Variable Order Research Articles

Impact of faults on the remote stress state

Abstract. The impact of faults on the contemporary stress field in the upper crust has been discussed in various studies. Data and models clearly show that there is an effect, but so far, a systematic study quantifying the impact as a function of distance from the fault is lacking. In the absence of data, here we use a series of generic 3-D models to investigate which component of the stress tensor is affected at which distance from the fault. Our study concentrates on the far field, located hundreds of metres from the fault zone. The models assess various techniques to represent faults, different material properties, different boundary conditions, variable orientation, and the fault's size. The study findings indicate that most of the factors tested do not have an influence on either the stress tensor orientation or principal stress magnitudes in the far field beyond 1000 m from the fault. Only in the case of oblique faults with a low static friction coefficient of μ=0.1 can noteworthy stress perturbations be seen up to 2000 m from the fault. However, the changes that we detected are generally small and of the order of lateral stress variability due to rock property variability. Furthermore, only in the first hundreds of metres to the fault are variations large enough to be theoretically detected by borehole-based stress data when considering their inherent uncertainties. This finding agrees with robust stress magnitude measurements and stress orientation data. Thus, in areas where high-quality and high-resolution data show gradual and continuous stress tensor rotations of >20∘ observed over lateral spatial scales of 10 km or more, we infer that these rotations cannot be attributed to faults. We hypothesize that most stress orientation changes attributed to faults may originate from different sources such as density and strength contrasts.

Variable-order porous media equations: Application on modeling the S&P500 and Bitcoin price return.

This article reveals a specific category of solutions for the 1+1 variable order (VO) nonlinear fractional Fokker-Planck equations. These solutions are formulated using VO q-Gaussian functions, granting them significant versatility in their application to various real-world systems, such as financial economy areas spanning from conventional stock markets to cryptocurrencies. The VO q-Gaussian functions provide a more robust expression for the distribution function of price returns in real-world systems. Additionally, we analyzed the temporal evolution of the anomalous characteristic exponents derived from our study, which are associated with the long-term (power-law) memory in time series data and autocorrelation patterns.

Both-strand gene coding in a plastome-like mitogenome of an enoplid nematode.

The phylum Nematoda remains very poorly sampled for mtDNA, with a strong bias toward parasitic, economically important or model species of the Chromadoria lineage. Most chromadorian mitogenomes share a specific order of genes encoded on one mtDNA strand. However, the few sequenced representatives of the Dorylaimia lineage exhibit a variable order of mtDNA genes encoded on both strands. While the ancestral arrangement of nematode mitogenome remains undefined, no evidence has been reported for Enoplia, the phylum's third early divergent major lineage. We describe the first mitogenome of an enoplian nematode, Campydora demonstrans, and contend that the complete 37-gene repertoire and both-strand gene encoding are ancestral states preserved in Enoplia and Dorylaimia versus the derived mitogenome arrangement in some Chromadoria. The C. demonstrans mitogenome is 17,018 bp in size and contains a noncoding perfect inverted repeat with 2013 bp-long arms, subdividing the mitogenome into two coding regions. This mtDNA arrangement is very rare among animals and instead resembles that of chloroplast genomes in land plants. Our report broadens mtDNA taxonomic sampling of the phylum Nematoda and adds support to the applicability of cox1 gene as a phylogenetic marker for establishing nematode relationships within higher taxa.

Processing characteristics correction of measuring systems by means a differintegral of variable order

The paper presents new methods for correcting the processing characteristics of measurement systems based on a modified Grünwald-Letnikov fractional calculus definition. The presented methods are based on the determination of the fractional order as an estimation factor. Two methods are presented: a fractional order array and a fractional order function. Both methods can be used in DSP systems as methods to correct the processing characteristics of systems with measuring transducers and measurement systems in general.

