This special issue is devoted to the advancement of fractional-order differential equations and their wide-ranging applications [...]

We study a discrete boundary value problem involving the Riemann-Liouville fractional operator. In particular, we introduce a discrete fractional summing boundary value problem that parallels the integral boundary value problem for fractional differential equations. By generalizing a commonly used fractional difference operator, we derive the corresponding Green's function and reformulate the problem as a fixed-point equation. The existence and multiplicity of positive solutions are established. Two illustrative examples are provided to demonstrate the applicability of the theoretical results, and numerical simulations are included for further validation.

This paper explores short-term stock price forecasting using high-frequency 2-minute interval data for Apple Inc. (AAPL), focusing on the application and evaluation of three predictive modeling approaches: Linear Regression, XGBoost, and Long Short-Term Memory (LSTM) neural networks. The dataset, collected via Yahoo Finance (YFinance API), was enriched with technical indicators, rolling statistics, and lag features to support predictive learning. The models were evaluated on their ability to predict the next closing price and directional movement. While Linear Regression achieved the highest R2 score (0.9778) and lowest MSE (0.0502), XGBoost demonstrated competitive performance and provided valuable insights into feature importance. LSTM, though promising for capturing sequential dependencies, underperformed with a higher MSE (0.2490) and reduced R2 (0.8932), likely due to the noisy and highly volatile nature of intraday data. All models showed limited directional accuracy (~52%), highlighting the challenge of predicting micro-movements in high-frequency financial time series. This study concludes with a discussion on the strengths and limitations of each approach and outlines future improvements including feature selection, hybrid modeling, and enhanced data preprocessing to boost intraday forecast reliability.

We study fixed-point theorems of contractive mappings in b-metric space, cone b-metric space, and the newly introduced extended b-metric space. To generalize an existence and uniqueness result for the so-called Φs functions in the b-metric space to the extended b-metric space and the cone b-metric space, we introduce the class of ΦM functions and apply the Hölder continuous condition in the extended b-metric space. The obtained results are applied to prove the existence and uniqueness of solutions and positive solutions for nonlinear integral equations and fractional boundary value problems. Examples and numerical simulation are given to illustrate the applications.

In this paper, the existence and nonexistence of solutions for a class of fractional differential equations are studied by using an interesting fixed point theorem on order intervals and the well-known Schauder's fixed point theorem. The theorems generalize some existing results on this topic.

Mathematical Methods in the Applied SciencesEarly View LETTER Recent advances in neural network methods for FDE and its application Feng Gao, Corresponding Author Feng Gao gaofeng@qut.edu.cn orcid.org/0000-0002-9546-462X Qingdao University of Technology, Qingdao, China Correspondence Feng Gao, Qingdao University of Technology, China. Email: gaofeng@qut.edu.cn Contribution: ConceptualizationSearch for more papers by this authorWenying Feng, Wenying Feng Trent University, Peterborough, Canada Contribution: MethodologySearch for more papers by this authorXinguang Zhang, Xinguang Zhang orcid.org/0000-0001-9250-6823 Curtin University of Technology, Perth, Australia Contribution: MethodologySearch for more papers by this authorFudong Ge, Fudong Ge China University of Geosciences, Wuhan, China Contribution: MethodologySearch for more papers by this author Feng Gao, Corresponding Author Feng Gao gaofeng@qut.edu.cn orcid.org/0000-0002-9546-462X Qingdao University of Technology, Qingdao, China Correspondence Feng Gao, Qingdao University of Technology, China. Email: gaofeng@qut.edu.cn Contribution: ConceptualizationSearch for more papers by this authorWenying Feng, Wenying Feng Trent University, Peterborough, Canada Contribution: MethodologySearch for more papers by this authorXinguang Zhang, Xinguang Zhang orcid.org/0000-0001-9250-6823 Curtin University of Technology, Perth, Australia Contribution: MethodologySearch for more papers by this authorFudong Ge, Fudong Ge China University of Geosciences, Wuhan, China Contribution: MethodologySearch for more papers by this author First published: 18 June 2022 https://doi.org/10.1002/mma.8500Read the full textAboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat No abstract is available for this article. Early ViewOnline Version of Record before inclusion in an issue RelatedInformation

