Particle-scale heat and mass transfer processes during the pyrolysis of millimeter-sized lignite particles with solid heat carriers

Fractional Diffusion Equation under Singular and Non-Singular Kernel and Its Stability

“…One limitation of the method is the entire dependence of the results from the integrals that appear in Equation (15), which can be hard to calculate in some cases. Another challenge is the numerical implementation.…”

“…The stability condition is D∆t/∆x 2 ≤ 1/2 [27]. To perform a similar analysis and obtain the stability condition for Equation (15), we considered, for simplicity, the absence of reaction terms, i.e., F(x i , t j ) = 0.…”

“…Besides Equation (15) admitting some forms of kernels, for the numerical examples, we start by focusing our analysis on:…”

“…For this case, we have K α (t) = t −α /Γ(1 − α) and Λ µ (x) = x 1−µ /Γ(2 − µ). Substituting these values in Equation (15) and performing some operations, we obtain…”

“…This equation has a wide variety of applications in different fields such as physics [3,4], epidemiology [5], rumor propagation [6], chemistry [7], the economy [8], and many others [9][10][11][12][13][14]. In physics, this equation is used, for example, to describe the heat, particles, mass diffusion in space [15], and the physics in semiconductors [16]. In biology, the diffusion process of molecules plays a crucial role in transporting nutrients and waste products within cells and tissues [17].…”