Variable Order Research Articles

The Stability of Solutions of the Variable-Order Fractional Optimal Control Model for the COVID-19 Epidemic in Discrete Time

This article introduces a discrete-time fractional variable order over a SEIQR model, incorporated for COVID-19. Initially, we establish the well-possedness of solution. Further, the disease-free and the endemic equilibrium points are determined. Moreover, the local asymptotic stability of the model is analyzed. We develop a novel discrete fractional optimal control problem tailored for COVID-19, utilizing a discrete mathematical model featuring a variable order fractional derivative. Finally, we validate the reliability of these analytical findings through numerical simulations and offer insights from a biological perspective.

Do ECB's rate hikes have spillover effects on the Hungarian BUBOR and the EUR/HUF exchange rate? A five‐variable VAR model approach using the Diebold‐Yilmaz spillover table

AbstractWe intend to show the directional volatility spillovers between the European short term interbank lending rates (3‐month Euro Interbank Offered Rate [EURIBOR] and Euro Short‐Term Rate [ESTR]) and the Hungarian Budapest Interbank Offered Rate (BUBOR) and Euro‐Hungarian Forint exchange rate. To determine the extent to which the variables affect each other's volatilities we build a five‐variable vector autoregression (VAR) and determine the spillover table like in Diebold‐Yilmaz's 2012 work. This methodology is preferred to a simple impulse response function (IRF) because we manage to avoid the problem of non‐orthogonal innovations via the generalized forecast error variance decomposition framework. The issue of variable ordering, therefore, does not arise. We focus on three episodes of increased volatility in Hungarian and European short‐term interest rates: Q3–Q4 of 2019, Q1 of 2020 and Q3 of 2022. These episodes correspond to volatility spikes in EU markets that to some extent had a measurable spillover effect on Hungarian interbank rates. We find that on average, across the entire sample of 957 observations, about 6.3% of the volatility forecast error variance in all five European and Hungarian variables comes from spillovers. The total and directional spillovers over the sample are extremely low. We conclude that the European Central Bank's surprise policy decisions have a marginal impact on Hungarian interbank rates. We also find that BUBOR is primarily a net receiver of spillovers from the MAX short‐term government bond benchmark rather than the EURIBOR—this disproved our initial considerations.

Exploring the stochastic patterns of hyperchaotic Lorenz systems with variable fractional order and radial basis function networks

Exploring the stochastic patterns of hyperchaotic Lorenz systems with variable fractional order and radial basis function networks

Probabilistic Context Neighborhood model for lattices

We present the Probabilistic Context Neighborhood model designed for two-dimensional lattices as a variation of a Markov random field assuming discrete values. In this model, the neighborhood structure has a fixed geometry but a variable order, depending on the neighbors’ values. Our model extends the Probabilistic Context Tree model, originally applicable to one-dimensional space. It retains advantageous properties, such as representing the dependence neighborhood structure as a graph in a tree format, facilitating an understanding of model complexity. Furthermore, we adapt the algorithm used to estimate the Probabilistic Context Tree to estimate the parameters of the proposed model. We illustrate the accuracy of our estimation methodology through simulation studies. Additionally, we apply the Probabilistic Context Neighborhood model to spatial real-world data, showcasing its practical utility.

An algorithmic exploration of variable order fractional partial integro-differential equations via hexic shifted Chebyshev polynomials

An algorithmic exploration of variable order fractional partial integro-differential equations via hexic shifted Chebyshev polynomials

Modern technique to study Cauchy-type problem of fractional variable order differential equations with infinite delay via phase space

In this paper, we use a novel approach to study the existence, uniqueness, and stability of solutions (EUSS) to a Cauchy-type problem of nonlinear fractional differential equations (FrDiEq) of variable order with infinite delay (CPNFDEVOID). Contrary to the techniques taken in the literature, which were centered on the usage of the concept of generalized intervals and the idea of piecewise constant functions, our approach is straightforward and based on a novel fractional operator that is more appropriate and demonstrates the solvability and stability of the main problem under less restrictive presumptions. The results are achieved in this paper by using Fixed Point Theory (FPT). The application, which includes an example and supporting images, concludes the paper.

