Novel Numerical Methods for Solving the Time-Space Fractional Diffusion Equation in Two Dimensions

Numerical investigation of three types of space and time fractional Bloch-Torrey equations in 2D

“…In this section, we utilize the matrix transfer technique proposed by Yang et al [27] to discretize the two-dimensional fractional Laplacian operator for solving Model-3, with initial and zero Dirichlet boundary conditions on a finite domain given by…”

“…as a sequential Riesz fractional order operator in space [2], and some authors [26][27][28][29] proposed to study the fractional Laplacian operator formulation replacing the Riesz fractional derivative. In this paper, we consider three types of space and time fractional Bloch-Torrey equations in two dimensions (ST-FBTE2D), namely, Model-1: ST-FBTE2D with the Riesz fractional derivative; Model-2: ST-FBTE2D with the one-dimensional fractional Laplacian operator, and Model-3: the space fractional BlochTorrey equation with a two-dimensional fractional Laplacian operator.…”

A class of fractional differential hemivariational inequalities with application to contact problem

“…The fractional calculus, as a natural generalization of the classical integer order calculus, provides a precise description of some physical phenomena for viscoelastic materials, for example, fractional Kelvin-Voigt constitutive laws and fractional Maxwell model [16,42,51]. Recent advances in the fractional calculus concern the fractional derivative modeling in applied science, see [2,9,38], the theory of fractional differential equations, see [21], numerical approaches for the fractional differential equations, see [26,55] and the references therein.…”

Generalized Boundary Conditions for the Time-Fractional Advection Diffusion Equation

“…The entropy approach to anomalous diffusion was used to analyze the magnetic resonance images in biological issues [16][17][18]. Fractional reaction-diffusion and advection-diffusion equations were studied by many authors (see [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34], among many others). Several numerical methods were used to solve the problem: the implicit and explicit difference schemes [19][20][21][22], the Adomian decomposition method [23], the homotopy perturbation method [24] and homotopy analysis method [25], the collocation methods [26,27], the finite element method [28].…”

“…Several numerical methods were used to solve the problem: the implicit and explicit difference schemes [19][20][21][22], the Adomian decomposition method [23], the homotopy perturbation method [24] and homotopy analysis method [25], the collocation methods [26,27], the finite element method [28]. The finite volume spatial discretization using the matrix transfer technique and the time discretization were employed in [29]. Jiang and Lin [30] constructed the orthonormal basis in the reproducing kernel space and gave a Fourier series representation of the solution.…”