Search Results (22)

The objective of this work is to discuss and thoroughly analyze the fractional variational principles of symmetric systems involving distributed-order Atangana–Baleanu derivatives. A component of distributed order, the fractional Euler–Lagrange equations of fractional Lagrangians for constrained systems are studied concerning Atangana–Baleanu derivatives. We give a general formulation and a solution technique for a class of fractional optimal control problems (FOCPs) for such systems. The dynamic constraints are defined by a collection of FDEs, and the performance index of an FOCP is considered a function of the control variables and the state. The formula for fractional integration by parts, the Lagrange multiplier, and the calculus of variations are used to obtain the Euler–Lagrange equations for the FOCPs. Full article

In this paper, we aim to investigate corrected Euler–Maclaurin inequalities involving pre-invex mappings within the framework of fractional calculus. We want to find a number of important results for differentiable pre-invex mappings and Riemann–Liouville (RL) fractional integrals so that we can make more accurate error estimates. Additionally, we present examples with graphical illustrations to substantiate our major findings and deduce several special cases under certain conditions. Afterwards, we introduce applications such as the linear combination of means, composite corrected Maclaurin’s rule, modified Bessel mappings, and novel iterative methods for solving nonlinear equations. Full article

We analyze the error estimates of a fully discrete scheme for solving a semilinear stochastic subdiffusion problem driven by integrated fractional Gaussian noise with a Hurst parameter $H\in (0,1)$. The covariance operator Q of the stochastic fractional Wiener process satisfies $\parallel A^{-\rho }{Q}^{1/2}{\parallel }_{HS}<\infty$ for some $\rho \in [0,1)$, where ${\parallel ·\parallel }_{HS}$ denotes the Hilbert–Schmidt norm. The Caputo fractional derivative and Riemann–Liouville fractional integral are approximated using Lubich’s convolution quadrature formulas, while the noise is discretized via the Euler method. For the spatial derivative, we use the spectral Galerkin method. The approximate solution of the fully discrete scheme is represented as a convolution between a piecewise constant function and the inverse Laplace transform of a resolvent-related function. By using this convolution-based representation and applying the Burkholder–Davis–Gundy inequality for fractional Gaussian noise, we derive the optimal convergence rates for the proposed fully discrete scheme. Numerical experiments confirm that the computed results are consistent with the theoretical findings. Full article

This paper aims to demonstrate that, beyond the small world of Riemann–Liouville and Caputo derivatives, there is a vast and rich world with many derivatives suitable for specific problems and various theoretical frameworks to develop, corresponding to different paths taken. The notions of time and scale sequences are introduced, and general associated basic derivatives, namely, right/stretching and left/shrinking, are defined. A general framework for fractional derivative definitions is reviewed and applied to obtain both known and new fractional-order derivatives. Several fractional derivatives are considered, mainly Liouville, Hadamard, Euler, bilinear, tempered, q-derivative, and Hahn. Full article

This paper aims to examine an approach that studies many Euler–Maclaurin-type inequalities for various function classes applying Riemann–Liouville fractional integrals. Afterwards, our results are provided by using special cases of obtained theorems and examples. Moreover, several Euler–Maclaurin-type inequalities are presented for bounded functions by fractional integrals. Some fractional Euler–Maclaurin-type inequalities are established for Lipschitzian functions. Finally, several Euler–Maclaurin-type inequalities are constructed by fractional integrals of bounded variation. Full article

We further extend the results of other researchers on existence theory to homogeneous fractional Cauchy–Euler equations ${\displaystyle \sum _{i=1}^m}{d}_i{x}^{{\alpha }_i}{D}^{{\alpha }_i}u\left(x\right)+\mu u\left(x\right)=0,{\alpha }_i>0,$ with the derivatives in Caputo or Riemann–Liouville sense. Unlike the existing works, we consider multi-term equations without any restrictions on the order of fractional derivatives. The results are based on the characteristic equations which generate the solutions. Depending on the roots of the characteristic equations (real, multiple, or complex), we construct the corresponding solutions and prove their linear independence. Full article

This work proposes a new numerical approach for dealing with fractional stochastic differential equations. In particular, a novel three-point fractional formula for approximating the Riemann–Liouville integrator is established, and then it is applied to generate approximate solutions for fractional stochastic differential equations. Such a formula is derived with the use of the generalized Taylor theorem coupled with a recent definition of the definite fractional integral. Our approach is compared with the approximate solution generated by the Euler–Maruyama method and the exact solution for the purpose of verifying our findings. Full article