Development and validation of a phenomenological model for hydrogen fueled PFI internal combustion engines considering Thermo-Diffusive effects on flame speed propagation

This paper proposes a 1D numerical/experimental study on a single-cylinder research ICE equipped with a hydrogen Port Fuel Injection (PFI) system. The experimental campaign covered tests at the fixed speed of 1500 rpm and different levels of load, from 5 bar to 11 bar IMEP, and of air excess, with a relative air-to-fuel ratio ranging from 1.4 to 4.0. A 0D/1D model of the investigated ICE is developed, including detailed combustion, turbulence, and NOx emission sub-models. In particular, combustion model is based on the fractal approach, k-K-T model is adopted for describing turbulent phenomena and NOx emissions are predicted by the Zeldovich mechanism. A sub-model was developed to assess the influence of Thermo-Diffusive (TD) instabilities on the freely propagating flame speed accounting for equivalence ratio, variable transport coefficients and reaction orders. This evaluation considered the impact of TD instabilities commonly observed in lean premixed hydrogen combustion on the flame front. The engine model calibration process enabled the comparison between numerical predictions and experimental observations, encompassing pressure cycles, burn rate traces, and main engine performance (air flow rate, IMEP, indicated efficiency, and timing of main combustion events). The model replicated the IMEP with an average error of 2.5%, gross indicated efficiency with an average error of 1.6%, main combustion angles with an average error of 3.8 CAD on CA50, and NOx emissions satisfactorily, showing high sensitivity to operational parameters. Nonetheless, less accurate predictions occur in cases with significantly elevated λ values (λ>3.6), which were attributed to the presence of thickened flames, where some assumptions inherent to the adopted combustion model may fail. Finally, a comparison between the proposed approach and the one proposed in Ballerini et al. (2022) is presented, and readers are provided with recommendations for further development of future models that account for Thermo-Diffusive instabilities in lean hydrogen combustion.

DHONE: Density-based higher-order network embedding

Studies have indicated that focusing solely on pairwise interactions between two nodes disregards the associativity among multi-nodes in the network’s local structure. This associativity can be seen as dependencies among nodes, where certain edges’ presence depends on the path leading to it. Examinations on diverse datasets have approved that the variable order of chained dependencies allows for the preservation of structure information, which enables the reconstruction of the original network into a Higher-Order Network (HON) with improved quality of network representation. This paper proposes a Density-based Higher-Order Network Embedding (DHONE) algorithm, which integrates the concept of higher-order density into the network-embedding process in order to classify the contribution of different orders of dependencies. Through the construction of a novel and effective higher-order adjacency matrix, DHONE steadily improves the accuracy of network representation learning. Experimental results demonstrate DHONEs proficiency in improving embedding accuracy and overall algorithm robustness. Furthermore, grounded in the concept of higher-order density proposed herein, numerous dependencies have been discerned within the network generated from trajectories, potentially indicating the role of multi-node structures in networks.

Analysis of Mesh Resolution Effect to Numerical Result of CFD-ROM: Turbulent Flow in Stationary Parallel Plate

Computational fluid dynamics (CFD) is extensively utilized to predict flow behaviour in various industries and applications. The Full Order Model (FOM) is a high-accuracy approach to flow modelling, but it requires significant computational resources due to its high order and thousands of variables. To address this problem, the Reduced Order Model (ROM) was developed. Despite the advancement brought by ROM, there is a notable gap in research concerning the impact of mesh configuration on CFD-ROM results. While the number of modes has been extensively studied for its influence on CFD-ROM, the mesh configuration, a critical aspect of the simulation process, has received relatively limited attention. This study investigates the effect of mesh resolution on numerical results in CFD-ROM concerning turbulent flow within stationary parallel plates. Employing rigorous methods, including Richardson Extrapolation, verification, validation, and error percentage. The results explicitly confirm that mesh resolution directly impacts the numerical results of the velocity field in CFD-ROM. It is found that there is a notable reduction in Convergence Grid Index (CGI) values for different mesh ratios: 6.401% for medium-to-coarse and 2.031% for fine-to-medium ratio. Thus, with the same mode number, mesh resolution selection can enhance the numerical result of the velocity field in CFD-ROM.

