Статті в журналах: "Fractional derivatives at zero"

1

Bouzeffour, Fethi. "Advancing Fractional Riesz Derivatives through Dunkl Operators." Mathematics 11, no. 19 (September 25, 2023): 4073. http://dx.doi.org/10.3390/math11194073.

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The aim of this work is to introduce a novel concept, Riesz–Dunkl fractional derivatives, within the context of Dunkl-type operators. A particularly noteworthy revelation is that when a specific parameter κ equals zero, the Riesz–Dunkl fractional derivative smoothly reduces to both the well-known Riesz fractional derivative and the fractional second-order derivative. Furthermore, we introduce a new concept: the fractional Sobolev space. This space is defined and characterized using the versatile framework of the Dunkl transform.

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2

HILFER, R. "FOUNDATIONS OF FRACTIONAL DYNAMICS." Fractals 03, no. 03 (September 1995): 549–56. http://dx.doi.org/10.1142/s0218348x95000485.

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Time flow in dynamical systems is reconsidered in the ultralong time limit. The ultralong time limit is a limit in which a discretized time flow is iterated infinitely often and the discretization time step is infinite. The new limit is used to study induced flows in ergodic theory, in particular for subsets of measure zero. Induced flows on subsets of measure zero require an infinite renormalization of time in the ultralong time limit. It is found that induced flows are given generically by stable convolution semigroups and not by the conventional translation groups. This could give new insight into the origin of macroscopic irreversibility. Moreover, the induced semigroups are generated by fractional time derivatives of orders less than unity, and not by a first order time derivative. Invariance under the induced semiflows therefore leads to a new form of stationarity, called fractional stationarity. Fractionally stationary states are dissipative. Fractional stationarity also provides the dynamical foundation for a previously proposed generalized equilibrium concept.

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3

Farr, Ricky E., Sebastian Pauli, and Filip Saidak. "zero-free region for the fractional derivatives of the Riemann zeta function." New Zealand Journal of Mathematics 50 (September 4, 2020): 1–9. http://dx.doi.org/10.53733/42.

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For any , we denote by the α-th Grunwald-Letnikov fractional derivative of the Riemann zeta function ζ(s). For these derivatives we show: inside the region | s − 1 | < 1. This result, the first of its kind, is proved by a careful analysis of integrals involving Bernoulli polynomials and bounds for fractional Stieltjes constants.

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4

NABER, MARK. "DISTRIBUTED ORDER FRACTIONAL SUB-DIFFUSION." Fractals 12, no. 01 (March 2004): 23–32. http://dx.doi.org/10.1142/s0218348x04002410.

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A distributed order fractional diffusion equation is considered. Distributed order derivatives are fractional derivatives that have been integrated over the order of the derivative within a given range. In this paper, sub-diffusive cases are considered. That is, the order of the time derivative ranges from zero to one. The equation is solved for Dirichlet, Neumann and Cauchy boundary conditions. The time dependence for each of the three cases is found to be a functional of the diffusion parameter. This functional is shown to have decay properties. Upper and lower bounds are computed for the functional. Examples are also worked out for comparative decay rates.

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5

Agarwal, Ravi P., Snezhana Hristova, and Donal O’Regan. "Asymptotic Behavior of Delayed Reaction-Diffusion Neural Networks Modeled by Generalized Proportional Caputo Fractional Partial Differential Equations." Fractal and Fractional 7, no. 1 (January 11, 2023): 80. http://dx.doi.org/10.3390/fractalfract7010080.

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In this paper, a delayed reaction-diffusion neural network model of fractional order and with several constant delays is considered. Generalized proportional Caputo fractional derivatives with respect to the time variable are applied, and this type of derivative generalizes several known types in the literature for fractional derivatives such as the Caputo fractional derivative. Thus, the obtained results additionally generalize some known models in the literature. The long term behavior of the solution of the model when the time is increasing without a bound is studied and sufficient conditions for approaching zero are obtained. Lyapunov functions defined as a sum of squares with their generalized proportional Caputo fractional derivatives are applied and a comparison result for a scalar linear generalized proportional Caputo fractional differential equation with several constant delays is presented. Lyapunov functions and the comparison principle are then combined to establish our main results.

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6

Diethelm, Kai, Roberto Garrappa, Andrea Giusti, and Martin Stynes. "Why fractional derivatives with nonsingular kernels should not be used." Fractional Calculus and Applied Analysis 23, no. 3 (June 25, 2020): 610–34. http://dx.doi.org/10.1515/fca-2020-0032.

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AbstractIn recent years, many papers discuss the theory and applications of new fractional-order derivatives that are constructed by replacing the singular kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e., bounded) kernel. It will be shown here, through rigorous mathematical reasoning, that these non-singular kernel derivatives suffer from several drawbacks which should forbid their use. They fail to satisfy the fundamental theorem of fractional calculus since they do not admit the existence of a corresponding convolution integral of which the derivative is the left-inverse; and the value of the derivative at the initial time t = 0 is always zero, which imposes an unnatural restriction on the differential equations and models where these derivatives can be used. For the particular cases of the so-called Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this restriction holds the derivative can be simply expressed in terms of integer derivatives and standard Caputo fractional derivatives, thus demonstrating that these derivatives contain nothing new.

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7

Luchko, Yuri. "General Fractional Integrals and Derivatives with the Sonine Kernels." Mathematics 9, no. 6 (March 10, 2021): 594. http://dx.doi.org/10.3390/math9060594.

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In this paper, we address the general fractional integrals and derivatives with the Sonine kernels on the spaces of functions with an integrable singularity at the point zero. First, the Sonine kernels and their important special classes and particular cases are discussed. In particular, we introduce a class of the Sonine kernels that possess an integrable singularity of power function type at the point zero. For the general fractional integrals and derivatives with the Sonine kernels from this class, two fundamental theorems of fractional calculus are proved. Then, we construct the n-fold general fractional integrals and derivatives and study their properties.

