Variable Order Research Articles

A New Operational Matrices Based-Technique for Fractional Integro Reaction-Diffusion Equation Involving Spatiotemporal Variable-Order Derivative

In this article, a novel generalized model of the nonlinear fractional integro-reaction-diffusion equation (NFIRDE) with spatiotemporal variable-order (SVO) is introduced, where the variable order derivatives are equipped with the Atangana–Baleanu–Caputo (ABC) sense. This model represents a great generalization of a significant type of NFIRDE and their applications. Moreover, A novel, efficient and fully spectral shifted Legendre tau technique is developed to solve the proposed model. Despite the difficulty of applying this mechanism to solve this type of equations, due to the presence of nonlinear terms and the SVO functions that appear in the traditional differential and integral operational matrices. We deduce some new operational matrices that play the fundamental role in facilitating the implementation of the tau method. These operational matrices represent the SVO ABC-derivative, the integro term within the model, as well as the vector multiplications with the space-time Kronecker product. As a result, the proposed model is restructured into a system of nonlinear algebraic equations, which simplifying the solving process. We illustrate our method’s effectiveness and validity with numerical examples with both smooth and non-smooth solutions. Our findings show that the proposed tau method delivers accurate results and exhibits non-local properties.

A new approach of generalized shifted Vieta-Fibonacci polynomials to solve nonlinear variable order time fractional Burgers-Huxley equations

A new approach of generalized shifted Vieta-Fibonacci polynomials to solve nonlinear variable order time fractional Burgers-Huxley equations

Higher Order Stable Numerical Algorithm for the Variable Order Time-Fractional Sub-diffusion Equation

Higher Order Stable Numerical Algorithm for the Variable Order Time-Fractional Sub-diffusion Equation

Existence results for variable-order fractional Kirchhoff equations with variable exponents

This study establishes the existence of two nontrivial weak solutions for a specific class of Kirchhoff-type equations. Utilizing the Mountain Pass Theorem and the Ekeland variational principle, alongside insights from fractional Sobolev spaces with variable orders and exponents, we provide a concise demonstration of our result.

The Impact of White Noise on Chaotic Behavior in a Financial Fractional System with Constant and VariableOrder: A Comparative Study

In this study, we investigated the impact of white noise on the chaotic dynamics of a financial system with Caputo fractional derivatives of constant- and variable-order. We solve thesefractional financial systems numerically. We analyze the chaotic behavior of these fractional-order hyperchaotic systems through simulations, study the effects of white noise on chaotic dynamics, and provide a comparison between fractional hyperchaotic systems of constant and variable orders. The study reveals that white noise can either amplify or suppress chaos, with significant differences observed between the constant- and variable-order cases. We obtained numerous intriguing graphical findings for the model by evaluating various scenarios. The findings provide useful information for better understanding financial market dynamics in the presence of noise.

Gronwall-Type Inequalities and Qualitative Studies onHigher-Variable Orders of Atangana-Baleanu Fractional Operators via Increasing Functions

This paper introduces a novel extension of Caputo-Atangana-Baleanu and Riemann-Atangana-Baleanu fractional derivatives from constant to increasing variable order. We generalize the fractional order from a fixed value in (0, 1] to a time-dependent function in (k, k + 1], where k ≥ 0. The corresponding Atangana-Baleanu fractional integral is also extended. Key properties ofthese new definitions are explored, including a generalized Gronwall inequality. We then delve into the analysis of higher-variable initial fractional differential equations using the Caputo-Atangana-Baleanu operator with an increasing function, establishing existence and uniqueness results via Picard’s iterative method. The findings presented in this work are expected to stimulate further research on inequalities and fractional differential equations related to Atangana-Baleanu fractional calculus with respect to increasing functions. Concrete examples are provided to illustrate the practical applications of our results.

Numerical Method for the Variable-Order Fractional Filtration Equation in Heterogeneous Media

This paper presents a study of the application of the finite element method for solving a fractional differential filtration problem in heterogeneous fractured porous media with variable orders of fractional derivatives. A numerical method for the initial-boundary value problem was constructed, and a theoretical study of the stability and convergence of the method was carried out using the method of a priori estimates. The results were confirmed through a comparative analysis of the empirical and theoretical orders of convergence based on computational experiments. Furthermore, we analyzed the effect of variable-order functions of fractional derivatives on the process of fluid flow in a heterogeneous medium, presenting new practical results in the field of modeling the fluid flow in complex media. This work is an important contribution to the numerical modeling of filtration in porous media with variable orders of fractional derivatives and may be useful for specialists in the field of hydrogeology, the oil and gas industry, and other related fields.