An analytical approach for shock response of fractional viscoelastic beams with tip mass

This paper specifically analyzes the behavior of a beam with fractional viscoelastic properties and a tip mass under shock. The shock is applied longitudinally to the beam, and a fractional Kelvin–Voigt model is employed to describe the viscoelastic properties. Additionally, the paper examines the fractional viscoelastic power as a function of time during the shock. The equation of motion for the fractional viscoelastic beam with a tip mass is derived using Hamilton’s principle. In order to ensure accurate analysis of the results, the graphs are carefully analyzed in both the time and frequency domains. To solve the equations, the special technique provided for differential equations of variable order (VO) is used. Two distinct methods, namely, absolute method and square root of the sum of squares method, are thoroughly examined to calculate the absolute acceleration. Subsequently, the most suitable method is selected based on the evaluation results. The study’s results demonstrate that the damping parameters of the viscoelastic beam have a significant impact on shock transmissibility. Additionally, the findings demonstrate that the ability to transfer shocks can be manipulated by modifying the system’s geometry and mass.

Examining the Impact of Layperson Rescuer Gender on the Receipt of Bystander CPR for Women in Cardiac Arrest.

Women who suffer a witnessed out-of-hospital cardiac arrest receive bystander cardiopulmonary resuscitation (CPR) less often than men. To understand this phenomenon, we queried whether there are differences in deterrents to providing CPR based on the rescuer's gender. Participants were surveyed using a national crowdsourcing platform. Participants ranked the following 5 previously identified themes as reasons: rescuers are afraid to injure or hurt women; rescuers might have a misconception that women do not suffer cardiac arrest; rescuers are afraid to be accused of sexual assault or sexual harassment; rescuers have a fear of touching women or that their touch might be inappropriate; and rescuers think that women are faking it or being overdramatic. Participants were adult US residents able to correctly define CPR. Participants ranked the themes if the rescuer was gender unidentified, a man, and a woman, in variable order. In November 2018, 520 surveys were completed. The respondents identified as 42.3% women, 74.2% White, 10.4% Black, and 6.7% Hispanic. Approximately half (48.1%) of the cohort knew how to perform CPR, but only 7.9% had ever performed CPR. When the rescuer was identified as a man, survey participants ranked fear of sexual assault or sexual harassment and fear of touching women or that the touch might be inappropriate as the top reasons (36.2% and 34.0% of responses, respectively). Conversely, when the rescuer was identified as a woman, survey respondents reported fear of hurting or injuring as the top reason (41.2%). Public perceptions as to why women receive less bystander CPR than men were different based on the gender of the rescuer. Participants reported that men rescuers would potentially be hindered by fears of accusations of sexual assault/harassment or inappropriate touch, while women rescuers would be deterred due to fears of causing physical injury.

Free vibration analysis of composite beams and laminated reinforced panels by refined dynamic stiffness method and CUF-based component-wise theory