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8

Mugbil, Ahmad, and Nasser-Eddine Tatar. "Hadamard-Type Fractional Integro-Differential Problem: A Note on Some Asymptotic Behavior of Solutions." Fractal and Fractional 6, no. 5 (May 15, 2022): 267. http://dx.doi.org/10.3390/fractalfract6050267.

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As a follow-up to the inherent nature of Hadamard-Type Fractional Integro-differential problem, little is known about some asymptotic behaviors of solutions. In this paper, an integro-differential problem involving Hadamard fractional derivatives is investigated. The leading derivative is of an order between one and two whereas the nonlinearities may contain fractional derivatives of an order between zero and one as well as some non-local terms. Under some reasonable conditions, we prove that solutions are asymptotic to logarithmic functions. Our approach is based on a generalized version of Bihari–LaSalle inequality, which we prove. In addition, several manipulations and crucial estimates have been used. An example supporting our findings is provided.

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9

Prodanov, Dimiter. "Generalized Differentiability of Continuous Functions." Fractal and Fractional 4, no. 4 (December 10, 2020): 56. http://dx.doi.org/10.3390/fractalfract4040056.

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Many physical phenomena give rise to mathematical models in terms of fractal, non-differentiable functions. The paper introduces a broad generalization of the derivative in terms of the maximal modulus of continuity of the primitive function. These derivatives are called indicial derivatives. As an application, the indicial derivatives are used to characterize the nowhere monotonous functions. Furthermore, the non-differentiability set of such derivatives is proven to be of measure zero. As a second application, the indicial derivative is used in the proof of the Lebesgue differentiation theorem. Finally, the connection with the fractional velocities is demonstrated.

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10

Area, I., J. Losada, and J. J. Nieto. "On Fractional Derivatives and Primitives of Periodic Functions." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/392598.

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11

Tarasov, Vasily E. "Fractional Dynamics with Depreciation and Obsolescence: Equations with Prabhakar Fractional Derivatives." Mathematics 10, no. 9 (May 4, 2022): 1540. http://dx.doi.org/10.3390/math10091540.

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In economics, depreciation functions (operator kernels) are certain decreasing functions, which are assumed to be equal to unity at zero. Usually, an exponential function is used as a depreciation function. However, exponential functions in operator kernels do not allow simultaneous consideration of memory effects and depreciation effects. In this paper, it is proposed to consider depreciation of a non-exponential type, and simultaneously take into account memory effects by using the Prabhakar fractional derivatives and integrals. Integro-differential operators with the Prabhakar (generalized Mittag-Leffler) function in the kernels are considered. The important distinguishing features of the Prabhakar function in operator kernels, which allow us to take into account non-exponential depreciation and fading memory in economics, are described. In this paper, equations with the following operators are considered: (a) the Prabhakar fractional integral, which contains the Prabhakar function as the kernels; (b) the Prabhakar fractional derivative of Riemann–Liouville type proposed by Kilbas, Saigo, and Saxena in 2004, which is left inverse for the Prabhakar fractional integral; and (c) the Prabhakar operator of Caputo type proposed by D’Ovidio and Polito, which is also called the regularized Prabhakar fractional derivative. The solutions of fractional differential equations with the Prabhakar operator and its special cases are suggested. The asymptotic behavior of these solutions is discussed.

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12

Albidah, Abdulrahman B. "Application of Riemann–Liouville Derivatives on Second-Order Fractional Differential Equations: The Exact Solution." Fractal and Fractional 7, no. 12 (November 28, 2023): 843. http://dx.doi.org/10.3390/fractalfract7120843.

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This paper applies two different types of Riemann–Liouville derivatives to solve fractional differential equations of second order. Basically, the properties of the Riemann–Liouville fractional derivative depend mainly on the lower bound of the integral involved in the Riemann–Liouville fractional definition. The Riemann–Liouville fractional derivative of first type considers the lower bound as a zero while the second type applies negative infinity as a lower bound. Due to the differences in properties of the two operators, two different solutions are obtained for the present two classes of fractional differential equations under appropriate initial conditions. It is shown that the zeroth lower bound implies implicit solutions in terms of the Mittag–Leffler functions while explicit solutions are derived when negative infinity is taken as a lower bound. Such explicit solutions are obtained for the current two classes in terms of trigonometric and hyperbolic functions. Some theoretical results are introduced to facilitate the solutions procedures. Moreover, the characteristics of the obtained solutions are discussed and interpreted.

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13

Luchko, Yuri. "General Fractional Integrals and Derivatives of Arbitrary Order." Symmetry 13, no. 5 (April 27, 2021): 755. http://dx.doi.org/10.3390/sym13050755.

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In this paper, we introduce the general fractional integrals and derivatives of arbitrary order and study some of their basic properties and particular cases. First, a suitable generalization of the Sonine condition is presented, and some important classes of the kernels that satisfy this condition are introduced. Whereas the kernels of the general fractional derivatives of arbitrary order possess integrable singularities at the point zero, the kernels of the general fractional integrals can—depending on their order—be both singular and continuous at the origin. For the general fractional integrals and derivatives of arbitrary order with the kernels introduced in this paper, two fundamental theorems of fractional calculus are formulated and proved.

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14

Zhang, Xiaoping, and Yongping Sun. "Monotone Iterative Methods of Positive Solutions for Fractional Differential Equations Involving Derivatives." Mathematical Problems in Engineering 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/254012.

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This paper studies the existence and computing method of positive solutions for a class of nonlinear fractional differential equations involving derivatives with two-point boundary conditions. By applying monotone iterative methods, the existence results of positive solutions and two iterative schemes approximating the solutions are established. The interesting point of our method is that the iterative scheme starts off with a known simple function or the zero function and the nonlinear term in the fractional differential equation is allowed to depend on the unknown function together with derivative terms. Two explicit numerical examples are given to illustrate the results.

