Legendre spectral element method for solving time fractional modified anomalous sub-diffusion equation

A Petrov–Galerkin spectral element method for fractional elliptic problems

“…We extend the notion of global linear system assembly by taking into account the presence of the history stiffness and mass matrices. We similarly impose the C 0 − continuity by employing the same "mapping arrays", map[e][p], defined in (42). Let us define the (P + 1) × (P + 1) matrix…”

“…They also developed a highly accurate discontinuous SEM for time-and space-fractional advection equation in [38]. Dehghan et al [42] considered Legendre SEM in space and FDM in time for solving time-fractional sub-diffusion equation. Su [43] provided a parallel spectral element method for the fractional Lorenz system and a comparison of the method with FEM and FDM.…”

Two‐dimensional shifted Legendre polynomial collocation method for electromagnetic waves in dielectric media via almost operational matrices

“…In recent years, there has been a growing interest in the areas of fractional calculus (e.g., ). Fractional derivative has application in physics, mathematics, engineering, and applied science as physical sciences phenomena in area like damping law , diffusion process , electrochemistry , arterial sciences , and the theory of ultra‐slow processes .…”

“…(5) into fractional partial integro-differential equation (FPIDE), then this FPIDE (with 1 ġ< < 3, in our case) is said to be Volterra type. Although the theory of Volterra FPIDEs has undergone rapid development during the last four decades, it remains wide open for further progress.In recent years, there has been a growing interest in the areas of fractional calculus (e.g., [7][8][9][10][11]). Fractional derivative has application in physics, mathematics, engineering, and applied science as physical sciences phenomena in area like damping law [12], diffusion process [13], electrochemistry [14], arterial sciences [15], and the theory of ultra-slow processes [16].…”

Modified Galerkin algorithm for solving multitype fractional differential equations

“…The principal advantage of spectral methods lies in their capacity to accomplish with high accurate outcomes. There are four popular methods of spectral methods; they are tau, collocation, Galerkin, and spectral element methods, see the studies of Rainville, Ezz-Eldien and Doha, Dehghan and Abbaszadeh, Dehghan, Abbaszadeh, and Mohebbi, Bernardi and Maday, and Canuto et al [37][38][39][40][41][42][43] The choice of the appropriate utilized spectral method suggested for solving such differential equations depends certainly on the type of the differential equation and also on the type of the initial or boundary conditions governed it, see the studies of Hafez et al, Abd-Elhameed and Youssri, Vanani and Aminataei, and Saker, Bhrawy, and Ezz-Eldien. [44][45][46][47] In recent decades, tau and collocation spectral approaches have been utilized for the solution of various types of FDEs, see the studies of Li et al, Khosravian-Arab, Dehghan, and Eslahchi, Saadatmandi and Dehghan, Ezz-Eldien, Dehghan, Manafian, and Saadatmandi, Kashkari and Syam, and Zaky et al [48][49][50][51][52][53][54] The essential purpose of the present article is to build a novel spectral algorithm based on the shifted Jacobi modified Galerkin method (SJMGM) for solving FDEs and system of FDEs (SFDEs) governed by homogeneous and nonhomogeneous initial and boundary conditions.…”