Documents published in Scipedia

• Solvability of Quadratic Fractional Integral Equations by Family of Measures of Noncompactness in Fréchet Algebra C(R+, L1(R+))

  H. Kayvanloo, M. Kahrobaeian, M. Mursaleen

  
  

  Abstract

  We define the Fréchet algebra C(R+,L1(R+)) and then define a new family of measures of noncompactness. We prove a fixed point theorem that generalizes the Darbo’s fixed point theoremin this space. By applying the technique of measures of noncompactness in conjunction with new fixed point theorem, we investigate the solvability of a certain quadratic fractional integral equations. Then, we state two examples to support our main results.

  Abstract
  We define the Fréchet algebra C(R+,L1(R+)) and then define a new family of measures of noncompactness. We prove a fixed point theorem that generalizes the Darbo’s [...]

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• Tuning Curvature in Quadratic Regression via Caputo Fractional Derivatives: Theory and Applications

  Y. Tandogdu, S. Ilgaz, M. Awadalla, Y. Yameni

  
  

  Abstract

  Classical regression can only examine the relation between response and predictor variables based on integer order calculus theory. What happens when non integer order calculus is considered is a field where a vast spectrum of studies can be undertaken. The purpose of this study introduces a novel fractional-order quadratic regression model grounded in the Caputo derivative framework, addressing the limitation and the rigidity of classical polynomial regression in adapting to the intrinsic curvature of data. The core innovation is the use of the fractional order ν as a tunable parameter for curvature-sensitive optimization. Our main contributions are fourfold: First, we establish a fundamental theoretical pillar by proving that the second-order Caputo derivative preserves the curvature direction of quadratic functions, enabling a principled optimization framework. Second, we rigorously demonstrate the model’s robustness by proving the existence and uniqueness of solutions via Banach’s fixed point theorem and establishing stability bounds through a fractional Grönwall inequality. Third, we develop a practical methodology to identify an optimal fractional order ν that minimizes the error-to-explained-variation ratio (SSE/SSR). Finally, we validate the framework on four diverse real-world datasets from air quality, soil science, education, and meteorology. The proposed model consistently outperforms classical quadratic regression, achieving a reduction in the SSE/SSR ratio by up to 21% in specific cases. The proposed method yields more efficient models with either lower estimation error or higher correlation coefficients, positioning Caputo fractional quadratic regression as a powerful and theoretically sound alternative for modeling cases where quadratic regression is considered appropriate.OPEN ACCESS Received: 10/09/2025 Accepted: 05/11/2025 Published: 23/01/2026

  Abstract
  Classical regression can only examine the relation between response and predictor variables based on integer order calculus theory. What happens when non integer order calculus [...]

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• Approximate Calculation of the Generalized Erdélyi-Kober Operator Using a Cubic Spline

  S. Karimov, E. Islamov, E. Shishkina

  
  

  Abstract

  This article investigates the problem of approximating the generalized Erdélyi-Kober fractional operator (often referred to as the Lowndes operator) using cubic splines. A method based on cubic spline interpolation is proposed for approximating the operator on a non-uniform grid. The convergence rate of the proposed method is proven, and its stability is analyzed. Error bounds are established for functions in the class C4[0; b], providing a mathematical justification for the accuracy of the approximation. The efficiency of the method is validated through practical examples using test functions such as f (x)= x4.7and f (x)= cos x, with results presented in graphical and numerical forms. This approach ensures high accuracy and flexibility in computing fractional integrals, which is of significant importance for solving fractional models used in physics, engineering, and other sciences. The article also provides an overview of the role of the generalized Erdélyi-Kober operator in modern fractional calculus and its applications.OPEN ACCESS Received: 06/06/2025 Accepted: 08/09/2025 Published: 27/10/2025

  Abstract
  This article investigates the problem of approximating the generalized Erdélyi-Kober fractional operator (often referred to as the Lowndes operator) using cubic splines. [...]

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• Bidirectional Fluid-Structure Coupling Analysis of Steering Tool Performance with Spring Stress Consideration

  X. Zhou, Y. Peng, W. Cheng, Y. Xue, H. Fang

  
  

  Abstract

  This paper investigates the automatic steering tool in the integrated injection-production tubing string and proposes an analysis method for bidirectional fluid-structure coupling problems, incorporating spring stress accumulation. The pre-stress field feature in ABAQUS software is used to model the variation in leaf spring stress. A ball valve acts as an intermediary, and FLUENT software is employed to simulate the transfer of forces between the flow field and the ball valve’s motion. The results indicate that the transition response time from steam injection to oil production under the designed operating conditions is approximately 0.0787 s. The motion path of the ball valve aligns with expectations, exhibiting stable movement. Additionally, an indoor simulation of the steering tool’s opening and closing performance was conducted, with high-speed cameras used for experimental validation. The simulation and experimental results show a discrepancy of less than 15%, confirming the accuracy of the numerical model and providing theoretical support for the practical application of integrated injection-production tubing string technology.OPEN ACCESS Received: 06/11/2024 Accepted: 01/04/2025 Published: 30/06/2025

  Abstract
  This paper investigates the automatic steering tool in the integrated injection-production tubing string and proposes an analysis method for bidirectional fluid-structure [...]