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15

Gorial, Iman I. "Numerical Simulation for Fractional Percolation Equation." Mathematical Modelling of Engineering Problems 8, no. 3 (June 24, 2021): 425–30. http://dx.doi.org/10.18280/mmep.080312.

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The aims of this paper are to propose approach of explicit finite difference mathod (EFDM), clarify the problem the mixed fractional derivative in one-dimensional fractional percolation equation (O-DFPE), and the study of consistency, stability, and convergence methods. Use of estimated Grunwald estimation in the analysis of mixed fractional derivatives. However, the given method is successfully applied to the mixed fractional derivative classes with the initial condition (IC) and derivative boundary conditions (DBC). To illustrate the efficiency and validity of the proposed algorithm, examples are given and the results are compared with the exact solution. From the figures shown for the examples in this work, the approximate solution values given by the EFDM for the various grid points are equivalent to the exact solution values with high-precision approximation. To show the effectiveness of the proposed method, where the error between the EFDM and the exact method is zero, the fractional derivative was used with various and random values. Using the package MATLAB and MathCAD 12 Figures were introduced.

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16

Salem, Ahmed, Lamya Almaghamsi, and Faris Alzahrani. "An Infinite System of Fractional Order with p-Laplacian Operator in a Tempered Sequence Space via Measure of Noncompactness Technique." Fractal and Fractional 5, no. 4 (October 25, 2021): 182. http://dx.doi.org/10.3390/fractalfract5040182.

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In the current study, a new class of an infinite system of two distinct fractional orders with p-Laplacian operator is presented. Our mathematical model is introduced with the Caputo–Katugampola fractional derivative which is considered a generalization to the Caputo and Hadamard fractional derivatives. In a new sequence space associated with a tempered sequence and the sequence space c0 (the space of convergent sequences to zero), a suitable new Hausdorff measure of noncompactness form is provided. This formula is applied to discuss the existence of a solution to our infinite system through applying Darbo’s theorem which extends both the classical Banach contraction principle and the Schauder fixed point theorem.

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17

Luchko, Yuri. "Operational calculus for the general fractional derivative and its applications." Fractional Calculus and Applied Analysis 24, no. 2 (April 1, 2021): 338–75. http://dx.doi.org/10.1515/fca-2021-0016.

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Abstract In this paper, we first address the general fractional integrals and derivatives with the Sonine kernels that possess the integrable singularities of power function type at the point zero. Both particular cases and compositions of these operators are discussed. Then we proceed with a construction of an operational calculus of the Mikusiński type for the general fractional derivatives with the Sonine kernels. This operational calculus is applied for analytical treatment of some initial value problems for the fractional differential equations with the general fractional derivatives. The solutions are expressed in form of the convolution series that generalize the power series for the exponential and the Mittag-Leffler functions.

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18

Tarasov, Vasily E. "Local Fractional Derivatives of Differentiable Functions are Integer-order Derivatives or Zero." International Journal of Applied and Computational Mathematics 2, no. 2 (April 18, 2015): 195–201. http://dx.doi.org/10.1007/s40819-015-0054-6.

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19

Zhang, Xianmin, Xianzhen Zhang, Zuohua Liu, Wenbin Ding, Hui Cao, and Tong Shu. "On the General Solution of Impulsive Systems with Hadamard Fractional Derivatives." Mathematical Problems in Engineering 2016 (2016): 1–12. http://dx.doi.org/10.1155/2016/2814310.

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This paper is concerned with the solution for impulsive differential equations with Hadamard fractional derivatives. The general solution of this impulsive fractional system is found by considering the limit case in which impulses approach zero. Next, an example is provided to expound the theoretical result.

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20

Alharthi, Nadiyah Hussain, Abdon Atangana, and Badr S. Alkahtani. "Numerical analysis of some partial differential equations with fractal-fractional derivative." AIMS Mathematics 8, no. 1 (2022): 2240–56. http://dx.doi.org/10.3934/math.2023116.

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<abstract> In this study, we expanded the partial differential equation framework to which fractal-fractional differentiation can be applied. For this, we employed the generalized Mittag-Leffler function, and the fractal-fractional derivatives based on the power-law kernel. A general partial differential equation with the fractal-fractional derivative, the power law kernel and the generalized Mittag-Leffler function was thoroughly examined. There is almost no numerical scheme for solving partial differential equations with fractal-fractional derivatives, as less investigation has been done in this direction in the last decades. In this work, therefore, we shall attempt to provide a numerical method that might be used to solve these equations in each circumstance. The heat equation was taken into consideration for the application and numerically solved using a few simulations for various values of fractional and fractal orders. It is observed that, when the fractal order is 1, one obtains fractional partial differential equations which have been known to replicate nonlocal behaviors. Meanwhile, if the fractional order is 1, one obtains fractal-partial differential equations. Thus, when the fractional order and fractal dimension are different from zero, nonlocal processes with similar features are developed. </abstract>

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21

Hristov, Jordan. "Double integral-balance method to the fractional subdiffusion equation: Approximate solutions, optimization problems to be resolved and numerical simulations." Journal of Vibration and Control 23, no. 17 (December 27, 2015): 2795–818. http://dx.doi.org/10.1177/1077546315622773.

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An approximate integral-balance solution of the fractional subdiffusion equation by a double-integration technique has been conceived. The time-fractional linear subdiffusion equation with Dirichlet boundary condition (and zero initial condition) has been chosen as a test example. Approximations of time-fractional Riemann-Liouville and Caputo derivatives when the distribution is assumed as a parabolic profile with unspecified exponent have been developed. Problems pertinent to determination of the optimal exponent of the parabolic profile and approximations of the time-fractional derivative of by different approaches have been formulated. Solved and unresolved problems in determination of the optimal exponents have been demonstrated. Examples with predetermined quadratic and cubic assumed profiles are analyzed, too. Comparative numerical studies with exact solutions expressed by the Mainardi function in terms of a similarity variable have been performed.

