Search Results (17)

In this study, we apply the Laplace Transform Homotopy Analysis Method (LTHAM) to numerically solve a fractional-order telegraph equation with a Bessel operator. The iterative scheme developed is tested on multiple examples to evaluate its efficiency. Our observations indicate that the method generates an approximate solution in series form, which converges rapidly to the analytic solution in each instance. The convergence of these series solutions is assessed both geometrically and numerically. Our results demonstrate that LTHAM is a reliable, powerful, and straightforward approach to solving fractional telegraph equations, and it can be effectively extended to solve similar types of equations. Full article

Time-fractional convection–diffusion equations are significant for their ability to model complex transport phenomena that deviate from classical behavior, with numerous applications in anomalous diffusion, memory effects, and nonlocality. This paper derives, for the first time, a unique series solution to a multiple time-fractional convection–diffusion equation with a non-homogenous source term, based on an inverse operator, a newly-constructed space, and the multivariate Mittag–Leffler function. Several illustrative examples are provided to show the power and simplicity of our main theorems in solving certain fractional convection–diffusions equations. Additionally, we compare these results with solutions obtained using the AI model DeepSeek-R1, highlighting the effectiveness and validity of our proposed methods and main theorems. Full article

As the popularity of smartphones, wearable devices, intelligent vehicles, and countless other devices continues to rise, the surging demand for mobile data traffic has resulted in an increasingly crowded electromagnetic spectrum. Spectrum sharing serves as a solution to optimize the utilization of wireless communication channels, allowing various types of users to share the same frequency band securely. This paper investigates spectrum allocation and power control problems in overlay spectrum sharing, with a focus on promoting green communication. Maximizing weighted sum energy efficiency (WSEE) requires solving complex multiple-ratio fractional programming (FP) problems. In contrast, weighted sum power (WSP) minimization offers a more straightforward approach. Moreover, because WSP is directly related to users’ power consumption, we can dynamically adjust their weights to balance their residual energy. We prioritize WSP minimization over the more common WSEE maximization. This choice not only simplifies computation but also maintains users’ quality of service (QoS) requirements. The joint optimization for multiple primary users (PUs) and secondary users (SUs) can be decomposed into two components: a weighted bipartite matching problem and a series of convex resource allocation problems. Utilizing Newton’s method, our system-level simulation results show that the proposed scheme achieves optimal performance with minimal computational time. We explore strategies to accelerate the proposed scheme by refining the selection of initial values for Newton’s method. Full article

The paper deals with the problem of representing special functions by branched continued fractions, particularly multidimensional A- and J-fractions with independent variables, which are generalizations of associated continued fractions and Jacobi continued fractions, respectively. A generalized Gragg’s algorithm is constructed that enables us to compute, by the coefficients of the given formal multiple power series, the coefficients of the corresponding multidimensional A- and J-fractions with independent variables. Presented below are numerical experiments for approximating some special functions by these branched continued fractions, which are similar to fractals. Full article

This research focuses on finding multiple solutions (MSs) to nonlinear fractional boundary value problems (BVPs) through a new development, namely the predictor Laplace fractional power series method. This method predicts the missing initial values by applying boundary or force conditions. This research provides a set of theorems necessary for deriving the recurrence relations to find the series terms. Several examples demonstrate the efficacy, convergence, and accuracy of the algorithm. Under Caputo’s definition of the fractional derivative with symmetric order, the obtained results are visualized numerically and graphically. The behavior of the generated solutions indicates that altering the fractional derivative parameters within their domain symmetrically changes these solutions, ultimately aligning them with the standard derivative. The results are compared with the homotopy analysis method and are presented in various figures and tables. Full article

This article describes the solution of two problems. First, based on the fractional diffusion equation, a boundary problem with arbitrary values of derivative indicators was formulated and solved, describing more general cases than existing solutions. Secondly, from the consideration of the probability schemes of transitions between states of the process, which can be observed in complex systems, a fractional-differential equation of the telegraph type with multiples is obtained (in time: β, 2β, 3β, … and state: α, 2α, 3α, …) using orders of fractional derivatives and its analytical solution for one particular boundary problem is considered. In solving edge problems, the Fourier method was used. This makes it possible to represent the solution in the form of a nested time series (one in time t, the second in state x), each of which is a function of the Mittag-Leffler type. The eigenvalues of the Mittag-Leffler function for describing states can be found using boundary conditions and the Fourier coefficient based on the initial condition and orthogonality conditions of the eigenfunctions. An analysis of the characteristics of time series of changes in the emotional color of users’ comments on published news in online mass media and the electoral campaigns of the US presidential elections showed that for the mathematical expectation of amplitudes of deviations of series levels from the size of the amplitude calculation interval (“sliding window”), a root dependence of fractional degree was observed; for dispersion, a power law with a fractional index greater than 1.5 was observed; and the behavior of the excess showed the presence of so-called “heavy tails”. The obtained results indicate that time series have unsteady non-locality, both in time and state. This provides the rationale for using differential equations with partial fractional derivatives to describe time series dynamics. Full article

