Research Article
Numerical Modeling of Fractional-order Diffusion for Complex Systems in Applied Mathematics
Majid Ghorbani*
Issue:
Volume 11, Issue 3, June 2025
Pages:
45-49
Received:
31 July 2025
Accepted:
12 August 2025
Published:
23 September 2025
Abstract: This study presents an enhanced and comprehensive approach to modeling fractional-order diffusion processes in complex systems using a numerical method based on the Grünwald-Letnikov (GL) approximation. The proposed model aims to bridge the theoretical foundations of fractional calculus with efficient simulation techniques applicable to heterogeneous and memory-dependent phenomena. Compared to classical integer-order models, fractional models offer greater flexibility in capturing anomalous diffusion, long-range interactions, and nonlocal behavior observed in real-world systems. The research investigates the influence of the fractional order parameter on diffusion dynamics across various applied scenarios, including heat conduction in porous media, pollutant transport in groundwater, epidemic spread in network structures, drug release through biological tissues, and petroleum flow in stratified reservoirs. Numerical simulations demonstrate that tuning the parameter allows for accurate modeling of both sub-diffusive and super-diffusive behaviors, improving the fidelity of results compared to classical models. The methodology employs an implicit Euler time integration scheme and adaptive mesh refinement to enhance stability, accuracy, and computational efficiency. The results confirm the robustness of the GL-based scheme in preserving mass conservation, achieving second-order spatial accuracy, and maintaining stability over a wide range of values. This approach provides practical tools for engineers, physicists, and biomedical researchers seeking precise numerical modeling of complex transport phenomena.
Abstract: This study presents an enhanced and comprehensive approach to modeling fractional-order diffusion processes in complex systems using a numerical method based on the Grünwald-Letnikov (GL) approximation. The proposed model aims to bridge the theoretical foundations of fractional calculus with efficient simulation techniques applicable to heterogeneo...
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Research Article
Mathematical Modeling of the Caspian Sea Geometry Under Water Level Decline: Evidence from Iran’s Mazandaran Coast
Majid Ghorbani*
Issue:
Volume 11, Issue 3, June 2025
Pages:
50-54
Received:
29 August 2025
Accepted:
12 September 2025
Published:
17 October 2025
DOI:
10.11648/j.ijtam.20251103.12
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Abstract: The Caspian Sea is Earth’s largest endorheic water body and a natural laboratory for studying coupled hydroclimatic forcing and coastal geomorphic response. Over the last century, the basin has undergone multi-decadal oscillations and an accelerated decline since the mid-1990s, with strong signals projected to continue through the twenty-first century. This meta-analysis synthesizes peer-reviewed evidence on horizontal shoreline migration, vertical (level) change, and areal transformation, and evaluates how these changes reconfigure the basin’s large-scale geometry—well approximated locally by parabolic or saddle-shaped (hyperbolic-paraboloid) surfaces—while cascading into socioeconomic risk. We combine quantitative shoreline metrics (Digital Shoreline Analysis System, DSAS), spectral water delineation (NDWI/MNDWI), sediment-transport theory (Exner equation), and water-balance diagnostics to: (i) characterize recent and projected Caspian water-level trajectories; (ii) resolve planform curvature and alongshore variability along Iran’s Mazandaran coast; (iii) contrast Caspian responses to those observed in China’s coastal systems (South China Sea littoral and the Yangtze River delta); and (iv) assess risk pathways for agriculture, shipping, fisheries, and wetlands. Results from the literature indicate a long-term negative water balance dominated by increased evaporation relative to precipitation and inflows, superimposed on high interannual variability; the Volga—regulated by reservoirs including Volgograd—remains the principal control on riverine supply. Shoreline retreat and shallow-water expansion are already disrupting Mazandaran’s port operations (e.g., Amir-Abad), accelerating maintenance dredging needs, and exposing wetlands such as Gorgan Bay to desiccation and dust-storm hazards. Comparative analysis shows that while China’s open-coast margins are influenced by marine processes, the Caspian’s closed-basin geometry transmits water-level anomalies more uniformly, amplifying parabolic/saddle-like morphodynamic adjustments. We conclude with actionable adaptation options for Mazandaran (channel realignment, dynamic zoning, nature-based buffers, and flexible port layouts) aligned with realistic twenty-first-century water-level scenarios.
Abstract: The Caspian Sea is Earth’s largest endorheic water body and a natural laboratory for studying coupled hydroclimatic forcing and coastal geomorphic response. Over the last century, the basin has undergone multi-decadal oscillations and an accelerated decline since the mid-1990s, with strong signals projected to continue through the twenty-first cent...
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