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22

Zhang, Xianmin, Wenbin Ding, Hui Peng, Zuohua Liu, and Tong Shu. "The general solution of impulsive systems with Riemann-Liouville fractional derivatives." Open Mathematics 14, no. 1 (January 1, 2016): 1125–37. http://dx.doi.org/10.1515/math-2016-0096.

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AbstractIn this paper, we study a kind of fractional differential system with impulsive effect and find the formula of general solution for the impulsive fractional-order system by analysis of the limit case (as impulse tends to zero). The obtained result shows that the deviation caused by impulses for fractional-order system is undetermined. An example is also provided to illustrate the result.

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23

Hanna, Latif A.-M., Maryam Al-Kandari, and Yuri Luchko. "Operational method for solving fractional differential equations with the left-and right-hand sided Erdélyi-Kober fractional derivatives." Fractional Calculus and Applied Analysis 23, no. 1 (February 25, 2020): 103–25. http://dx.doi.org/10.1515/fca-2020-0004.

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AbstractIn this paper, we first provide a survey of some basic properties of the left-and right-hand sided Erdélyi-Kober fractional integrals and derivatives and introduce their compositions in form of the composed Erdélyi-Kober operators. Then we derive a convolutional representation for the composed Erdélyi-Kober fractional integral in terms of its convolution in the Dimovski sense. For this convolution, we also determine the divisors of zero. These both results are then used for construction of an operational method for solving an initial value problem for a fractional differential equation with the left-and right-hand sided Erdélyi-Kober fractional derivatives defined on the positive semi-axis. Its solution is obtained in terms of the four-parameters Wright function of the second kind. The same operational method can be employed for other fractional differential equation with the left-and right-hand sided Erdélyi-Kober fractional derivatives.

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24

Luchko, Yuri. "Special Functions of Fractional Calculus in the Form of Convolution Series and Their Applications." Mathematics 9, no. 17 (September 2, 2021): 2132. http://dx.doi.org/10.3390/math9172132.

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In this paper, we first discuss the convolution series that are generated by Sonine kernels from a class of functions continuous on a real positive semi-axis that have an integrable singularity of power function type at point zero. These convolution series are closely related to the general fractional integrals and derivatives with Sonine kernels and represent a new class of special functions of fractional calculus. The Mittag-Leffler functions as solutions to the fractional differential equations with the fractional derivatives of both Riemann-Liouville and Caputo types are particular cases of the convolution series generated by the Sonine kernel κ(t)=tα−1/Γ(α),0<α<1. The main result of the paper is the derivation of analytic solutions to the single- and multi-term fractional differential equations with the general fractional derivatives of the Riemann-Liouville type that have not yet been studied in the fractional calculus literature.

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25

Kleiner, Tillmann, and Rudolf Hilfer. "Sequential generalized Riemann–Liouville derivatives based on distributional convolution." Fractional Calculus and Applied Analysis 25, no. 1 (February 2022): 267–98. http://dx.doi.org/10.1007/s13540-021-00012-0.

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AbstractSequential generalized fractional Riemann–Liouville derivatives are introduced as composites of distributional derivatives on the right half axis and partially defined operators, called Dirac-function removers, that remove the component of singleton support at the origin of distributions that are of order zero on a neighborhood of the origin. The concept of Dirac-function removers allows to formulate generalized initial value problems with less restrictions on the orders and types than previous approaches to sequential fractional derivatives. The well-posedness of these initial value problems and the structure of their solutions are studied.

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26

Malik, Amita, та Arindam Roy. "On the distribution of zeros of derivatives of the Riemann ξ-function". Forum Mathematicum 32, № 1 (1 січня 2020): 1–22. http://dx.doi.org/10.1515/forum-2018-0081.

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AbstractFor the completed Riemann zeta function {\xi(s)}, it is known that the Riemann hypothesis for {\xi(s)} implies the Riemann hypothesis for {\xi^{(m)}(s)}, where m is any positive integer. In this paper, we investigate the distribution of the fractional parts of the sequence {(\alpha\gamma_{m})}, where α is any fixed non-zero real number and {\gamma_{m}} runs over the imaginary parts of the zeros of {\xi^{(m)}(s)}. We also obtain a zero density estimate and an explicit formula for the zeros of {\xi^{(m)}(s)}. In particular, all our results hold uniformly for {0\leq m\leq g(T)}, where the function {g(T)} tends to infinity with T and {g(T)=o(\log\log T)}.

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27

Schmitt, Joseph M. "Fractional Derivative Analysis of Diffuse Reflectance Spectra." Applied Spectroscopy 52, no. 6 (June 1998): 840–46. http://dx.doi.org/10.1366/0003702981944580.

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Fractional differentiation is introduced as a mathematical tool for analysis of diffuse-reflectance spectra. The quantity —log10 e/R dqR/ dλ q, where R is the measured reflectance and q is a real number greater than zero, is defined and shown to have properties analogous to those of the integer-order derivatives of log10(1/ R) that are commonly employed in near-infrared spectroscopy. Like conventional derivative spectroscopy, fractional derivative spectroscopy (FDS) is effective for reducing baseline variations and separating overlapping peaks. FDS has the additional benefit that it enables the user to control the weight given to the slope and curvature of spectral features and, therefore, provides greater flexibility in the choice of wavelengths for regression. FDS also enables the user to adjust the relative sensitivities of the regressions to constant offsets and high-frequency noise. An example is given in which FDS is used to estimate the concentration of hemoglobin in a scattering liquid containing a large background concentration of water.

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28

Dong, Qixiang, Can Liu, and Zhenbin Fan. "Weighted fractional differential equations with infinite delay in Banach spaces." Open Mathematics 14, no. 1 (January 1, 2016): 370–83. http://dx.doi.org/10.1515/math-2016-0035.