Infinitesimal Prolongation for Fractional Derivative $$\varPsi $$-Caputo Variable Order and Applications

Infinitesimal Prolongation for Fractional Derivative $$\varPsi $$-Caputo Variable Order and Applications

Legendre Polynomials and Techniques for Collocation in the Computation of Variable-Order Fractional Advection-Dispersion Equations

The paper discusses a numerical approach to solving complicated partial differential equations, with a particular emphasis on fractional advection-dispersion equations of space-time variable order. With the use of fractional derivative matrices, Legendre polynomials, and numerical examples and comparisons, it surpasses current methods by utilizing spectral collocation techniques. It resolves equations involving spatial and time variables that are variable-order fractional advection–dispersion (VOFADE). Legendre polynomials serve as basis functions in this method, whereas Legendre operational matrices are employed for fractional derivatives. The technique is more computationally efficient since it reduces fractional advection–dispersion equations to systems of algebraic equations. Numerical examples and a comparison with current approaches illustrate the method’s superior performance in solving complicated partial differential equations, especially in the context of transport processes.

Guided structure learning of DAGs for count data

In this paper, we tackle structure learning of directed acyclic graphs (DAGs), with the idea of exploiting available prior knowledge of the domain at hand to guide the search of the best structure. In particular, we assume to know the topological ordering of variables in addition to the given data. We study a new algorithm for learning the structure of DAGs, proving its theoretical consistency in the limit of infinite observations. Furthermore, we experimentally compare the proposed algorithm to several popular competitors, to study its behaviour in finite samples. Biological validation of the algorithm is presented through the analysis of non-small cell lung cancer data.

Analysis of Caputo fractional variable order multi-point initial value problems: existence, uniqueness, and stability

In this paper, we examine the existence, uniqueness, and stability of solutions for a Caputo variable order ϑ-initial value problem (ϑ-IVP) with multi-point initial conditions. The proofs for uniqueness and existence leverage Sadovski’s and Banach’s fixed point theorems, along with the Kuratowski measure of noncompactness. Furthermore, we explore the Ulam–Hyers–Rassias (UHR) stability of the solution. To validate our findings, we present a numerical example.

Who gets better after surgery for degenerative cervical myelopathy? A responder analysis from the multicenter Canadian spine outcomes and research network

Degenerative cervical myelopathy (DCM) is the most common cause of acquired non-traumatic spinal cord injury worldwide. Surgery is a common treatment for DCM; however, outcomes often vary across patients. To inform preoperative education and counseling, we performed a responder analysis to identify factors associated with treatment response. An observational cohort study was conducted utilizing prospectively collected data from the Canadian Spine Outcomes Research Network (CSORN) registry collected between 2015-2022. We included all surgically treated DCM patients with complete 12-month follow-up and patient-reported outcomes (PROs) available at 1-year. Treatment response was measured using the minimal clinically important difference (MCID) in PROs including the Neck Disability Index (NDI) and EuroQol-5D (EQ-5D) at 12 months post-surgery. A Least Absolute Shrinkage and Selection Operator (LASSO) machine learning model was used to identify significant associations between 14 pre-operative patient factors and likelihood of treatment response measured by achievement of the MCID in NDI, and EQ-5D. Variable importance was measured using standardized coefficients. To test robustness of findings we trained a separate XGBOOST model, with variable importance measured using SHAP values. Among the 554 DCM patients included, 229 (41.3%) and 330 (59.6%) patients responded to treatment by meeting or surpassing MCID thresholds for NDI and EQ-5D at 1-year, respectively. LASSO regression for likelihood of treatment response measured through NDI found the variable importance rank order to be baseline NDI (OR 1.06 per 1 point increase; 95% CI 1.04 - 1.07), then symptom duration (OR 0.65; 95% CI 0.44-0.97). For EQ-5D, the variable importance rank order was baseline EQ-5D (OR 0.16 per 0.1-point increase; 95% CI 0.03 - 0.78), living independently (OR 2.17; 95% CI 1.22 - 3.85), symptom duration (OR 0.62; 95% CI 0.40 - 0.97), then number of levels affected (OR 0.80 per additional level; 95% CI 0.67 - 0.96). A separate XGBoost model of treatment response measured through NDI, corroborated findings that patients with higher baseline NDI, and shorter symptom duration were more likely to respond to treatment, and additionally found older patients, and those with kyphosis on baseline upright x-ray were less likely to respond. Similarly, an XGBoost model for treatment response measured through EQ-5D corroborated findings that patients with higher baseline EQ-5D, shorter symptom duration, living independently, with fewer affected levels were more likely to respond to treatment, and additionally found older patients were less likely to respond. Our findings suggest patients with shorter symptom duration, higher baseline patient NDI, lower EQ-5D, younger age, living independently, without kyphosis on pre-operative x-ray, and fewer affected levels are more likely to respond to treatment. Timing of surgery with respect to patient symptoms is underscored as a crucial and modifiable patient factor associated with improved surgical outcomes in DCM.

