Current Issue : July-September Volume : 2026 Issue Number : 3 Articles : 5 Articles
In this paper, we set out to implement a code for processing the edges of a plane domain with a complex geometric shape. The algorithm that enabled us to write this code is based on the classical properties of analytic geometry. The coordinates of the vertices of the polygon-like domain must first be given. The code developed automatically generates indices for interior and edge points. Any point located at a distance defined by user from the edge of a domain is considered an edge point. To this condition, we add a second condition that requires the edge point to be inside the circle with a diameter of two consecutive vertices of the polygon. An illustration is included to clarify the readers of this paper....
Analytic solutions for elastostatic displacement due to vertical/shear traction applied to the faces of a single crack in a 2D plane are reviewed in Section 2. The purpose of this review is to introduce mathematics relevant to specifying multi-crack elastostatic displacements. In Section 3, the relation between multicrack elastostatic displacement fields and Riemann surface geometry is discussed. Section 4 explains why 2D multi-crack solutions appear limited to describe idealized cracks in 2 spatial dimensions, and conjectures how 3D fracture networks in the lithosphere can be described in terms of singular spectrum analysis of crack phase fields....
This study introduces a mathematical framework for analyzing unsteady gas filtration in porous media with pressure-dependent porosity variations. The physical process is formulated as a nonlinear parabolic boundary value problem that captures the coupled interaction between pressure evolution and porosity changes during gas production. To solve the equation, a numerical strategy is developed by integrating the Alternating Direction Implicit (ADI) scheme with quasi-linearization iterations, employing finite difference discretization on a two-dimensional spatial grid. Extensive computational experiments are performed to investigate the influence of key reservoir parameters—including porosity coefficient, permeability, gas viscosity, and well production rate—on the spatiotemporal behavior of pressure and porosity during long-term extraction. The results indicate significant porosity variations near the wellbore driven by local pressure depletion, reflecting strong sensitivity of the system to formation properties. The validated numerical model provides valuable quantitative insights for optimizing reservoir management and improving production forecasting in gas field development. Overall, the proposed methodology serves as a practical tool for oil and gas engineers to assess long-term reservoir performance under diverse operational conditions and to design efficient extraction strategies that incorporate pressure-dependent formation property changes....
In this paper, we study a class of second-order fractional boundary value problems involving Θ-Caputo derivatives of different orders. By reformulating the problem to an integral equation, we introduce an appropriate notion of a mild solution in the Θ-fractional framework. Existence results are obtained via Krasnoselskii’s fixed point theorem, while uniqueness is established using the Banach contraction principle under suitable Lipschitztype conditions. The obtained results extend several earlier works on Caputo, Hadamard– Caputo, and Riemann–Liouville fractional derivatives. Two examples are presented to illustrate the applicability of the theoretical results....
The chain rule is a foundational concept in calculus, critical for differentiating composite functions, especially those appearing in modern AI techniques. Its extension to fractional calculus presents challenges due to the integral-based nature and intrinsic memory effects of these fractional operators. This survey provides a review of chain-rule formulations across major known FDs, including Riemann-Liouville (RL), Caputo, Caputo-Fabrizio (CF), Atangana-Baleanu-Riemann (ABR), Atangana-Baleanu-Caputo (ABC), and Caputo- Fabrizio with Gaussian kernel (CFG). The main contribution here is the introduction of a unified criterion, denoted as C, which synthesizes and extends previous classification frameworks for systematically formulating the chain rule across different operators. Each chain rule is examined in terms of its derivation, operator structure, and scope of applicability. In addition, the survey analyzes series-based approximations that appear in computing these derivatives, highlighting the minimum number of terms required to achieve acceptable mean absolute error (MAE). By consolidating theoretical developments, derivation methods, and numerical strategies, this paper provides a comprehensive resource for researchers and practitioners working with fractional-order models....
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