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AbstractThis paper is devoted to the study of fractional differential equations with Riemann-Liouville fractional derivatives and infinite delay in Banach spaces. The weighted delay is developed to deal with the case of non-zero initial value, which leads to the unboundedness of the solutions. Existence and uniqueness results are obtained based on the theory of measure of non-compactness, Schaude’s and Banach’s fixed point theorems. As auxiliary results, a fractional Gronwall type inequality is proved, and the comparison property of fractional integral is discussed.

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29

Jamil, Muhammad, and Najeeb Alam Khan. "Slip Effects on Fractional Viscoelastic Fluids." International Journal of Differential Equations 2011 (2011): 1–19. http://dx.doi.org/10.1155/2011/193813.

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Unsteady flow of an incompressible Maxwell fluid with fractional derivative induced by a sudden moved plate has been studied, where the no-slip assumption between the wall and the fluid is no longer valid. The solutions obtained for the velocity field and shear stress, written in terms of Wright generalized hypergeometric functions , by using discrete Laplace transform of the sequential fractional derivatives, satisfy all imposed initial and boundary conditions. The no-slip contributions, that appeared in the general solutions, as expected, tend to zero when slip parameter is . Furthermore, the solutions for ordinary Maxwell and Newtonian fluids, performing the same motion, are obtained as special cases of general solutions. The solutions for fractional and ordinary Maxwell fluid for no-slip condition also obtained as limiting cases, and they are equivalent to the previously known results. Finally, the influence of the material, slip, and the fractional parameters on the fluid motion as well as a comparison among fractional Maxwell, ordinary Maxwell, and Newtonian fluids is also discussed by graphical illustrations.

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30

Liaqat, Muhammad Imran, Adnan Khan, Ali Akgül, and Md Shajib Ali. "A Novel Numerical Technique for Fractional Ordinary Differential Equations with Proportional Delay." Journal of Function Spaces 2022 (July 7, 2022): 1–21. http://dx.doi.org/10.1155/2022/6333084.

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Some researchers have combined two powerful techniques to establish a new method for solving fractional-order differential equations. In this study, we used a new combined technique, known as the Elzaki residual power series method (ERPSM), to offer approximate and exact solutions for fractional multipantograph systems (FMPS) and pantograph differential equations (PDEs). In Caputo logic, the fractional-order derivative operator is measured. The Elzaki transform method and the residual power series method (RPSM) are combined in this novel technique. The suggested technique is based on a new version of Taylor’s series that generates a convergent series as a solution. Establishing the coefficients for a series, like the RPSM, necessitates computing the fractional derivatives each time. As ERPSM just requires the concept of a zero limit, we simply need a few computations to get the coefficients. The novel technique solves nonlinear problems without the need for He’s and Adomian polynomials, which is an advantage over the other combined methods based on homotopy perturbation and Adomian decomposition methods. The relative, recurrence, and absolute errors of the problems are analyzed to evaluate the efficiency and consistency of the presented method. Graphical significances are also identified for various values of fractional-order derivatives. As a result, the procedure is quick, precise, and easy to implement, and it yields outstanding results.

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31

Khan, Adnan, Muhammad Imran Liaqat, Muhammad Younis, and Ashraful Alam. "Approximate and Exact Solutions to Fractional Order Cauchy Reaction-Diffusion Equations by New Combine Techniques." Journal of Mathematics 2021 (December 16, 2021): 1–12. http://dx.doi.org/10.1155/2021/5337255.

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In this paper, we present a simple and efficient novel semianalytic method to acquire approximate and exact solutions for the fractional order Cauchy reaction-diffusion equations (CRDEs). The fractional order derivative operator is measured in the Caputo sense. This novel method is based on the combinations of Elzaki transform method (ETM) and residual power series method (RPSM). The proposed method is called Elzaki residual power series method (ERPSM). The proposed method is based on the new form of fractional Taylor’s series, which constructs solution in the form of a convergent series. As in the RPSM, during establishing the coefficients for a series, it is required to compute the fractional derivatives every time. While ERPSM only requires the concept of the limit at zero in establishing the coefficients for the series, consequently scarce calculations give us the coefficients. The recommended method resolves nonlinear problems deprived of utilizing Adomian polynomials or He’s polynomials which is the advantage of this method over Adomain decomposition method (ADM) and homotopy-perturbation method (HTM). To study the effectiveness and reliability of ERPSM for partial differential equations (PDEs), absolute errors of three problems are inspected. In addition, numerical and graphical consequences are also recognized at diverse values of fractional order derivatives. Outcomes demonstrate that our novel method is simple, precise, applicable, and effectual.

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32

Kostic, Marko. "Asymptotically almost periodic solutions of fractional relaxation inclusions with Caputo derivatives." Publications de l'Institut Math?matique (Belgrade) 104, no. 118 (2018): 23–41. http://dx.doi.org/10.2298/pim1818023k.

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We analyze asymptotically almost periodic solutions for a class of (semilinear) fractional relaxation inclusions with Stepanov almost periodic coefficients. As auxiliary tools, we use subordination principles, fixed point theorems and the well known results on the generation of infinitely differentiable degenerate semigroups with removable singularities at zero. Our results are well illustrated and seem to be not considered elsewhere even for fractional relaxation equations with almost sectorial operators.

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33

Krasnoshchok, Mykola. "Strong solution of a hydrodinamics problem with memory." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 34 (April 24, 2021): 62–74. http://dx.doi.org/10.37069/1683-4720-2020-34-7.