Temporal Second-order Scheme for a Hidden-memory Variable Order Time Fractional Diffusion Equation with an Initial Singularity

Temporal Second-order Scheme for a Hidden-memory Variable Order Time Fractional Diffusion Equation with an Initial Singularity

Research on Image Encryption Method based on the Chaotic Iteration of a Ternary Nonlinear Function

Objective: Considering that some image encryption algorithms have the disadvantages of complex structure and high computational cost, and there are not many commonly used chaotic systems, which are easy to crack by attacks, to solve these problems, this paper proposes an image encryption algorithm based on three-dimensional nonlinear functions to solve these problems. Methods: The algorithm mainly combines the sinusoidal chaotic map with the ternary nonlinear function system to encrypt the image. Firstly, multiple ternary nonlinear function chaotic systems are designed. Then, the function iteration system is changed to invoke the computation of a specific expression under a random number; it is a chaotic sequence generated according to a chaotic mapping such as sine, and then the value of this chaotic sequence is used to select a ternary nonlinear function for iteration to obtain a chaotic sequence. Finally, the chaotic sequence performs the XOR and scrambling operations on the grey image Results: The algorithm has a simple structure, a better encryption effect, and more incredible difficulty deciphering. Moreover, through the phase diagram and bifurcation diagram, it can be seen that the system has good chaotic characteristics Conclusion: The method in this paper is novel; this method is a random variable order composite operation, which can not only be applied to image encryption but also can be used for fractal map generation and so on, and in some other chaotic fields will have a wide range of applications. It has essential research value.

Implementing Sobol's Global Sensitivity Analysis to SFRC's Flexural Strength Predictive Equation

Steel fibers are essential for SFRC since they strengthen the material’s resistance to bending and cracking stresses and guarantee its endurance. However, the uncertainty of input parameters such as concrete compositions and steel fibers causes the stochasticity of flexural strength. The study uses big data to analyze the influence of fibers and other concrete compositions on the flexural strength properties of SFRC. For this purpose, the study focuses on developing predictive models for SFRC flexural properties based on a comprehensive database comprising two hundred and seven experimental results recorded by seventeen researchers. Bayesian Model Averaging is employed to identify significant components that influence the overall flexural strength and to develop a predictive flexural strength model. Monte Carlo simulation generates big data by utilizing the probability distribution of input variables and the predictive flexural strength model. The study used Sobol’s global sensitivity analysis method to assess various input parameters’ sensitivity to SFRC flexural strength based on the generated database. The impact order of input variables on the flexural strength is identified, as determined by the Sobol’ Indice.

Review of approximation formulas for fractional derivatives of variable order

Fractional differential equations of variable order, depending on a time or space variable, are successfully used to study time and/or spatial dynamics. The purpose of this review is to consider the latest research and results related to the basic definitions and approximation formulas for fractional derivatives of variable order. The review begins with an examination of existing definitions based on various physical and practical knowledge. The following are formulas for discretizing fractional derivatives, since they play a key role in numerical modeling, providing an effective means of describing complex systems with long-term memory, heterogeneous parameters and non-local interactions. Their use not only improves the accuracy of models, but also facilitates the numerical solution of differential equations, which is essential for analyzing and predicting the behavior of real processes in various fields of science and engineering. This review is intended to help readers select appropriate definitions and discretization formulas to effectively solve specific physical and engineering problems.