Beamspace MIMO-NOMA is an effective way to improve spectral efficiency. This paper focuses on a downlink non-orthogonal multiple access (NOMA) transmission scheme for a beamspace multiple-input multiple-output (MIMO) system. To increase the sum rate, we jointly optimize precoding and power allocation, which presents a non-convex problem. To solve this difficulty, we employ an alternating algorithm to optimize the precoding and power allocation. Regarding the precoding subproblem, we demonstrate that the original optimization problem can be transformed into an unconstrained optimization problem. Drawing inspiration from fraction programming (FP), we reconstruct the problem and derive a closed-form expression of the optimization variable. In addition, we effectively reduce the complexity of precoding by utilizing Neumann series expansion (NSE). For the power allocation subproblem, we adopt a dynamic power allocation scheme that considers both the intra-beam power optimization and the inter-beam power optimization. Simulation results show that the energy efficiency of the proposed beamspace MIMO-NOMA is significantly better than other conventional schemes. Full article

It is generally considered that the representation of a double layer supercapacitor (DLSC) cannot be performed with the usual capacitance and resistance series connected, as it induces a relatively high level of inaccuracy in the results. In multiple previous studies, more advanced models have been developed with very different approaches: models with distributed parameter circuits, based on artificial neural networks (ANNs), fractional order, etc. A non-linear model, less complex than the previous ones and whose behavior adequately represents the DLSCs, is the one formed by a variable capacitance, dependent on its internal voltage. This paper presents a mathematical study to obtain analytical expressions of all the electrical variables of DLSCs, voltage, current, dissipated power and so on, by means of a previous model. This study is carried out considering that the DLSC is charged and discharged through a voltage source and also discharged through a resistor. In later sections, the operational conditions of the DLSC in numerous industrial applications are presented. Finally, a comparative analysis is made between the results produced by the conventional model, with constant capacitance, and the developed model. This analysis is finally followed by the conclusions. Full article

This is the first review of conductive electrets (unpoled carbons and metals), which provide a new avenue for low-power electronics. The electret provides low DC voltage (μV) while allowing low DC current (μA) to pass through. Ohm’s Law is obeyed. The voltage scales with the inter-electrode distance. Series connection of multiple electret components provides a series voltage that equals the sum of the voltages of the components if there is no bending at the connection between the components. Otherwise, the series voltage is below the sum. Bending within the component also diminishes the voltage because of the polarization continuity decrease. The electret originates from the interaction of a tiny fraction of the carriers with the atoms. This interaction results in the charge in the electret. Dividing the electret charge by the electret voltage V’ provides the electret-based capacitance C’, which is higher than the permittivity-based capacitance (conventional) by a large number of orders of magnitude. The C’ governs the electret energy (1/2 C’V’²) and electret discharge time constant (RC’, where R = resistance), as shown for metals. The discharge time is promoted by a larger inter-electrode distance. The electret discharges occur upon short-circuiting and charge back upon subsequent opencircuiting. The discharge or charge of the electret amounts to the discharge or charge of C’. Full article

With the growing interest in owning electric vehicles due to increased environmental awareness and uncertain energy security together with the development of Li-ion batteries, quietness, and trouble-free operation, it is urgent to develop charging stations that are fast enough to supply the vehicles with energy conveniently, as in case of conventional petrol stations. The main reason that hinders the spread of fast charging stations is the installation cost, comprising the infrastructure and converter costs. In this article, a multiport DC-DC converter with differential power processing stages is proposed for Electric Vehicle (EV) fast charging stations, which results in a considerable reduction in the cost of using converters while achieving high efficiency. The proposed topology consists of two paths for the power flow (outer and inner loops) for EV battery charging with main and auxiliary DC-DC converters in the outer loop; all the ports are connected in series with the main supply, where the bulk power is being transferred. The main DC-DC converter injects a series voltage to control the power in the outer loop. The auxiliary DC-DC converters are rated at a fractional power that controls the partial power supplied to each port through the inner loops. Thanks to the fractional power processed by the auxiliary converter with the remaining power fed to the battery through the main converter, the proposed architecture enables simultaneous charging of multiple electric vehicles with better efficiency, lower cost, and the capability of providing a fault tolerance feature. A PWM control scheme for the converters to achieve bi-directional power flow in the partially rated DC-DC converters is discussed for the proposed system. Moreover, a practical down-scaled hardware prototype is designed to validate the functionality, control scheme, and effectiveness of the proposed topology in different case studies being investigated. The efficiency of the proposed converter is compared to the conventional configuration. Full article