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In the last few years, the concepts of fractional calculus were frequently applied to other disciplines. Recently, this subject has been extended in various directions such as signal processing, applied mathematics, bio-engineering, viscoelasticity, fluid mechanics, and fluid dynamics. In fluid dynamics, the fractional derivative models were used widely in the past for the study of viscoelastic materials such as polymers in the glass transition and in the glassy state. Recently, it has increasingly been seen as an efficient tool through which a useful generalization of physical concepts can be obtained. The fractional derivatives used most are the Riemann--Liouville fractional derivative and the Caputo fractional derivative. It is well known that these operators exhibit difficulties in applications. For example, the Riemann--Liouville derivative of a constant is not zero. We deal with so called temporal fractional derivative as a prototype of general fractional derivative. We prove the global strong solvability of a linear and quasilinear initial-boundary value problems with a singular complete monotone kernels. Our main tool is a theory of evolutionary integral equations. An abstract fractional order differential equation is studied, which contains as particular case the Rayleigh–Stokes problem for a generalized second-grade fluid with a fractional derivative model. This paper concerns with an initial-boundary value problem for the Navier--Stokes--Voigt equations describing unsteady flows of an incompressible viscoelastic fluid. We give the strong formulation of this problem as a nonlinear evolutionary equation in Sobolev spaces. Using the Galerkin method with a special basis of eigenfunctions of the Stokes operator, we construct a global-in-time strong solution in two-dimensional domain. We also establish an $L_2$ decay estimate for the velocity field under the assumption that the external forces field is conservative.

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34

Bayındır, Cihan, Sofi Farazande, Azmi Ali Altintas, and Fatih Ozaydin. "Petviashvili Method for the Fractional Schrödinger Equation." Fractal and Fractional 7, no. 1 (December 23, 2022): 9. http://dx.doi.org/10.3390/fractalfract7010009.

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In this paper, we extend the Petviashvili method (PM) to the fractional nonlinear Schrödinger equation (fNLSE) for the construction and analysis of its soliton solutions. We also investigate the temporal dynamics and stabilities of the soliton solutions of the fNLSE by implementing a spectral method, in which the fractional-order spectral derivatives are computed using FFT (Fast Fourier Transform) routines, and the time integration is performed by a 4th order Runge–Kutta time-stepping algorithm. We discuss the effects of the order of the fractional derivative, α, on the properties, shapes, and temporal dynamics of the soliton solutions of the fNLSE. We also examine the interaction of those soliton solutions with zero, photorefractive and q-deformed Rosen–Morse potentials. We show that for all of these potentials, the soliton solutions of the fNLSE exhibit a splitting and spreading behavior, yet their dynamics can be altered by the different forms of the potentials and noise considered.

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35

Lopushanska, H., and A. Lopushansky. "REGULAR SOLUTION OF THE INVERSE PROBLEM WITH INTEGRAL CONDITION FOR A TIME-FRACTIONAL EQUATION." Bukovinian Mathematical Journal 8, no. 2 (2020): 103–13. http://dx.doi.org/10.31861/bmj2020.02.09.

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Direct and inverse problems for equations with fractional derivatives are arising in various fields of science and technology. The conditions for classical solvability of the Cauchy and boundary-value prob\-lems for diffusion-wave equations with fractional derivatives are known. Estimates of components of the Green's vector-function of the Cauchy problem for such equations are known. We study the inverse problem of determining the space-dependent component of the right-hand side of the equation with a time fractional derivative and known functions from Schwartz-type space of smooth rapidly decreasing functions or with values in them. We also consider such a problem in the case of data from some wider space of smooth, decreasing to zero at infinity functions or with values in them. We find sufficient conditions for unique solvability of the inverse problem under the time-integral additional condition \[\frac{1}{T}\int_{0}^{T}u(x,t)\eta_1(t)dt=\Phi_1(x), \;\;\;x\in \Bbb R^n\] where $u$ is the unknown solution of the Cauchy problem, $\eta_1$ and $\Phi_1$ are the given functions. Using the method of the Green's vector function, we reduce the problem to solvability of an integrodifferential equation in a certain class of smooth, decreasing to zero at infinity functions. We prove its unique solvability. There are various methods for the approximate solution of direct and inverse problems for equations with fractional derivatives, mainly for the one-dimensional spatial case. It follows from our results the method of constructing an approximate solution of the inverse problem in the multidimensional spatial case. It is based on the use of known methods of constructing the numerical solutions of integrodifferential equations. The application of the Fourier transform by spatial variables is effective for constructing a numerical solution of the obtained integrodifferential equation, since the Fourier transform of the components of the Green's vector function can be explicitly written.

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36

Madamlieva, Ekaterina, Hristo Kiskinov, Milena Petkova, and Andrey Zahariev. "On the Preservation with Respect to Nonlinear Perturbations of the Stability Property for Nonautonomous Linear Neutral Fractional Systems with Distributed Delays." Mathematics 10, no. 15 (July 28, 2022): 2642. http://dx.doi.org/10.3390/math10152642.

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In the present paper, sufficient conditions are obtained under which the Cauchy problem for a nonlinearly perturbed nonautonomous neutral fractional system with distributed delays and Caputo type derivatives has a unique solution in the case of initial functions with first-kind discontinuities. For this system, by applying a formula for the integral presentation of the solution of the nonhomogeneous linear neutral fractional system, we found some additional natural conditions to ensure that from the global asymptotically stability of the zero solution of the linear part of the nonlinearly perturbed system, global asymptotic stability of the zero solution of the whole nonlinearly perturbed system follows.

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37

Prodanov, Dimiter. "Characterization of the Local Growth of Two Cantor-Type Functions." Fractal and Fractional 3, no. 3 (August 21, 2019): 45. http://dx.doi.org/10.3390/fractalfract3030045.

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The Cantor set and its homonymous function have been frequently utilized as examples for various physical phenomena occurring on discontinuous sets. This article characterizes the local growth of the Cantor’s singular function by means of its fractional velocity. It is demonstrated that the Cantor function has finite one-sided velocities, which are non-zero of the set of change of the function. In addition, a related singular function based on the Smith–Volterra–Cantor set is constructed. Its growth is characterized by one-sided derivatives. It is demonstrated that the continuity set of its derivative has a positive Lebesgue measure of 1/2.