We study bike sharing systems using machine learning techniques and statistics data analytics. Using data from the state of Minnesota, we investigate number of trips from each station and trip length in time affected by weather condition, weekday, and weekend. Correlations among the parameters are obtained and discussed. Machine learning algorithms including neural networks and gradient tree boosting are applied to predict the number of start trips using the input dataStation ID, number of docks, date, and weather variables. Our results show the efficiency of the learning algorithms in this applied area.

In this paper, we study the dynamic behavior of a stochastic tungiasis model for public health education. First, the existence and uniqueness of global positive solution of stochastic models are proved. Secondly, by constructing Lyapunov function and using It o ^ formula, sufficient conditions for disease extinction and persistence in the stochastic model are proved. Thirdly, under the condition of disease persistence, the existence and uniqueness of an ergodic stationary distribution of the model is obtained. Finally, the importance of public health education in preventing the spread of tungiasis is illustrated through the combination of theoretical results and numerical simulation.

With the increased dimensionality of datasets, high-dimensional data decomposition models have become essential data analysis tools. However, the decomposition method usually suffers from the overfitting problem and, consequently, cannot achieve state-of-the-art performance. This motivates the introduction of various regularization terms. The commonly applied Ridge regression has limited applicability for the asperity dataset and reduces performance for sparse data, while the Lasso regression has higher efficiency in the sparse dataset. To address this challenge, we propose a modified regularization term designed by integrating both the Lasso and Ridge regressions. The different roles of these two regressions are analyzed. By adjusting the weights of the regression in the regularization term, the existing decomposition method can be applied to the dataset with different degrees of sparsity. The experiments show that the modified regularization term yields consistent improvement in the performance of existing benchmarks.

This paper studies a class of integral boundary value problem of fractional q-difference equations. We first give an explicit expression for the associated Green's function and obtain an important property of the function. The new property allows us to prove sufficient conditions for the existence of positive solutions based on the associated parameter. The results are derived from the application of a fixed point theorem on order intervals.

Accurate prediction of future electricity demand is important in the energy industry. Machine learning for time series prediction provides solutions for short term energy forecasting through a variety of algorithms, such as LSTM, SVR, Xgboost, and Facebook Prophet. However, many companies primarily rely on univariate time series algorithms, while numerous external data, e.g. weather data, are available as input features for energy forecasting. In this paper, we study the impact of external features on the performance of univariate and multivariate time series algorithms for Short-term Energy Forecasting using a standard benchmark energy data set. Quantitative comparisons on prediction accuracy measured by Root Mean Square Error (RMSE) and Mean Absolute Percentage Error (MAPE) for the models are obtained. It is found that multivariate algorithms using external features outperform univariate algorithms, and that multivariate algorithms achieve reasonable accuracy even without using past step energy consumption as an input feature.

We study a class of nonlinear operators that can be written as the composition of a linear operator and a nonlinear map. We obtain results on fixed point index based on parameters that are related to the definitions of nonlinear spectra. As a particular case, existence of positive solutions for a second-order differential equation with separated boundary conditions is proved. The result also provides a spectral interval for the corresponding Hammerstein integral operator.

In this paper, we evaluate two deep learning models which integrate convolutional and recurrent neural networks. We implement both sequential and parallel architectures for fine-grain musical subgenre classification. Due to the exceptionally low signal to noise ratio (SNR) of our low level mel-spectrogram dataset, more sensitive yet robust learning models are required to generate meaningful results. We investigate the effects of three commonly applied optimizers, dropout, batch regularization, and sensitivity to varying initialization distributions. The results demonstrate that the sequential model specifically requires the RMSprop optimizer, while the parallel model implemented with the Adam optimizer yielded encouraging and stable results achieving an average F1 score of 0.63. When all factors are considered, the optimized hybrid parallel model outperformed the sequential in classification accuracy and system stability.