Research on the application of Breiman's algorithm integrated with the Random Forest in determining the importance of input factors to landslide formation in Son La province

Landslide susceptibility maps are effective and intuitive tools in natural disaster management, helping to minimize damage caused by natural disasters due to the specific spatial information they provide. However, the performance of these susceptibility maps depends mainly on the number and importance of input factors. Determining the importance and order of influencing factors often receives little attention in landslide prediction studies. Breiman's algorithm, integrated into the Random Forest method, can comprehensively determine the importance and order of input variables by considering the correlation relationship between the landslide inventory map and these input variables. Consequently, this study utilized Breiman's algorithm within the Random Forest technique to assess the importance of 16 input factors influencing the formation of landslide events in Son La province. The results obtained from this study serve as the foundation for selecting appropriate input factors to enhance the construction and accuracy of landslide susceptibility maps within the study area, especially in the context of climate change.

Role of shift constant in Energy Shifted SAV for Hamiltonian Systems

In recent years, considerable work has been devoted to the design of energy-stable numerical methods, a class of geometrical integration technique in which the discrete energy or pseudo-energy remains conserved. This property finds practical applications in systems for which the energy is bounded from below since the growth of the solutions is then itself bounded, yielding a form of stability. Such a property may be further exploited to transform the energy overall into a quadratic form, representing the core of recent numerical techniques such as the invariant energy quadratization (IEQ) and the scalar auxiliary variable (SAV) approaches. These methods have been applied to a large class of problems due to their remarkable efficiency. Yet, several aspects of such techniques have seen little investigation. In this work, the role of the “shift” constant in the expression of the potential energy in Hamiltonian systems is investigated. This is a positive gauge constant that may be used to augment the expression of the energy. Since Hamilton’s equations are given in terms of gradients of the Hamiltonian, such a constant has no influence on the resulting dynamics in the continuous system. However, empirical evidence has suggested that the convergence properties of the associated SAV approaches are affected by the magnitude of the shift factor. In this work, the behaviour of the relative global error of SAV schemes in the cases of the harmonic and Duffing oscillators is numerically investigated. The results reveal an optimal shift factor that increases the convergence rate. Using the optimal shift factor, the proposed method displays variable order accuracy ranging from second to twelfth order.

A mixed-method to numerical simulation of variable order stochastic advection diffusion equations

The study of stochastic problems is very important and there is an increasing demand for investigating the behavior of a number of sophisticated dynamical systems in different areas of science as well as in engineering and finance. In this paper, we present a method which is based on the collocation method (CM) and the operational matrix method (OMM) by using Hosoya polynomials (HPs). This mixed-method helps us to convert the stochastic problems into a system of algebraic equations to access a solution of such problems. One of the most important advantages of this method is the simulation of the noise terms at the collocation points. Finally, two types of the VO–SADEs are simulated and tested via the proposed technique to confirm the superiority and accuracy.

Stimulus variability improves generalization following response inhibition training.

The present study examined the effect of stimulus variability and practice order on generalization to novel stimuli following a single session of response inhibition training. Ninety-six young adults practiced the Go/No-go task online in three training conditions: (1) constant (N = 32)-inhibition practiced on one stimulus; (2) variable-blocked (N = 32)-inhibition practiced on 6 stimuli, each in a separate block; and (3) variable-random (N = 32)-inhibition practiced on 6 stimuli in random order. Generalization was measured by comparing groups on inhibition of novel stimuli and a trained stimulus immediately and 24h after training. Consistent with our hypothesis, the variable-random and the variable-blocked groups showed better generalization to the novel items than the constant group, demonstrating the benefit of stimulus variability. The variable-random group also showed better generalization than the variable-blocked group, demonstrating the benefit of presenting stimuli in random order. Participants' capacity for working memory maintenance was found to modulate the effect of practice order. While the benefit of variability was retained 24h after training, the effect of order was not. Results also show generalization to (1) different type of stimuli using the same task and (2) the same stimuli on a different response inhibition task (the Stop-Signal Task), however, the effect of variable practice and order were not evident in these cases. The study findings illustrate the advantage of using variable stimuli presented in random order for generalization and suggest that these principles of motor learning can be applied to learning of cognitive skills.