Flood events have long been very frequent along the Yangtze River in Chongqing, China. A complete sedimentary sequence of alluvia, found in the Yuxi profile (YXP) was applied to explore features of the palaeoflood layers that maintained records related to the contexts of flooding hydroclimate. The AMS¹⁴C dating results dependent on animal bones from the YXP validate that the chronology of the palaeoflood layers was dated, between ca. 8200 and 6400 a BP, and multiple cultural layers were intercut among these palaeoflood layers. By means of particle size and end-member analyses for the palaeoflood sediments, the fractions of fine silt and clay in deposits account for a high proportion of the flood sediments, suggesting that the overbank flood was the main power in building the palaeoflood layers. Due to the climatic episodes defined by pollen assemblages, the thickness of the flood layers is positively correlated with soil erosion because of different hydrothermal conditions. The wavelet spectra of the mean particle-size series also suggest that there may be two major palaeoflooding cycles of ~700 and ~30 years. Despite the sustained palaeoflooding, the Yuxi Culture grew from small to big, and was never broken off, in terms of the findings of artificial remains in the YXP. Full article

We develop a fractional return-mapping framework for power-law visco-elasto-plasticity. In our approach, the fractional viscoelasticity is accounted for through canonical combinations of Scott-Blair elements to construct a series of well-known fractional linear viscoelastic models, such as Kelvin–Voigt, Maxwell, Kelvin–Zener, and Poynting–Thomson. We also consider a fractional quasi-linear version of Fung’s model to account for stress/strain nonlinearity. The fractional viscoelastic models are combined with a fractional visco-plastic device, coupled with fractional viscoelastic models involving serial combinations of Scott-Blair elements. We then develop a general return-mapping procedure, which is fully implicit for linear viscoelastic models, and semi-implicit for the quasi-linear case. We find that, in the correction phase, the discrete stress projection and plastic slip have the same form for all the considered models, although with different property and time-step-dependent projection terms. A series of numerical experiments is carried out with analytical and reference solutions to demonstrate the convergence and computational cost of the proposed framework, which is shown to be at least first-order accurate for general loading conditions. Our numerical results demonstrate that the developed framework is more flexible and preserves the numerical accuracy of existing approaches while being more computationally tractable in the visco-plastic range due to a reduction of $50\%$ in CPU time. Our formulation is especially suited for emerging applications of fractional calculus in bio-tissues that present the hallmark of multiple viscoelastic power-laws coupled with visco-plasticity. Full article

In this article, an attractive numeric–analytic algorithm, called the fractional residual power series algorithm, is implemented for predicting the approximate solutions for a certain class of fractional systems of partial differential equations in terms of Caputo fractional differentiability. The solution methodology combines the residual function and the fractional Taylor’s formula. In this context, the proposed algorithm provides the unknown coefficients of the expansion series for the governed system by a straightforward pattern as well as it presents the solutions in a systematic manner without including any restrictive conditions. To enhance the theoretical framework, some numerical examples are tested and discussed to detect the simplicity, performance, and applicability of the proposed algorithm. Numerical simulations and graphical plots are provided to check the impact of the fractional order on the geometric behavior of the fractional residual power series solutions. Moreover, the efficiency of this algorithm is discussed by comparing the obtained results with other existing methods such as Laplace Adomian decomposition and Iterative methods. Simulation of the results shows that the fractional residual power series technique is an accurate and very attractive tool to obtain the solutions for nonlinear fractional partial differential equations that occur in applied mathematics, physics, and engineering. Full article

In this paper, we consider the resource allocation problem to maximize the minimum (max–min) user’s secure energy efficiency (SEE) in downlink massive multiple-input multiple-output (MIMO) systems with simultaneous wireless information and power transfer (SWIPT). First, transmission power and power splitting ratio are designed to achieve the max–min user’s SEE subject to harvested energy threshold, the constraints of transmission power, and power splitting ratio. Secondly, the optimization problem is non-convex and very difficult to tackle. In order to solve the optimization problem, we converted to a series of parameter optimization subproblems by fractional programming. Then, we employ the first-order Taylor expansion and successive convex approximation (SCA) method to solve parameter optimization problems. Next, a secure energy-efficient resource allocation (SERA) algorithm with the bisection method is proposed to find the max–min SEE of the system. Finally, simulation results show the effectiveness and superiority of the SERA algorithm. Full article

The Newell–Whitehead–Segel equation is one of the most nonlinear amplitude equations that plays a significant role in the modeling of various physical phenomena arising in fluid mechanics, solid-state physics, optics, plasma physics, dispersion, and convection system. In this analysis, a recent numeric-analytic technique, called the fractional residual power series (FRPS) approach, was successfully employed in obtaining effective approximate solutions to the Newell–Whitehead–Segel equation of the fractional sense. The proposed algorithm relies on a generalized classical power series under the Caputo sense and the concept of an error function that systematically produces an analytical solution in a convergent fractional power series form with accurately computable structures, without the need for any unphysical restrictive assumptions. Meanwhile, two illustrative applications are included to show the efficiency, reliability, and performance of the proposed technique. Plotted and numerical results indicated the compatibility between the exact and approximate solution obtained by the proposed technique. Furthermore, the solution behavior indicates that increasing the fractional parameter changes the nature of the solution with a smooth sense symmetrical to the integer-order state. Full article