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38

Wang, Weiqian, Yuanhua Qiao, Jun Miao, and Lijuan Duan. "Dynamic Analysis of Fractional-Order Recurrent Neural Network with Caputo Derivative." International Journal of Bifurcation and Chaos 27, no. 12 (November 2017): 1750181. http://dx.doi.org/10.1142/s0218127417501814.

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In this paper, fractional-order recurrent neural network models with Caputo Derivative are investigated. Firstly, we mainly focus our attention on Hopf bifurcation conditions for commensurate fractional-order network with time delay to reveal the essence that fractional-order equation can simulate the activity of neuron oscillation. Secondly, for incommensurate fractional-order neural network model, we prove the stability of the zero equilibrium point to show that incommensurate fractional-order neural network still converges to zero point. Finally, Hopf bifurcation conditions for the incommensurate fractional-order neural network model are first obtained using bifurcation theory based on commensurate fractional-order system.

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39

Zahariev, Andrey, and Hristo Kiskinov. "Asymptotic Stability of the Solutions of Neutral Linear Fractional System with Nonlinear Perturbation." Mathematics 8, no. 3 (March 10, 2020): 390. http://dx.doi.org/10.3390/math8030390.

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In this article existence and uniqueness of the solutions of the initial problem for neutral nonlinear differential system with incommensurate order fractional derivatives in Caputo sense and with piecewise continuous initial function is proved. A formula for integral presentation of the general solution of a linear autonomous neutral system with several delays is established and used for the study of the stability properties of a neutral autonomous nonlinear perturbed linear fractional differential system. Natural sufficient conditions are found to ensure that from global asymptotic stability of the zero solution of the linear part of a nonlinearly perturbed system it follows global asymptotic stability of the zero solution of the whole nonlinearly perturbed system.

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40

Pauli, Sebastian, and Filip Saidak. "Zero-free regions of the fractional derivatives of the Riemann zeta function." Lithuanian Mathematical Journal 62, no. 1 (January 2022): 99–112. http://dx.doi.org/10.1007/s10986-022-09551-2.

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41

Khan, Najeeb Alam, Muhammad Jamil, and Asmat Ara. "Approximate Solutions to Time-Fractional Schrödinger Equation via Homotopy Analysis Method." ISRN Mathematical Physics 2012 (March 28, 2012): 1–11. http://dx.doi.org/10.5402/2012/197068.

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We construct the approximate solutions of the time-fractional Schrödinger equations, with zero and nonzero trapping potential, by homotopy analysis method (HAM). The fractional derivatives, in the Caputo sense, are used. The method is capable of reducing the size of calculations and handles nonlinear-coupled equations in a direct manner. The results show that HAM is more promising, convenient, efficient and less computational than differential transform method (DTM), and easy to apply in spaces of higher dimensions as well.

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42

Liao, Zhiwu. "A New Definition of Fractional Derivatives Based on Truncated Left-Handed Grünwald-Letnikov Formula with0<α<1and Median Correction". Abstract and Applied Analysis 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/914386.

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We propose a new definition of fractional derivatives based on truncated left-handed Grünwald-Letnikov formula with0<α<1and median correction. Analyzing the difficulties to choose the fractional orders and unsatisfied processing results in signal processing using fractional-order partial differential equations and related methods; we think that the nonzero values of the truncated fractional order derivatives in the smooth regions are major causes for these situations. In order to resolve the problem, the absolute values of truncated parts of the G-L formula are estimated by the median of signal values of the remainder parts, and then the truncated G-L formula is modified by replacing each of the original signal value to the differences of the signal value and the median. Since the sum of the coefficients of the G-L formula is zero, the median correction can reduce the truncated errors greatly to proximate G-L formula better. We also present some simulation results and experiments to support our theory analysis.

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43

Torres-Hernandez, A., F. Brambila-Paz, and J. J. Brambila. "A Nonlinear System Related to Investment under Uncertainty Solved using the Fractional Pseudo-Newton Method." Journal of Mathematical Sciences: Advances and Applications 63, no. 1 (October 10, 2020): 41–53. http://dx.doi.org/10.18642/jmsaa_7100122150.

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A nonlinear algebraic equation system of two variables is numerically solved, which is derived from a nonlinear algebraic equation system of four variables, that corresponds to a mathematical model related to investment under conditions of uncertainty. The theory of investment under uncertainty scenarios proposes a model to determine when a producer must expand or close, depending on his income. The system mentioned above is solved using a fractional iterative method, valid for one and several variables, that uses the properties of fractional calculus, in particular the fact that the fractional derivatives of constants are not always zero, to find solutions of nonlinear systems.

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44

García-Fiñana, Marta, and Luis M. Cruz-Orive. "FRACTIONAL TREND OF THE VARIANCE IN CAVALIERI SAMPLING." Image Analysis & Stereology 19, no. 2 (May 3, 2011): 71. http://dx.doi.org/10.5566/ias.v19.p71-79.

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Cavalieri sampling is often used to estimate the volume of an object with systematic sections a constant distance T apart. The variance of the corresponding estimator can be expressed as the sum of the extension term (which gives the overall trend of the variance and is used to estimate it), the 'Zitterbewegung' (which oscillates about zero) and higher order terms. The extension term is of order T2m+2 for small T, where m is the order of the first non-continuous derivative of the measurement function f, (namely of the area function if the target is the volume). A key condition is that the jumps of the mth derivative f (m) of f are finite. When this is not the case, then the variance exhibits a fractional trend, and the current theory fails. Indeed, in practice the mentioned trend is often of order T2q+2, typically with 0 <q <1. We obtain a general representation of the variance, and thereby of the extension term, by means of a new Euler-MacLaurin formula involving fractional derivatives of f. We also present a new and general estimator of the variance, see Eq. 26a, b, and apply it to real data (white matter of a human brain).

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45

Wang, Zhi-Bo, Da-Yan Liu, and Driss Boutat. "Algebraic estimation for fractional integrals of noisy acceleration based on the behaviour of fractional derivatives at zero." Applied Mathematics and Computation 430 (October 2022): 127254. http://dx.doi.org/10.1016/j.amc.2022.127254.

