Implicit Difference Approximation Research Articles

Compact implicit difference approximation for time-fractional diffusion-wave equation

In this article, developed the compact implicit difference method based Grünwald Letnikov formula (GLF) to compute the solution of the time-fractional diffusion-wave equation (TFDWE) describing wave propagation phenomenan having order α(1<α<2). The fractional derivative is in Caputo sense. The theoretical analysis such as stability, consistency, convergence and solvability of the said scheme are discussed and proved that the scheme is conditionally stable and convergent with the order (τ2+(Δx)4). The numerical results compared with the recent existed method. The results of the numerical examples show that the GLF and the proposed method is very accurate and efficient for time fractional diffusion-wave equation.

Analytical and numerical solutions of a two‐dimensional multi‐term time‐fractional Oldroyd‐B model

In this paper, we consider a two‐dimensional multi‐term time‐fractional Oldroyd‐B equation on a rectangular domain. Its analytical solution is obtained by the method of separation of variables. We employ the finite difference method with a discretization of the Caputo time‐fractional derivative to obtain an implicit difference approximation for the equation. Stability and convergence of the approximation scheme are established in the L∞‐norm. Two examples are given to illustrate the theoretical analysis and analytical solution. The results indicate that the present numerical method is effective for this general two‐dimensional multi‐term time‐fractional Oldroyd‐B model.

The analytical solution and numerical solutions for a two-dimensional multi-term time fractional diffusion and diffusion-wave equation

In this paper we consider the analytical and numerical solutions for a two-dimensional multi-term time-fractional diffusion and diffusion-wave equation. We derive the analytical solution for the equation using the method of separation of variables and properties of the multivariate Mittag-Leffler function. An implicit difference approximation is constructed. Stability and convergence analysis of the numerical scheme are proved by the energy method. Numerical examples are constructed to evaluate the working of the numerical scheme as compared to theoretical analysis. • A new two-dimensional multi-term time-fractional diffusion and diffusion. • Wave equation (2D-MT-TFD-DWE) is considered. • The analytical solution for the 2D-MT-TFD-DWE is derived. • A novel implicit difference method (IDM) for 2D-MT-TFD-DWE is proposed. • The stability and convergence of the IDM are proved by the energy method. • Numerical examples are given.

Modified fractional neutron point kinetics equations for finite and infinite medium of bar reactor core

Fractional order neutron point kinetics equations are generalization of the classical neutron point kinetics equations. The system of the fractional neutron point kinetics equations is treated in different context by many authors ignoring an important term related to the derivative of the reactivity. In this paper, we derive a time fractional neutron point kinetics equations (FNPK) model considering a new term for the time derivative of the reactivity. Fractional Leibenz rule is used for such derivation. The effect resulting from the additional term of the reactivity derivative is discussed. It has been found that without this additional term, the modified model will be compliant with the earlier published models for the FNPK. The proposed model of FNPK describes infinite and finite bar reactor cores according to the nonleakage probability and the geometric buckling values. We have applied an effective implicit difference approximation to solve the modified model for finite cylindrical reactor core for both average one group and six groups of delayed neutron precursors. The neutron density results with different values of fractional order for step, ramp and oscillatory reactivities are shown and compared with the classical neutron point kinetics equations.

Numerical solution of an inverse filtration problem

A numerical method is suggested for solving the multidimensional inverse problem of finding the flow rates of wells from given bottomhole pressures in the multidimensional flow model of a weakly compressible liquid in a poroelastic medium. Spatial approximation is based on the finite element method, which allows one to employ unstructured grids with refinements near the location of wells. Time discretization is performed with the use of implicit difference approximation. The numerical results for two- and three-dimensional test problems are presented.

A novel numerical method for the time variable fractional order mobile–immobile advection–dispersion model

Evolution equations containing fractional derivatives can provide suitable mathematical models for describing anomalous diffusion and transport dynamics in complex systems that cannot be modeled accurately by normal integer order equations. Recently, researchers have found that many physical processes exhibit fractional order behavior that varies with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider the mobile–immobile advection–dispersion model with the Coimbra variable time fractional derivative which is preferable for modeling dynamical systems and is more efficient from the numerical standpoint. A novel implicit numerical method for the equation is proposed and the stability of the approximation is investigated. As for the convergence of the numerical method, we only consider a special case, i.e., the time fractional derivative is independent of the time variable t. The case where the time fractional derivative depends on both the time variable t and the space variable x will be considered in a future work. Finally, numerical examples are provided to show that the implicit difference approximation is computationally efficient.

The analytical solution and numerical solution of the fractional diffusion-wave equation with damping

Fractional partial differential equations have been applied to many problems in physics, finance, and engineering. Numerical methods and error estimates of these equations are currently a very active area of research. In this paper we consider a fractional diffusion-wave equation with damping. We derive the analytical solution for the equation using the method of separation of variables. An implicit difference approximation is constructed. Stability and convergence are proved by the energy method. Finally, two numerical examples are presented to show the effectiveness of this approximation.