It is well known that the existence of roots theorem for continuous functions not only is important in Mathematical Analysis, but also has wide applications. In this paper, we generalize the existing theorems in this area to compounded functions. Application of the obtained results is also shown by examples.

Building safe, reliable, fully automated energy smart grid systems requires a trustworthy electric load forecasting system. Recent work has shown the efficacy of Long Short-Term Memory neural networks in energy load forecasting. However, such predictions do not come with an estimate of uncertainty, which can be dangerous when critical decisions are being made autonomously in energy production and distribution. In this paper, we present methods for evaluating the uncertainty in short-term electrical load predictions for both deep learning and gradient tree boosting. We train Bayesian deep learning and gradient boosting models with real electric load data and show that an uncertainty estimate may be obtained alongside the prediction itself with minimal loss of accuracy. We find that the uncertainty estimates obtained are robust to changes in the input features. This result is an important step in building reliable autonomous smart grids.

We study a two-point Boundary Value Problem depending on two parameters that represents a mathematical model arising from the combustion theory. Applying fixed point theorems for concave operators, we prove uniqueness, existence, upper, and lower bounds of positive solutions. In addition, we give an estimation for the value of λ* such that, for the parameter λ∈[λ*,λ*], there exist exactly three positive solutions. Numerical examples are presented to illustrate various cases. The results complement previous work on this problem.

We study a class of nonlinear fractional difference equations with nonlocal boundary conditions at resonance. The system is inspired by the three‐point boundary value problem for differential equations that have been extensively studied. It is also an extension to a fractional difference equation arising from real‐world applications. Converting the problem to an equivalent system corresponding to the integral operator and Green's function for differential equations, we are able to apply the coincidence degree theory for semilinear operators to obtain sufficient conditions for the existence of solutions. In addition, we prove a new property of the Gamma function and construct a family of examples to illustrate the applications of the results.

Errors are always present in predictions produced by machine learning models. Producing a quantitative estimate of the uncertainty in a model’s output is crucial for many fields, especially those where predictive models drive important decisions. In this paper we discuss two methods for producing prediction intervals for neural network, random forest, and gradient boosted tree models. We then evaluate the prediction intervals produced by each algorithm by predicting the expected ride length for a NYC taxi trip dataset. We show that inductive conformal prediction produces the most reliable intervals for all machine learning models investigated.

We study the existence of positive solutions for second-order differential equations with separated integral boundary conditions. The nonlinear part of the equation involves the derivative and may be singular for the second and third space variables. The result ensures existence of a positive solution when the parameters are in certain ranges. The proof depends on general properties of the associated Green's function and the Krasnosel'skii–Guo fixed point theorem applied to a perturbed Hammerstein integral operator. Both numerical and analytical examples are constructed to illustrate applications of the theorem to a group of equations. The result generalizes previous work.

Many statistical and machine learning models for prediction make use of historical data as an input and produce single or small numbers of output values. To forecast over many timesteps, it is necessary to run the program recursively. This leads to a compounding of errors, which has adverse effects on accuracy for long forecast periods. In this paper, we show this can be mitigated through the addition of generating features which can have an “anchoring” effect on recurrent forecasts, limiting the amount of compounded error in the long term. This is studied experimentally on a benchmark energy dataset using two machine learning models LSTM and XGBoost. Prediction accuracy over differing forecast lengths is compared using the forecasting MAPE. It is found that for LSTM model the accuracy of short term energy forecasting by using a past energy consumption value as a feature is higher than the accuracy when not using past values as a feature. The opposite behavior takes place for the long term energy forecasting. For the XGBoost model, the accuracy for both short and long term energy forecasting is higher when not using past values as a feature.