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46

Plociniczak, Lukasz. "ON ASYMPTOTICS OF SOME FRACTIONAL DIFFERENTIAL EQUATIONS." Mathematical Modelling and Analysis 18, no. 3 (June 1, 2013): 358–73. http://dx.doi.org/10.3846/13926292.2013.804888.

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In this paper we study the large-argument asymptotic behaviour of certain fractional differential equations with Caputo derivatives. We obtain exponential and algebraic asymptotic solutions. The latter, decaying asymptotics differ significantly from the integer-order derivative equations. We verify our theorems numerically and find that our formulas are accurate even for small values of the argument. We analyze the zeros of fractional oscillations and find the approximate formulas for their distribution. Our methods can be used in studying many other fractional equations.

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47

Gomoyunov, Mikhail I. "Differential Games for Fractional-Order Systems: Hamilton–Jacobi–Bellman–Isaacs Equation and Optimal Feedback Strategies." Mathematics 9, no. 14 (July 15, 2021): 1667. http://dx.doi.org/10.3390/math9141667.

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The paper deals with a two-person zero-sum differential game for a dynamical system described by differential equations with the Caputo fractional derivatives of an order α∈(0,1) and a Bolza-type cost functional. A relationship between the differential game and the Cauchy problem for the corresponding Hamilton–Jacobi–Bellman–Isaacs equation with fractional coinvariant derivatives of the order α and the natural boundary condition is established. An emphasis is given to construction of optimal positional (feedback) strategies of the players. First, a smooth case is studied when the considered Cauchy problem is assumed to have a sufficiently smooth solution. After that, to cope with a general non-smooth case, a generalized minimax solution of this problem is involved.

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48

Kittisopaporn, Adisorn, and Pattrawut Chansangiam. "Approximate solutions of the $ 2 $D space-time fractional diffusion equation via a gradient-descent iterative algorithm with Grünwald-Letnikov approximation." AIMS Mathematics 7, no. 5 (2022): 8471–90. http://dx.doi.org/10.3934/math.2022472.

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<abstract>We consider the two-dimensional space-time fractional differential equation with the Caputo's time derivative and the Riemann-Liouville space derivatives on bounded domains. The equation is subjected to the zero Dirichlet boundary condition and the zero initial condition. We discretize the equation by finite difference schemes based on Grünwald-Letnikov approximation. Then we linearize the discretized equations into a sparse linear system. To solve such linear system, we propose a gradient-descent iterative algorithm with a sequence of optimal convergence factor aiming to minimize the error occurring at each iteration. The convergence analysis guarantees the capability of the algorithm as long as the coefficient matrix is invertible. In addition, the convergence rate and error estimates are provided. Numerical experiments demonstrate the efficiency, the accuracy and the performance of the proposed algorithm.</abstract>

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49

Laksaci, Noura, Ahmed Boudaoui, Seham Mahyoub Al-Mekhlafi, and Abdon Atangana. "Mathematical analysis and numerical simulation for fractal-fractional cancer model." Mathematical Biosciences and Engineering 20, no. 10 (2023): 18083–103. http://dx.doi.org/10.3934/mbe.2023803.

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<abstract>The mathematical oncology has received a lot of interest in recent years since it helps illuminate pathways and provides valuable quantitative predictions, which will shape more effective and focused future therapies. We discuss a new fractal-fractional-order model of the interaction among tumor cells, healthy host cells and immune cells. The subject of this work appears to show the relevance and ramifications of the fractal-fractional order cancer mathematical model. We use fractal-fractional derivatives in the Caputo senses to increase the accuracy of the cancer and give a mathematical analysis of the proposed model. First, we obtain a general requirement for the existence and uniqueness of exact solutions via Perov's fixed point theorem. The numerical approaches used in this paper are based on the Grünwald-Letnikov nonstandard finite difference method due to its usefulness to discretize the derivative of the fractal-fractional order. Then, two types of stabilities, Lyapunov's and Ulam-Hyers' stabilities, are established for the Incommensurate fractional-order and the Incommensurate fractal-fractional, respectively. The numerical results of this study are compatible with the theoretical analysis. Our approaches generalize some published ones because we employ the fractal-fractional derivative in the Caputo sense, which is more suitable for considering biological phenomena due to the significant memory impact of these processes. Aside from that, our findings are new in that we use Perov's fixed point result to demonstrate the existence and uniqueness of the solutions. The way of expressing the Ulam-Hyers' stabilities by utilizing the matrices that converge to zero is also novel in this area.</abstract>

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50

Lozynskyy, Orest, Yaroslav Marushchak, Andriy Lozynskyy, Bohdan Kopchak, and Lidiya Kasha. "Application of frequency stability criterion for analysis of dynamic systems with characteristic polynomials formed in j1/3 basis." Computational Problems of Electrical Engineering 10, no. 1 (May 12, 2020): 11–18. http://dx.doi.org/10.23939/jcpee2020.01.011.

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This paper considers the stability of dynamical systems described by differential equations with fractional derivatives. In contrast to a number of works, where the differential equation describing the system may have a set of different values of fractional derivatives, and the characteristic polynomial is formed on the basis of the least common multiple for the denominators of these indicators, this article proposes forming such a polynomial in a specific j¹/³ basis and studying the stability of systems with such fractional description based on the resulting rotation angles of Hn(jl/mω) vector at a frequency change from zero to infinity. This technique is similar to the investigation of system stability by frequency criteria used for a similar problem in describing the system by differential equations in integer derivatives. The application of characteristic polynomials formed in the j¹/³ basis for the description of the processes in dynamic systems and the analysis of the stability of such systems on the basis of the frequency criterion are the essence of the scientific novelty of this paper. The article contains the following sections: problem statement, work purpose, presentation of the research material, conclusions, list of references.

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