Implicit difference approximation for the time fractional heat equation with the nonlocal condition

In this work, a method for solving inhomogeneous nonlocal fractional heat equations is proposed. The method is based on the modified Gauss elimination method. It is proved by using matrix stability approach that the method is unconditionally stable. Numerical results are provided to illustrate the accuracy and efficiency of the proposed method.

The fundamental solution and numerical solution of the Riesz fractional advection-dispersion equation

In this paper, we consider a Riesz fractional advection-dispersion equation (RFADE), which is derived from the kinetics of chaotic dynamics. The RFADE is obtained from the standard advection-dispersion equation by replacing the first-order and second-order space derivatives by the Riesz fractional derivatives of order a e (0, 1) and fi e (1, 2], respectively. We derive the fundamental solution for the Riesz fractional advection-dispersion equation with an initial condition (RFADE-IC). We investigate a discrete random walk model based on an explicit finite-difference approximation for the RFADE-IC and prove that the random walk model belongs to the domain of attraction of the corresponding stable distribution. We also present explicit and implicit difference approximations for the Riesz fractional advection-dispersion equation with initial and boundary conditions (RFADE-IBC) in a finite domain. Stability and convergence of these numerical methods for the RFADE-IBC are discussed. Some numerical examples are given to show that the numerical results are in good agreement with our theoretical analysis.

Implicit difference approximation of the Galilei invariant fractional advection diffusion equation

A Galilei invariant fractional advection diffusion equation with initial-boundary conditions is considered. An implicit difference approximation for solving the Galilei invariant fractional advection diffusion equation is presented. We introduce a new Fourier method for analyzing the stability and convergence of the implicit difference approximation. Finally, some numerical examples are given. The numerical results are in good agreement with our theoretical analysis. This method and supporting theoretical techniques can also be extended to a larger class of fractional integro-differential equations. References T. A. M. Langlands and B. I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comp. Phys., 205 (2005), 719--736. doi:10.1016/j.jcp.2004.11.025 F. Liu, V. Anh, I. Turner, Numerical Solution of the Space Fractional Fokker--Planck Equation, J. Comp. Appl. Math., 166 (2004), 209--219. doi:10.1016/j.cam.2003.09.028 Ch. Lubich, Discretized fractional calculus, SIAM J. Math. Anal., 17 (1986), 704--719. R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: a fractional dynamics approach, Phys. Reports, 339,(2000), 1--77. doi:10.1016/S0370-1573(00)00070-3 M. Meerschaert, C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comp. Appl. Math., 172 (2004), 65--77. doi:10.1016/j.cam.2004.01.033 I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999. http://people.tuke.sk/igor.podlubny/fde.html I. M. Sokolov, J. Klafter and A. Blumen, Fractional Kinetics, Physics Today, 55(11) (2002) 48--54. S. B. Yuste and L. Acedo, An explicit finite difference method and a new Von Neumann-type stability analysis for fractional diffusion equations, SIAM J. Numer. Anal., 42(5) (2005), 1862--1874. P. Zhuang and F. Liu, Implicit difference approximation for the time fractional diffusion equation, J. Appl. Math. Comp., 22(3) (2006) 87--99. http://jamc.net/contents/table_contents_view.php?idx=580

A Fourier method for the fractional diffusion equation describing sub-diffusion

In this paper, a fractional partial differential equation (FPDE) describing sub-diffusion is considered. An implicit difference approximation scheme (IDAS) for solving a FPDE is presented. We propo...

Implicit difference approximation for the two-dimensional space-time fractional diffusion equation

In this paper, we consider a two-dimensional fractional spacetime diffusion equation (2DFSTDE) on a finite domain. We examine an implicit difference approximation to solve the 2DFSTDE. Stability and convergence of the method are discussed. Some numerical examples are presented to show the application of the present technique.

A Fourier method for the fractional diffusion equation describing sub-diffusion

In this paper, a fractional partial differential equation (FPDE) describing sub-diffusion is considered. An implicit difference approximation scheme (IDAS) for solving a FPDE is presented. We propose a Fourier method for analyzing the stability and convergence of the IDAS, derive the global accuracy of the IDAS, and discuss the solvability. Finally, numerical examples are given to compare with the exact solution for the order of convergence, and simulate the fractional dynamical systems.

Finite Difference Approximation for Two-Dimensional Time Fractional Diffusion Equation

Fractional diffusion equations have recently been used to model problems in physics, hydrology, biology and other areas of application. In this paper, we consider a two-dimensional time fractional diffusion equation (2D-TFDE) on a finite domain. An implicit difference approximation for the 2D-TFDE is presented. Stability and convergence of the method are discussed using mathematical induction. Finally, a numerical example is given. The numerical result is in excellent agreement with our theoretical analysis.

