Scipedia: RIMNI SPECIAL ISSUE - Numerical Methods for Fractional Equations in Engineering

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

Scipedia content — Thu, 14 May 2026 10:40:03 +0200

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.

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

Scipedia content — Thu, 14 May 2026 10:33:03 +0200

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

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

Scipedia content — Thu, 14 May 2026 10:29:03 +0200

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

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

Scipedia content — Thu, 14 May 2026 10:22:13 +0200

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