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• Statistical Inference of Step Stress Partially Accelerated Life-Testing for Insulating Fluid between Electrodes under Censored Data and Different Loss Functions

  R. Alotaibi, H. Rezk, E. Almetwally

  
  

  Abstract

  Long testing times are usually required for the life testing of very reliable products or materials. The testing process can be hastened by using accelerated life tests. The lifespan of the items that accelerated life tests inspect is reduced since they test products in more severe circumstances than those found in regular use scenarios. Data that was censored and disclosed the precise timings of failure may point to accelerated life tests where all units assigned to test are unknown, or where all units assigned to test have not failed for a few reasons, including challenges with technology, tools, costs, and schedules. The step-stress partially accelerated life test was examined in this work using the type-I progressive hybrid censoring scheme and the type-II progressive censoring scheme. The influence of the stress shift is explained using the tempered random variable model, where the failure times of the items are assumed to follow the alpha power Lomax distribution. The unknown parameters are estimated using the maximum likelihood estimation and Bayesian methods. The asymptotic theory of maximum likelihood estimation is also employed in the construction of the approximate confidence intervals. While the point estimates under two censoring schemes are compared in terms of absolute biases and root mean squared errors, approximate confidence intervals and coverage probabilities are compared in terms of their lengths and coverage probabilities. Additionally, three possible optimal test strategies are investigated using different optimal criteria. The performance of the estimators was evaluated and contrasted with two censoring techniques with various sample sizes using a simulation study. Finally, a numerical example for insulating fluid between electrodes data is presented to illustrate how the methods will work in real-world scenarios.OPEN ACCESS Received: 11/06/2025 Accepted: 29/07/2025 Published: 23/01/2026

  Abstract
  Long testing times are usually required for the life testing of very reliable products or materials. The testing process can be hastened by using accelerated life tests. The [...]

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• Generalized Convergence of Sequences of Fuzzy Numbers by Means of Modulus Functions

  I. Ibrahim, D. Baleanu, M. Yousif, P. Mohammed, T. Abdeljawad

  
  

  Abstract

  In this paper, we extend the concepts of statistical convergence and strong summability for the sequences of fuzzy numbers using modulus functions. By introducing appropriate conditions on the modulus functions, we generalize and refine existing notions of convergence within the fuzzy setting. Additionally, we establish several interrelationships between these extended concepts, thereby contributing to the deeper understanding of summability and convergence behavior in the sequences of fuzzy numbers.OPEN ACCESS Received: 16/06/2025 Accepted: 20/08/2025 Published: 27/11/2025

  Abstract
  In this paper, we extend the concepts of statistical convergence and strong summability for the sequences of fuzzy numbers using modulus functions. By introducing appropriate [...]

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• Optical Soliton Dynamics in Nonlinear Evolution Equations: Modified Kawahara and Modified Benjamin Bona Mahony Models

  H. Mehmood, D. Baleanu, M. Abbas, M. Yousif, P. Mohammed, F. Abdullah, I. Ibrahim

  
  

  Abstract

  This paper explores the dynamic behavior of optical soliton solutions for the modified Kawahara (mK) equation and the modified BenjaminBona-Mahony (mBBM) equation, two significant nonlinear evolution equations. Using an advanced analytical approach, a diverse set of soliton solutions is derived, including bell-shaped, anti-bell-shaped, W-shaped, M-shaped, and periodic waveforms. These solutions unveil the intricate nonlinear dynamics underlying the equations. The robustness of the method is demonstrated through comprehensive 2D, 3D, and contour visualizations, offering clear insights into the physical significance of the solitons. The study enhances the existing catalog of soliton solutions, contributing to a deeper understanding of nonlinear wave propagation and its potential applications in fields such as optical communication and fluid dynamics.OPEN ACCESS Received: 06/05/2025 Accepted: 09/06/2025 Published: 15/08/2025

  Abstract
  This paper explores the dynamic behavior of optical soliton solutions for the modified Kawahara (mK) equation and the modified BenjaminBona-Mahony (mBBM) equation, two significant [...]

