and Applied Analysis 3 3. Numerical Solution Environmental phenomena, such as groundwater flow described by variational order derivative, are highly complex phenomena, which do not lend themselves readily to analysis of analytical models. The discussion presented in this section will therefore be devoted to the derivation of numerical solution to the modified Hantush equation (6). Solving diffi...

this study develops and analyzes preconditioned krylov subspace methods to solve linear systemsarising from discretization of the time-independent space-fractional models. first, we apply shifted grunwald formulas to obtain a stable finite difference approximation to fractional advection-diffusion equations. then, we employee two preconditioned iterative methods, namely, the preconditioned gene...

An unsteady problem is considered for a space-fractional diffusion equation in a bounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary conditions of Robin type. Finite element approximation in space is employed. To construct approximation in time, regularized twolevel schemes are used. The numeri...

In this paper, we present a numerical solution to an ordinary differential equation of a fractional order in one-dimensional space. The solution to this equation can describe a steady state of the process of anomalous diffusion. The process arises from interactions within complex and non-homogeneous background. We present a numerical method which is based on the finite differences method. We co...

Burgers’ equation frequently appears in the study of turbulence theory, as well as some other scientific fields. High and low Reynolds numbers play important roles in both modeling and numerical simulation. In this paper, we apply a numerical scheme to solve a two-dimensional time-fractional Burgers equation. The key feature of the proposed method is formed by combining the discontinuous Galerk...

Abstract In this paper, we introduce a two-point nonlinear boundary value problem for finite fractional difference equation. An associated Green’s function is constructed as series of functions and some its properties are obtained. Some existence results deduced from fixed point theory lower upper solutions.

This article provides a practical overview of numerical solutions to the heat equation using the finite difference method. The forward time, centered space (FTCS), the backward time, centered space (BTCS), and Crank-Nicolson schemes are developed, and applied to a simple problem involving the one-dimensional heat equation. Complete, working Matlab codes for each scheme are presented. The result...

Abst rac t . The Crank-Nicolson scheme is widely used to solve numerically the diffusion equation, because of its good stability properties. It is, however, ill-behaved when large time-steps are used: the short wave-lengths may happen to be less damped than the long ones. A detailed analysis of this flaw is performed and an Mternative scheme is proposed, which removes this difficulty while pres...

Abstract. The traditional wave equation models wave propagation in an ideal conducting medium. For characterizing the wave propagation in inhomogeneous media with frequency dependent power-law attenuation, the spacetime fractional wave equation appears; further incorporating the additive white Gaussian noise coming from many natural sources leads to the stochastic spacetime fractional wave equa...