Nonlinear variable order system of multi-point boundary conditions with adaptive finite-time fractional-order sliding mode control

Nonlinear variable order system of multi-point boundary conditions with adaptive finite-time fractional-order sliding mode control

Existence and Stability Results for Differential Equations with a Variable-Order Generalized Proportional Caputo Fractional Derivative

An initial value problem for a scalar nonlinear differential equation with a variable order for the generalized proportional Caputo fractional derivative is studied. We consider the case of a piecewise constant variable order of the fractional derivative. Since the order of the fractional integrals and derivatives depends on time, we will consider several different cases. The argument of the variable order could be equal to the current time or it could be equal to the variable of the integral determining the fractional derivative. We provide three different definitions of generalized proportional fractional integrals and Caputo-type derivatives, and the properties of the defined differentials/integrals are discussed and compared with what is known in the literature. Appropriate auxiliary systems with constant-order fractional derivatives are defined and used to construct solutions of the studied problem in the three cases of fractional derivatives. Existence and uniqueness are studied. Also, the Ulam-type stability is defined in the three cases, and sufficient conditions are obtained. The suggested approach is more broadly based, and the same methodology can be used in a number of additional issues.

Problem Involving the Riemann–Liouville Derivative and Implicit Variable-Order Nonlinear Fractional Differential Equations

The problem of boundary values for implicit differential equations with nonlinear fractions involving the variable order and the Riemann–Liouville derivative is examined in this article along with its existence and stability. Specifically, the locally solvability, which is equivalent to the existence of solutions, is related to the symmetry of a transformation of a nonlinear equations system. To demonstrate the reliability of the found results, we design an example.

A novel variable-order fractional chaotic map and its dynamics

In recent years, fractional-order chaotic maps have been paid more attention in publications because of the memory effect. This paper presents a novel variable-order fractional sine map (VFSM) based on the discrete fractional calculus. Specially, the order is defined as an iterative function that incorporates the current state of the system. By analyzing phase diagrams, time sequences, bifurcations, Lyapunov exponents and fuzzy entropy complexity, the dynamics of the proposed map are investigated comparing with the constant-order fractional sine map. The results reveal that the variable order has a good effect on improving the chaotic performance, and it enlarges the range of available parameter values as well as reduces non-chaotic windows. Multiple coexisting attractors also enrich the dynamics of VFSM and prove its sensitivity to initial values. Moreover, the sequence generated by the proposed map passes the statistical test for pseudorandom number and shows strong robustness to parameter estimation, which proves the potential applications in the field of information security.

On numerical solution of two-dimensional variable-order fractional diffusion equation arising in transport phenomena

<abstract><p>In recent years, the application of variable-order (VO) fractional differential equations for describing complex physical phenomena ranging from biology, hydrology, mechanics and viscoelasticity to fluid dynamics has become one of the most hot topics in the context of scientific modeling. An interesting aspect of VO operators is their capability to address the behavior of scientific and engineering systems with time and spatially varying properties. The VO fractional diffusion equation is a fundamental model that allows transitions among sub-diffusive, diffusive and super-diffusive behaviors without altering the underlying governing equations. In this paper, we considered the two-dimensional fractional diffusion equation with the Caputo time VO derivative, which is essential for describing anomalous diffusion in real-world complex systems. A new Crank-Nicolson (C-N) difference scheme and an efficient explicit decoupled group (EDG) method were proposed to solve the problem under consideration. The proposed EDG method is based on a skewed difference scheme in conjunction with a grouping procedure of the solution grid points. Special attention was devoted to investigating the stability and convergence of the proposed methods. Three numerical examples with known exact analytical solutions were provided to illustrate our considerations. The proposed methods were shown to be stable and convergent theoretically as well as numerically. In addition, a comparative study was done between the EDG method and the C-N difference scheme. It was found that the proposed methods are accurate in simulating the considered problem, while the EDG method is superior to the C-N difference method in terms of Central Processing Unit (CPU) timing, verifying the efficiency of the former method in solving the VO problem.</p></abstract>