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• Computational Investigation of Novel Half-Normal Data Using Improved Type-II Adaptive Progressive Censoring and Its Application to the Shasta Reservoir

  O. Alqasem, M. Nassar, M. Elwahab

  
  

  Abstract

  Shasta Reservoir is the largest in California, formed by Shasta Dam on the Sacramento River, and plays a major role in the Central Valley Project (CVP) by providing water storage, flood control, hydroelectric power, and irrigation. This study employs advanced statistical methods to evaluate the reservoir’s reliability and operational risks using censored hydrological data. We propose an improved adaptive progressive censoring plan and apply established statistical techniques, maximum likelihood and maximum product of spacings, alongside Bayesian estimation. The Bayes estimates are obtained through the squared error loss function and based on two sources for the observed data, namely the likelihood and spacing functions. The focus is on estimating the distribution’s scale parameter and two critical reliability metrics: the reliability function and the hazard rate function. The approximate confidence intervals based on the two classical approaches of the scale parameter and reliability metrics are studied. The highest posterior density credible intervals are also discussed. A simulation study evaluates the model’s accuracy under diverse data scenarios, and its practical utility is demonstrated through real-world data from Shasta Reservoir. The problem of optimizing data collection strategies is discussed with the same real data. The findings underscore the model’s value in enhancing reservoir reliability assessments, offering actionable insights for hydrology, disaster preparedness, and sustainable resource management.OPEN ACCESS Received: 15/03/2025 Accepted: 12/06/2025 Published: 22/09/2025

  Abstract
  Shasta Reservoir is the largest in California, formed by Shasta Dam on the Sacramento River, and plays a major role in the Central Valley Project (CVP) by providing water [...]

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• Hybrid Fractional Coupled Systems with Generalized Caputo Derivatives

  D. Alhwikem, M. Awadalla

  Rev. int. métodos numér. cálc. diseño ing. (2026). Vol. Online First, 117

  
  

  Abstract

  This paper establishes a comprehensive theoretical framework for a novel coupled system of nonlinear hybrid fractional differential equations involving generalized Caputo derivatives. The system’s hybrid nature, coupled with the generality of the fractional operators, allows for modeling complex interdependent processes with memory effects that cannot be adequately captured by existing models. Using Krasnoselskii’s fixed point theorem, we prove the existence of at least one solution under explicit coupling conditions. Under appropriate Lipschitz conditions, we establish uniqueness via Banach’s contraction principle, deriving a quantitative condition involving the fractional orders, generalization parameters, and Lipschitz constants. We also conduct a rigorous analysis of Ulam-Hyers and Ulam-Hyers-Rassias stability, obtaining explicit stability constants that provide quantitative bounds on how perturbations in the input affect the solution. The theoretical results are validated through three carefully constructed numerical examples with explicit parameter values demonstrating existence, uniqueness, and Ulam-Hyers stability. A parameter sensitivity analysis confirms the robustness of the uniqueness condition across variations in fractional orders, generalization parameters, and interval lengths. The paper concludes with a discussion of limitations and directions for future research, including extensions to higher fractional orders, delay and impulsive effects, and numerical methods.

  Abstract
  This paper establishes a comprehensive theoretical framework for a novel coupled system of nonlinear hybrid fractional differential equations involving generalized Caputo [...]

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• Integrating Vector Computation and Numerical Methods for Complex Surface Design in Engineering via MATLAB

  J. Ruge, V. García, J. Vásquez, M. Salazar

  
  

  Abstract

  The teaching and practical use of vector calculus in engineering often face challenges rooted in mathematical abstraction and the limited availability of tools capable of supporting three-dimensional geometric analysis. These constraints hinder precision when designing complex structural surfaces. Addressing this gap, the present study proposes the development and implementation of an interactive computational tool—built in MATLAB App Designer that integrates vector-based formulations with numerical methods to parameterize, visualize, and compute the surface area of three-dimensional geometries, with a particular focus on sizing geomembranes for circular aquaculture ponds. The research methodology comprised theoretical, numerical, and experimental components. Exact vector parameterizations were formulated, symbolic integration and discretization algorithms were implemented, and the resulting computations were assessed through error estimation and convergence analysis. The findings demonstrate a close match between analytical and numerical solutions, with relative errors below 0.1%, stable computational behavior under moderate discretization settings, and distortion-free threedimensional visualizations. Overall, the study shows that combining exact vector modeling with adaptive numerical techniques and interactive visualization provides an efficient and low-cost framework for surface-area computation and structural design. This approach offers a practical alternative to conventional CAD platforms and delivers meaningful benefits for both engineering education and industrial applications within sustainable production systems.OPEN ACCESS Received: 10/11/2025 Accepted: 14/01/2026 Published: 16/04/2026

  Abstract
  The teaching and practical use of vector calculus in engineering often face challenges rooted in mathematical abstraction and the limited availability of tools capable of [...]

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