustrate our main results.

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