STUDY ON THE INITIAL BOUNDARY VALUE PROBLEM FOR AFRACTIONAL DIFFERENTIAL EQUATION WITH A FRACTIONALDERIVATIVE OF VARIABLE ORDER

This article studies the convergence of a numerical method for solving an initial-boundary value problem of a fractional differential equation with a variable order of the fractional derivative. In the generalized fractional differential filtration equation with a transitional filtration law in heterogeneous porous media, it is assumed that the order of the fractional derivative depends on the spatial variable. The main attention is paid to the development and theoretical justification of a method that provides high accuracy and efficiency of calculations with a variable order of the fractional derivative. For the numerical solution, an approximation was developed that combines the finite difference method for the time derivative and the finite element method for the spatial variable. The fractional derivative of variable order in the sense of Caputo is approximated by a formula of second order in time. The convergence of the constructed method is proven with order O(τ2+hk+1) for the case α(x)ϵ(0,1). The results of computational experiments for various functions of the order of the fractional derivative are presented, confirming the reliability of the theoretical analysis. The conclusions drawn emphasize the importance and relevance of the further development of numerical methods for fractional differential equations of variable order in modern mathematics and applied sciences, including the modeling of complex processes.

Analytical solutions for a class of variable-order fractional Liu system under time-dependent variable coefficients

In this article, we present a nonlinear model of the Liu system that includes fractional derivatives of variable-order. Due to the nonlocality of the dynamical system, we introduce the fractional derivative with power laws, exponential decay laws, and generalized Mittag-Leffler functions as kernels. We provide a detailed analysis of the existence and uniqueness of the proposed model, as well as the stability of these equations. Due to the existence of time-varying fractional derivatives, the proposed variable-order fractional system exhibits more complex characteristics and more degrees of freedom than an integer or conventional constant fractional-order chaotic Liu oscillator. Different chaotic behaviors can be obtained by using different smooth functions defined within the interval (0,1] as variable orders for fractional derivatives in the simulations. Furthermore, simulations demonstrate that fractional chaotic systems with variable orders can be synchronized.

Variable–Order Model of Cardiac Fibrillation

Atrial fibrillation is a cardiac disorder marked by rapid and disorganized electrical activity, leading to atrial mechanical dysfunction. The alterations in electrophysiological properties during this arrhythmia are not solely attributed to electrical remodeling, structural changes in atrial tissue are also involved. This work aims to formulate a mathematical model for fibrillatory electrical conduction through the implementation of variable-order fractional derivatives. The adoption of such an operator is intended to represent the process of structural remodeling, which has been related to the course of atrial fibrillation. Simulations are performed using a simplified model of the ionic kinetics of the cardiac cell membrane, which allows for distinct electrophysiological properties through its parameterization. For the variable order, a fluctuating function is adopted that can be interpreted as the progression of structural remodeling when the order decreases, and the reverse process when the order increases. Fibrillatory propagation is initiated by generating reentrant conduction, also known as a rotor. We observed that, on the one hand, electrical conduction becomes chaotic as the fractional order decreases, and persists under such dynamics while the order increases. On the other hand, rotational activity persists during the fluctuation of the fractional order. Such mesoscopic outcomes depend on the sensitivity of the microscopic electrophysiological properties to the variations of the fractional order. Moreover, both propagation patterns can be associated with the known course of clinical AF. These results suggest that the fractional order model of cardiac electrophysiology may provide insight into the AF underlying mechanisms.

Existence of chaos and the approximate solution of the Lorenz–Lü–Chen system with the Caputo fractional operator

Dynamical systems and fractional differential equations can be modeled using variable-order differential operators. In this study, the dynamics of a variable-order fractional Lorenz–Lü–Chen system with variable-order and constant-order derivatives are examined. We propose a generalized numerical scheme for simulating fractional differential operators with power-law kernels. The numerical scheme is based on the fundamental theorem of fractional calculus and the Lagrange polynomial interpolation. Furthermore, we examine how random parameter variations affect an approximate unified chaotic model with variable order. Specifically, we examine chaos disappearance in systems that rapidly switch from one family to another. In our simulation study, we show that increasing the key parameter and reducing its reciprocal fractional order together can suppress chaotic fluctuations much earlier than if the parameters remain fixed. Different simulation techniques have also been explored for fractional order switched chaotic systems with random parameters. The proposed system can also be solved using power series methods. The novelty of this paper is not to examine weak chaos masked by additive noise, but rather to describe the dynamics of systems that are subjected to such noise as parameter switching. By adjusting the structure, parameters, and order of the system, different chaotic oscillations and special dynamic behaviors of the Lorenz–Lü–Chen family chaos system are discovered and analyzed.