In this article, we introduce the generalized Bessel–Maitland function (EGBMF) using the extended beta function and some important properties obtained. Thus, we first show interesting relationships of this function with Laguerre polynomials and the Whittaker functions. We also introduce and prove some properties of the derivatives associated with EGBMF. In this sense, we establish a result relative to the extended fractional derivatives of Riemann–Liouville. Furthermore, the Mellin transform of this function is evaluated in terms of the generalized Wright hypergeometric function, and its Euler transform is also obtained. Finally, we derive several graphical representations using the Gauss quadrature and the Laguerre–Gauss quadrature methods, which show that the numerical and theoretical simulations are consistent. The results derived from this research can be potentially useful in applications in several fields, in particular, physics, applied mathematics, and engineering. Full article

In this article, Euler’s technique was employed to solve the novel post-pandemic sector-based investment mathematical model. The solution was established within the framework of the new generalized Caputo-type fractional derivative for the system under consideration that serves as an example of the investment model. The mathematical investment model consists of a system of four fractional-order nonlinear differential equations of the generalized Liouville–Caputo type. Moreover, the existence and uniqueness of solutions for the above fractional order model under pandemic situations were investigated using the well-known Schauder and Banach fixed-point theorem technique. The stability analysis in the context of Ulam—Hyers and generalized Ulam—Hyers criteria was also discussed. Using the investment model under consideration, a new analysis was conducted. Figures that depict the behavior of the classes of the projected model were used to discuss the obtained results. The demonstrated results of the employed technique are extremely emphatic and simple to apply to the system of non-linear equations. When a generalized Liouville–Caputo fractional derivative parameter ($\rho$) is changed, the results are asymmetric. The current work can attest to the novel generalized Caputo-type fractional operator’s suitability for use in mathematical epidemiology and real-world problems towards the future pandemic circumstances. Full article

In the past few years, the world has suffered from an untreated infectious epidemic disease (COVID-19), caused by the so-called coronavirus, which was regarded as one of the most dangerous and viral infections. From this point of view, the major objective of this intended paper is to propose a new mathematical model for the coronavirus pandemic (COVID-19) outbreak by operating the Caputo fractional-order derivative operator instead of the traditional operator. The behavior of the positive solution of COVID-19 with the initial condition will be investigated, and some new studies on the spread of infection from one individual to another will be discussed as well. This would surely deduce some important conclusions in preventing major outbreaks of such disease. The dynamics of the fractional-order COVID-19 mathematical model will be shown graphically using the fractional Euler Method. The results will be compared with some other concluded results obtained by exploring the conventional model and then shedding light on understanding its trends. The symmetrical aspects of the proposed dynamical model are analyzed, such as the disease-free equilibrium point and the endemic equilibrium point coupled with their stabilities. Through performing some numerical comparisons, it will be proved that the results generated from using the fractional-order model are significantly closer to some real data than those of the integer-order model. This would undoubtedly clarify the role of fractional calculus in facing epidemiological hazards. Full article

The Earth’s revolution is modified by changes in inclination of its rotation axis. Its trajectory is not closed and the equinoxes drift. Changes in polar motion and revolution are coupled through the Liouville–Euler equations. Milanković (1920) argued that the shortest precession period of solstices is 20,700 years: the summer solstice in one hemisphere takes place alternately every 11,000 year at perihelion and at aphelion. Milanković assumed that the planetary distances to the Sun and the solar ephemerids are constant. There are now observations that allow one to drop these assumptions. We have submitted the time series for the Earth’s pole of rotation, global mean surface temperature and ephemeris to iterative Singular Spectrum Analysis. iSSA extracts from each a trend a 1 year and a 60 year component. Both the apparent drift of solstices of Earth around the Sun and the global mean temperature exhibit a strong 60 year oscillation. We monitor the precession of the Earth’s elliptical orbit using the positions of the solstices as a function of Sun–Earth distance. The “fixed dates” of solstices actually drift. Comparing the time evolution of the winter and summer solstices positions of the rotation pole and the first iSSA component (trend) of the temperature allows one to recognize some common features. A basic equation from Milankovic links the derivative of heat received at a given location on Earth to solar insolation, known functions of the location coordinates, solar declination and hour angle, with an inverse square dependence on the Sun–Earth distance. We have translated the drift of solstices as a function of distance to the Sun into the geometrical insolation theory of Milanković. Shifting the inverse square of the 60 year iSSA drift of solstices by 15 years with respect to the first derivative of the 60 year iSSA trend of temperature, that is exactly a quadrature in time, puts the two curves in quasi-exact superimposition. The probability of a chance coincidence appears very low. Correlation does not imply causality when there is no accompanying model. Here, Milankovic’s equation can be considered as a model that is widely accepted. This paper identifies a case of agreement between observations and a mathematical formulation, a case in which an element of global surface temperature could be caused by changes in the Earth’s rotation axis. It extends the range of Milankovic cycles and resulting global temperature variations to shorter periods (1–100 year range), with a major role for the 60-year oscillation). Full article