This paper presents a comprehensive analysis of the free vibration behavior of composite beams and laminated reinforced panels. Employing high-order theories with displacement variables only, the investigation combines dynamic stiffness method (DSM) and the Carrera unified formulation (CUF). The 3D displacement field can be expanded in the framework of CUF as any order of generic unknown variables over the cross-section, in the case of beam theories. Specifically, Lagrange expansions (LE) of cross-sectional displacement variables are considered, enabling the refinement modeling of complex cross-sections with different layers (layer-wise) and components (component-wise). The governing differential equations and natural boundary conditions are derived using the principle of virtual displacements. Subsequently, an exact dynamic stiffness matrix is developed by establishing a relationship between the amplitudes of harmonically varying loads and the corresponding responses. The Wittrick–Williams algorithm is employed to solve the transcendental eigenvalue problem resulting from this approach. The Lagrange-based CUF(LE)-DSM outperforms other polynomials (such as Taylor)-based CUF in analyzing composite structures, enabling detailed analysis with various geometries, lamination schemes and boundary conditions. It delivers accurate results in reinforced panel analysis, requiring only 13% of the DOFs compared to 3D FE models, thereby reducing computational costs significantly, as confirmed by comparative studies and validations.

Моделирование транспортных потоков методом системной динамики

The article discusses the application of the system dynamics methodology for simulation of controlled traffic flows. In the classical formulation, system dynamics models represent a system of finite-difference equations, which is an approximation of the Cauchy problem by the Euler method. The implementation of finite-difference equations in system dynamics by J. Forrester is carried out by the DYNAMO specialized graphical tools. It is proposed that the Cauchy problem be implemented using structural diagrams in which a one-to-one correspondence of specific structural links to system dynamics paradigms is established. As a result of the research, it was shown that an aperiodic link with a typical nonlinear “saturation” function identifies with a sufficient accuracy the basic paradigms of system dynamics, such as rates, levels and delays. It is shown that the resulting system of ordinary differential equations, when imposing control conditions, has discontinuities of the first kind in derivatives and requires integration methods with a variable step and variable order for effective solution. The proposed approach to modeling traffic flows, in contrast to the classical one, allows the use of modern modeling tools with a graphical language of structural diagrams and a library of methods for numerical analysis of complex dynamic processes. The implementation of a specific test scenario for modeling traffic flows was carried out in the SimInTech software package. The efficiency metrics of integration methods are obtained. A comparative analysis of the effectiveness of integration methods in solving this problem was carried out. It is shown that the most effective in solving problems of the considered class are the explicit adaptive integration methods ARK21, AM61 with a variable step from the SimInTech library, which are much more effective than the explicit Euler method in classical system dynamics.

Hardness of Random Reordered Encodings of Parity for Resolution and CDCL

Parity reasoning is challenging for Conflict-Driven Clause Learning (CDCL) SAT solvers. This has been observed even for simple formulas encoding two contradictory parity constraints with different variable orders (Chew and Heule 2020). We provide an analytical explanation for their hardness by showing that they require exponential resolution refutations with high probability when the variable order is chosen at random. We obtain this result by proving that these formulas, which are known to be Tseitin formulas, have Tseitin graphs of linear treewidth with high probability. Since such Tseitin formulas require exponential resolution refutations, our result follows. We generalize this argument to a new class of formulas that capture a basic form of parity reasoning involving a sum of two random parity constraints with random orders. Even when the variable order for the sum is chosen favorably, these formulas remain hard for resolution. In contrast, we prove that they have short DRAT refutations. We show experimentally that the running time of CDCL SAT solvers on both classes of formulas grows exponentially with their treewidth.

Compact inclusions between variable Hölder spaces

We give a sufficient condition for the inclusion between two variable order Hölder spaces be compact. In some cases, mostly asking log-Hölder continuity in one of the order functions, we find a criterion for the inclusion between variable order Hölder spaces to be compact. We consider two variable order functions as well as one variable order functions.

Variable-Order Fractional Linear Systems with Distributed Delays—Existence, Uniqueness and Integral Representation of the Solutions

In this work, we study a general class of retarded linear systems with distributed delays and variable-order fractional derivatives of Caputo type. We propose an approach consisting of finding an associated one-parameter family of constant-order fractional systems, which is “almost” equivalent to the considered variable-order system in an appropriate sense. This approach allows us to replace the study of the initial problem (IP) for variable-order fractional systems with the study of an IP for these one-parameter families of constant-order fractional systems. We prove that the initial problem for the variable-order fractional system with a discontinuous initial function possesses a unique continuous solution on the half-axis when the function describing the variable order of differentiation is locally bounded, Lebesgue integrable and has an appropriate decomposition similar to the Lebesgue decomposition of functions with bounded variation. The obtained results lead to the existence and uniqueness of a fundamental matrix for the studied variable-order fractional homogeneous system. As an application of the obtained results, we establish an integral representation of the solutions of the studied IP.