Periodic Points, Stability, Bifurcations, and Transition to Chaos in Generalized Fractional Maps

This paper is a review of the recent advances in the investigation of the properties of generalized fractional maps. These maps are Volterra difference equations of the convolution type with power-law-like kernels. They include fractional and fractional difference maps of a constant or variable order. The main objective of this article is to describe the criteria of stability, equations defining periodic points and bifurcations, and transition to chaos that are common for all generalized fractional maps.

Investing Neonatal Septic Arthritis: A Systematic Review

Background and Objectives: Septic arthritis is an uncommon condition in neonates. It is a serious disorder, especially because of the possibility of sequelae, if not identified and treated early. In addition, due to the rarity of this condition in neonates and the paucity of signs and symptoms, the diagnosis of septic arthritis is more difficult in older children. The published literature is limited to case series, which is why the purpose of this systematic review is to provide a comprehensive summary of neonatal septic arthritis based on the existing literature. This study identifies the changing trends over time, specifically focusing on intravenous cannulation as a major risk factor that led to the undertaking of the study. Methods: The preferred reporting items for systematic reviews and meta-analyses protocol guidelines were used in this study. Three search engines were used for 239 articles. A total of 26 studies were screened in full text, of which 16 articles underwent quantitative analysis. Due to limited data and heterogeneous reporting, the data were summarized descriptively. Results: The total number of babies and joints studied was 307 and 313, respectively, of which the proportion of male babies was mentioned in 12 case series, accounting for 136(228) (60% [42%-71%]) subjects. The most common presenting symptoms were swelling at the involved joint, erythema, and decreased range of motion in variable order. The most common site of involvement was the hip joint (42%), followed by the knee joint (27%). Staphylococcus aureus (40%) was the most commonly cultured pathogen, followed by Klebsiella (18%). Antibiotics were reported in only 12% (38 babies) of the cases. Open surgery was performed on 16% of the 49 babies. The rest of the babies responded to joint aspiration and irrigation, with or without immobilization. Prompt laboratory and radiographic evaluations could help reduce delays in diagnosis and improve outcomes. Blood and tissue cultures were positive in most of the cases. Previous culture reports for the same unit guide the addition of empirical antibiotics. Conclusions: In our case series, the majority of the babies were premature. The most common joint involved is the hip joint. The most common underlying cause is sepsis and catheterization. Intravenous canulation is also a major risk factor because of the continuity of metaphysis and epiphysial vessels. The most common presenting symptom is local signs. Unexplained fever and irritability make it better to look for a local examination of joints. Follow-up is mandatory as a long-term sequel is more.

Piecewise implicit coupled system under ABC fractional differential equations with variable order

<abstract><p>This research paper presented a novel investigation into an implicit coupled system of fractional variable order, which has not been previously studied in the existing literature. The study focused on establishing and developing sufficient conditions for the existence and uniqueness of solutions, as well as the Ulam-Hyers stability, for the proposed coupled system without using semigroup property. By extending the existing conclusions examined for the Atangana-Baleanu-Caputo (ABC) operator, we contributed to advancing the understanding of variable-order fractional differential equations. The paper provided a solid theoretical foundation for further analysis, numerical simulations, and practical applications. The obtained results have implications for designing and controlling systems modeled using fractional variable order equations and serve as a basis for addressing a wide range of dynamical problems. The transformation techniques, qualitative analysis, and illustrative examples presented in this work highlight its unique contributions and potential to serve as a foundation for future research.</p></abstract>