Variations in sea-level, based on tide gauge data (GSLTG) and on combining tide gauges and satellite data (GSLl), are subjected to singular spectrum analysis (SSA) to determine their trends and periodic or quasi-periodic components. GLSTG increases by 90 mm from 1860 to 2020, a contribution of 0.56 mm/yr to the mean rise rate. Annual to multi-decadal periods of ∼90/80, 60, 30, 20, 10/11, and 4/5 years are found in both GSLTG and GSLl. These periods are commensurable periods of the Jovian planets, combinations of the periods of Neptune (165 yr), Uranus (84 yr), Saturn (29 yr) and Jupiter (12 yr). These same periods are encountered in sea-level changes, the motion of the rotation pole RP and evolution of global pressure GP, suggesting physical links. The first SSA components comprise most of the signal variance: 95% for GSLTG, 89% for GSLl, 98% for GP and 75% for RP. Laplace derived the Liouville–Euler equations that govern the rotation and translation of the rotation axis of any celestial body. He emphasized that one must consider the orbital kinetic moments of all planets in addition to gravitational attractions and concluded that the Earth’s rotation axis should undergo motions that carry the combinations of periods of the Sun, Moon and planets. Almost all the periods found in the SSA components of sea-level (GSLl and GSLTG), global pressure (GP) and polar motion (RP), of their modulations and their derivatives can be associated with the Jovian planets. The trends themselves could be segments of components with still longer periodicities (e.g., 175 yr Jose cycle). Full article

Differences in formulation of the equations of celestial mechanics may result in differences in interpretation. This paper focuses on the Liouville-Euler system of differential equations as first discussed by Laplace. In the “modern” textbook presentation of the equations, variations in polar motion and in length of day are decoupled. Their source terms are assumed to result from redistribution of masses and torques linked to Earth elasticity, large earthquakes, or external forcing by the fluid envelopes. In the “classical” presentation, polar motion is governed by the inclination of Earth’s rotation pole and the derivative of its declination (close to length of day, lod). The duration and modulation of oscillatory components such as the Chandler wobble is accounted for by variations in polar inclination. The “classical” approach also implies that there should be a strong link between the rotations and the torques exerted by the planets of the solar system. Indeed there is, such as the remarkable agreement between the sum of forces exerted by the four Jovian planets and components of Earth’s polar motion. Singular Spectral Analysis of lod (using more than 50 years of data) finds nine components, all with physical sense: first comes a “trend”, then oscillations with periods of ∼80 yrs (Gleissberg cycle), 18.6 yrs, 11 yrs (Schwabe), 1 year and 0.5 yr (Earth revolution and first harmonic), 27.54 days, 13.66 days, 13.63 days and 9.13 days (Moon synodic period and harmonics). Components with luni-solar periods account for 95% of the total variance of the lod. We believe there is value in following Laplace’s approach: it leads to the suggestion that all the oscillatory components with extraterrestrial periods (whose origin could be found in the planetary and solar torques), should be present in the series of sunspots and indeed, they are. Full article

In this paper, we are concerned with approach solutions for Sturm–Liouville problems (SLP) using variational problem (VP) formulation of regular SLP. The minimization problem (MP) is also set forth, and the connection between the solution of each formulation is then proved. Variational estimations (the variational equation associated through the Euler–Lagrange variational principle and Nehari’s method, shooting method and bisection method) and iterative variational methods (He’s method and HPM) for regular RSL are unitary presented in final part of the paper, which ends with applications. Full article

In this paper, a numerical analysis of the oscillation equation with a derivative of a fractional variable Riemann–Liouville order in the dissipative term, which is responsible for viscous friction, is carried out. Using the theory of finite-difference schemes, an explicit finite-difference scheme (Euler’s method) was constructed on a uniform computational grid. For the first time, the issues of approximation, stability and convergence of the proposed explicit finite-difference scheme are considered. To compare the results, the Adams–Bashford–Moulton scheme was constructed as an experimental method. The theoretical results were confirmed using test examples, the computational accuracy of the method was evaluated, which is consistent with the theoretical one, and the simulation results were visualized. Using the example of a fractional Duffing oscillator, waveforms and phase trajectories, as well as its amplitude–frequency characteristics, were constructed using a finite-difference scheme. To identify chaotic regimes, the spectra of maximum Lyapunov exponents and Poincaré points were constructed. It is shown that an explicit finite-difference scheme can be acceptable under the condition of a step of the computational grid. Full article