Enhancing Road Traffic Prediction Using Data Preprocessing Optimization

Traffic prediction is essential for transportation planning, resource allocation, congestion management and enhancing travel experiences. This study optimizes data preprocessing techniques to improve machine learning-based traffic prediction models. Data preprocessing is critical in preparing the data for machine learning models. This study proposes an approach that optimizes data preprocessing techniques, focusing on flow-based analysis and optimization, to enhance traffic prediction models. The proposed approach explores fixed and variable orders of data preprocessing using a genetic algorithm across five diverse datasets. Evaluation metrics such as root mean squared error (RMSE), mean absolute error (MAE) and R-squared error assess model performance. The results indicate that the genetic algorithm’s variable order achieves the best performance for the ArcGIS Hub and Frementon Bridge Cycle datasets, fixed order one preprocessing for the Traffic Prediction dataset and variable order using the genetic algorithm for the PeMS08 dataset. Fixed order 2 preprocessing yields the best performance for the XI AN Traffic dataset. These findings highlight the importance of selecting the appropriate data preprocessing flow order for each dataset, improving traffic prediction accuracy and reliability. The proposed approach advances traffic prediction methodologies, enabling more precise and reliable traffic forecasts for transportation planning and management applications.

Hidden Chaos in a new memristor-based discrete system with commensurate, incommensurate and variable fractional orders

Memristors special qualities and potential to completely transform computing and memory technology have made them indispensable parts of many applications , which have attracted a lot of interest in complex dynamics. The manuscript focuses on an exhaustive exploration of the chaotic dynamical behaviors exhibited by memristor-based discrete systems under various fractional orders. Specifically, it scrutinizes three distinct variants: the memristor-based discrete system with commensurate fractional order, the memristor-based discrete system with incommensurate fractional order, and the memristor-based discrete system with variable fractional order. The study highlights how different fractional order cases can produce different dynamic patterns and chaotic behavior in the model. Each variant presents unique characteristics, such as the absence of a fixed point and the presence of multiple chaotic hidden attractors. To substantiate the theoretical assertions, a comprehensive array of numerical methods is employed, including bifurcation charts, phase portraits, and the computation of Lyapunov exponents. These techniques serve to validate the presence of chaotic attractors within the proposed variable fractional versions of the memristor-based discrete system. MATLAB R2024a simulations are conducted to further bolster the obtained findings, providing empirical validation and robustness to the theoretical conjectures put forth in the manuscript. Through this approach, the work offers valuable insights into the complex dynamics of memristor-based discrete systems under fractional orders, paving the way for future research and practical applications in the field of nonlinear dynamics and chaos theory.

Affine Projection Algorithms with Novel Schemes of Variable Projection Order

Affine Projection Algorithms with Novel Schemes of Variable Projection Order

Stripe Noise Removal Algorithm for Infrared Remote Sensing Images Based on Adaptive Weighted Variable Order Model

In the infrared remote sensing imaging system, the output of the detector is uneven, resulting in noticeable stripe noise in the image, which significantly reduces the image quality. Therefore, in response to the problem of establishing a variational model for the image globally and fixing the fractional order of the regularization term in the existing stripe noise removal algorithms based on model optimization, a stripe noise removal algorithm for infrared remote sensing images based on an adaptive weighted variable order model is proposed. In this algorithm, the vertical and approximate components containing stripe noise in the image are first separated through multi-level and multi-scale wavelet transform, and only these two components are processed; secondly, the global sparsity and the gradient sparsity of stripe noise and the variable order gradient sparsity of the information component are constrained by L1 norm, so as to establish the stripe noise removal model. The order of the fractional derivative is adaptively assigned to each pixel of the information component through local variance; then, an adaptive weight operator is introduced, which can assign different weights to the pixels in the variable order derivative of the information component on the basis of the image gradient information. Finally, through the ADMM algorithm, we can obtain the optimal solution of this model. The processing results of simulated and actual data indicate that the proposed algorithm performs well in all indexes and has obvious advantages in removing stripe noise and preserving image details.