A Convergent Legendre Spectral Collocation Method for the Variable-Order Fractional-Functional Optimal Control Problems

In this paper, a numerical method is applied to approximate the solution of variable-order fractional-functional optimal control problems. The variable-order fractional derivative is described in the type III Caputo sense. The technique of approximating the optimal solution of the problem using Lagrange interpolating polynomials is employed by utilizing the shifted Legendre–Gauss–Lobatto collocation points. To obtain the coefficients of these interpolating polynomials, the problem is transformed into a nonlinear programming problem. The proposed method offers a significant advantage in that it does not require the approximation of singular integral. In addition, the matrix differentiation is calculated accurately and efficiently, overcoming the difficulties posed by variable-order fractional derivatives. The convergence of the proposed method is investigated, and to validate the effectiveness of our proposed method, some examples are presented. We achieved an excellent agreement between numerical and exact solutions for different variable orders, indicating our method’s good performance.

A new one-dimensional consolidation creep model for clays

Creep deformation is a prevalent form of deformation infill projects, and accurately characterizing the creep of fill soils is essential for ensuring the safe and reliable operation of these projects. To develop a novel variable-order fractional creep model that describes the consolidation creep behavior of fill soils, this research enhances the fractional order algorithm to achieve more precise fractional orders and determine the variable order function. Consequently, a novel variable-order fractional element featuring viscoelastic memory effects is introduced and formulated as a model. Unlike the traditional Maxwell model, the new model is simpler, has fewer parameters, and includes a memory effect that reveals the characteristics of mechanical properties over time during the creep process. In addition, this study validates the proposed model's applicability and modeling benefits through one-dimensional consolidation creep tests on red clay at the fill site. The findings indicated that the change in fractional order in the new model should be an exponential function. The trend of decreasing fractional order with time is consistent with the soil consolidation creep-hardening behavior, suggesting that the variable fractional order function represents the changing nature of the soil. In this study, exploring fractional order laws reveals the physical significance of the fractional order element and enables the composed model to make effective predictions of relevant working conditions.

Some properties of convex and increasing convex orders under Archimedean copula

Abstract In this paper, the ordering properties of convex and increasing convex orders of the dependent random variables are studied. Some closure properties of the convex and increasing convex orders under independent random variables are extended to the dependent random variables under the Archimedean copula. Two applications are provided to illustrate our results.

Diversity enforced Genetic Algorithm (GA) for Binary Decision Diagram (BDD) reordering

Binary Decision Diagrams (BDDs) have become the state-of-art data structures in numerous fields, wherein, small-sized BDDs are required to reduce the companion cost. Since BDD size is sensitive to the order of variables for the Boolean function in use, evolutionary algorithms have been extensively exploited to solve the BDD reordering problem which is provably an NP-hard problem. However, getting trapped in a local minima is more likely as the population gets homogeneous during the evolution of the algorithm. This paper compares various diversity measures correlated to BDDs and studies the impact of different crossover and mutations used in the Genetic Algorithm (GA) on those diversity measures. Thereafter, we propose an enhanced GA-based reordering algorithm for BDD, wherein, the chosen diversity measure is calculated per generation to steer the application of proper variation operators, in such a way that an acceptable level of equilibrium between exploration and exploitation is preserved. Finally, we utilize a new heuristic Cyclic Crossover (HCX) that further improves the performance of the proposed algorithm following the fitness value. Experimental results on public benchmarks show that our proposed algorithm outperforms other algorithms from the literature when switching between HCX and swap mutation operators is made following the measured diversity metric.