Solutions of a three-dimensional multi-term fractional anomalous solute transport model for contamination in groundwater

Abstract

The study presents a meshless computational approach for simulating the three-dimensional multi-term time-fractional mobile-immobile diffusion equation in the Caputo sense. The methodology combines a stable Crank-Nicolson time-integration scheme with the definition of the Caputo derivative to discretize the problem in the temporal direction. The spatial function derivative is approximated using the inverse multiquadric radial basis function. The solution is approximated on a set of scattered or uniform nodes, resulting in a sparse and well-conditioned coefficient matrix. The study highlights the advantages of meshless method, particularly their simplicity of implementation in higher dimensions. To validate the accuracy and efficacy of the proposed method, we performed numerical simulations and compared them with analytical solutions for various test problems. These simulations were carried out on computational domains of both rectangular and non-rectangular shapes. The research highlights the potential of meshless techniques in solving complex diffusion problems and its successful applications in groundwater contamination and other relevant fields.

Efficient computational hybrid method for the solution of 2D multi-term fractional order advection-diffusion equation

“…Equations including multi-term time-fractional derivatives offer for more accurate modeling of the dispersion of pollutants in air and groundwater in environmental engineering by taking into consideration the influence of several transport modes, including advection, diffusion, and dispersion, as well as long-term memory effects. Partial differential equations including multi term time fractional derivatives finds numerous applications in many scientific and engineering domains, for example, the contamination in groundwater [17], the oxygen diffusion from capillary to tissues [18], the anomalous diffusion phenomena [19], the diffusive waves in viscoelastic solids [20], and many other phenomena in physics and biology [21].…”

Exploring fractional-order new coupled Korteweg-de Vries system via improved Adomian decomposition method

“…In recent decades, notable advancements have been made in exploring fractional differential equations (FDEs) and fractional calculus, carrying implications across diverse domains of applied sciences and engineering [15][16][17]. Fractional calculus, which addresses derivatives and integrals of non-integer order, has garnered substantial attention due to its capacity to accurately describe complex behaviors and phenomena in areas such as electromagnetic fields, acoustics, viscoelasticity, electrochemistry, cosmology, and material science [18][19][20][21].…”

An Efficient Numerical Solution of a Multi-Dimensional Two-Term Fractional Order PDE via a Hybrid Methodology: The Caputo–Lucas–Fibonacci Approach with Strang Splitting

“…Fractional calculus, which deals with derivatives and integrals of non-integer order, finds broad applications in science and technology. In physics, it models complex systems with non-local behavior, such as anomalous diffusion in fluid dynamics and memory effects in viscoelastic materials [3]. Engineering benefits from fractional calculus are found in signal processing, control theory, and optimization, offering robustness and efficiency in dynamic systems.…”