Numerical Solutions of Space-Fractional Advection–Diffusion–Reaction Equations

Salomoni, Valentina; Marchi, Niccolò

doi:10.3390/fractalfract6010021

Cited by 7 publications

(1 citation statement)

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“…Gao et al [22], in their study, have analyzed the behavior of solute transport in convectiondispersion transport model as compared to homogeneous and heterogeneous porous media. Other robust methods for solving the MI model can be found in [6,23,24] and there references. The finite volume method, finite element method, and finite difference method are all mesh-based methods and have powerful features, but in these methods the need to create a polygonization, either in the domain or on its boundary is a common drawback.…”

Section: Introductionmentioning

confidence: 99%

On the Numerical Approximation of Mobile-Immobile Advection-Dispersion Model of Fractional Order Arising from Solute Transport in Porous Media

et al. 2022

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The fractional mobile/immobile solute transport model has applications in a wide range of phenomena such as ocean acoustic propagation and heat diffusion. The local radial basis functions (RBFs) method have been applied to many physical and engineering problems because of its simplicity in implementation and its superiority in solving different real-world problems easily. In this article, we propose an efficient local RBFs method coupled with Laplace transform (LT) for approximating the solution of fractional mobile/immobile solute transport model in the sense of Caputo derivative. In our method, first, we employ the LT which reduces the problem to an equivalent time-independent problem. The solution of the transformed problem is then approximated via the local RBF method based on multiquadric kernels. Afterward, the desired solution is represented as a contour integral in the left half complex along a smooth curve. The contour integral is then approximated via the midpoint rule. The main advantage of the LT-RBFs method is the avoiding of time discretization technique due which overcomes the time instability issues, second is its local nature which overcomes the ill-conditioning of the differentiation matrices and the sensitivity of the shape parameter, since the local RBFs method only considers the discretization points in each local domain around the collocation point. Due to this, sparse and well-conditioned differentiation matrices are produced, and third is the low computational cost. The convergence and stability of the numerical scheme are discussed. Some test problems are performed in one and two dimensions to validate our numerical scheme. To check the efficiency, accuracy, and efficacy of the scheme the 2D problems are solved in complex domains. The numerical results confirm the stability and efficiency of the method.

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