Numerical approach for modeling fractional heat conduction in porous medium with the generalized Cattaneo model

• Published in: Applied Mathematical Modelling Vol. 100; pp. 107 - 124
• Main Authors: Nikan, O., Avazzadeh, Z., Tenreiro Machado, J.A.
• Format: Journal Article
• Language: English
• Published: New York Elsevier Inc 01.12.2021; Elsevier BV
• Subjects: Algorithms; Caputo fractional derivative; Conduction heating; Conductive heat transfer; Convergence; Finite difference; Finite difference method; Finite element method; Fractional Cattaneo equation; Heat transmission; Ill-conditioned problems (mathematics); Mathematical analysis; Meshless methods; Porous media; Radial basis function; RBF-PU; Robustness (mathematics); Stability; Stability analysis; Finite difference; Fractional Cattaneo equation; RBF-PU; Stability; Caputo fractional derivative; Convergence
• ISSN: 0307-904X; 1088-8691; 0307-904X
• DOI: 10.1016/j.apm.2021.07.025

•The fractional Cattaneo model is formulated to describe the heat flow in a porous medium.•The hybrid scheme based on the RBF-PU method is proposed to approximate the model.•Unconditional stability and convergence of the time discretization formulation are proved using energy method.•Numerical results are conducted to validate the theoretical findings. The generalized Cattaneo model describes the heat conduction system in the perspective of time-nonlocality. This paper proposes an accurate and robust meshless technique for approximating the solution of the time fractional Cattaneo model applied to the heat flow in a porous medium. The fractional derivative is formulated in the Caputo sense with order 1<α<2. First, a finite difference technique of convergence order O(δt3−α) is adopted to achieve the temporal discretization. The unconditional stability of the method and its convergence are analysed using the discrete energy technique. Then, a local meshless method based on the radial basis function partition of unity collocation is employed to obtain a full discrete algorithm. The matrices produced using this localized scheme are sparse and, therefore, they are not subject to ill-conditioning and do not pose a large computational burden. Two examples illustrate in computational terms of the accuracy and effectiveness of the proposed method.