Implicit difference approximation for the time fractional diffusion equation

In this paper, we consider a time fractional diffusion equation on a finite domain. The equation is obtained from the standard diffusion equation by replacing the first-order time derivative by a fractional derivative (of order $0<\alpha<1$ ). We propose a computationally effective implicit difference approximation to solve the time fractional diffusion equation. Stability and convergence of the method are discussed. We prove that the implicit difference approximation (IDA) is unconditionally stable, and the IDA is convergent with $O(\tau+h^2)$, where $\tau$ and $h$ are time and space steps, respectively. Some numerical examples are presented to show the application of the present technique.

A fractional-order implicit difference approximation for the space-time fractional diffusion equation

We consider a space-time fractional diffusion equation on a finite domain. The equation is obtained from the standard diffusion equation by replacing the second order space derivative by a Riemann--Liouville fractional derivative of order between one and two, and the first order time derivative by a Caputo fractional derivative of order between zero and one. A fractional order implicit finite difference approximation for the space-time fractional diffusion equation with initial and boundary values is investigated. Stability and convergence results for the method are discussed, and finally, some numerical results show the system exhibits diffusive behaviour.

Eddy current loss calculation in thin rectangular plates at cryogenic temperatures and its experimental verification

Eddy current losses in conducting plates placed in a non-uniform alternating magnetic held at cryogenic temperatures have been calculated taking into account temperature dependencies of resistivity, heat capacity and heat-transfer coefficient. The calculation method is based on a coupled solution of a quasi-stationary electromagnetic problem and a non-stationary heat-conduction equation. A simplification of the task is achieved by neglecting eddy current reaction and free space charge. In this case the problem is solved in two stages: (1) the calculation of electromagnetic field; (2) the determination of a temperature distribution in the plate. As the first boundary condition for heat-conduction equation it is assumed that the heat transfer from edges of the plate equals zero. The second boundary condition is determined as a solution of the integral equation describing the law of conservation of energy relevant to the plate. The nonlinear two-dimensional heat conduction equation has been solved using partition of coordinates, implicit difference approximation and modified Thomas algorithm for obtained finite-difference equations. The numerical results remain in good agreement with data obtained by means of an experimental model of the cryogenic device, while neglecting thermal processes results in substantial errors amounting up to several hundreds percent. >

Vertical transport and fate of low-water-solubility chemicals in unsaturated soils

Adsorption, dispersion and changes in moisture content control the rate of transport of solutes in an unsaturated soil column. A one-dimensional mathematical model has been developed to examine the simultaneous transport of heat, moisture and chemical mass in unsaturated soils. Chemical concentration is assumed to be low. The field equation for chemical transport is strongly coupled to the field equations for moisture content and temperature while the converse is not true. Vertical temperature, moisture content and chemical concentration profiles were determined using the iterative Thomas algorithm procedure on implicit backward difference approximations to the defining set of nonlinear partial differential equations. For a specified set of hydraulic properties, initial distributions of chemicals in the soil column and external weather conditions, chemical concentration profiles for 100-days simulations illustrated the influence of the combined effects of adsorption and dispersion on chemical transport in unsaturated soils. Numerical dispersion was minimized by keeping nodal spacing small in the vicinity of steep gradients of moisture content, temperature and chemical concentration. Numerical dispersion occurred more frequently in the solution for chemical transport, particularly when dispersivities were low. For this situation the criterion for nodal spacing was Δz < 2 a L, where a L was dispersivity.

Stability analysis of numerical boundary conditions and implicit difference approximations for hyperbolic equations

Implicit noniterative finite-difference schemes have recently been developed by several authors for multidimensional systems of nonlinear hyperbolic partial differential equations. When applied to linear model equations with periodic boundary conditions those schemes are unconditionally stable (A-stable). As applied in practice the algorithms often face a severe time-step restriction. A major source of the difficulty is the treatment of the numerical boundary conditions. One conjecture has been that unconditional stability requires implicit numerical boundary conditions. An apparent counterexample was the space-time extrapolation considered by Gustafsson, Kreiss, and Sundstrom. In this paper we examine space (implicit) and space-time (explicit) extrapolation using normal mode analysis for a finite and infinite number of spatial mesh intervals. The results indicate that for unconditional stability with a finite number of spatial mesh intervals the numerical boundary conditions must be implicit.

Boundary Approximations for Implicit Schemes for One-Dimensional Inviscid Equations of Gasdynamics

The applicability to practical calculations of recent theoretical developments in the stability analysis of difference approximations is examined for initial boundary-value problems of the hyperbolic type. For the experiments the one-dimensional inviscid gasdynamic equations in conservation law form are selected. A class of implicit schemes based on linear multistep methods for ordinary differential equations is chosen and the use of space or space-time extrapolations as implicit or explicit schemes is emphasized. Some examples with various inflow-outflow conditions highlight the commonly discussed issues: explicit vs implicit schemes, and unconditionally stable schemes. HEN finite-difference schemes are used to solve initial boundary-value problems for the equations of fluid dynamics, it is well known that most methods require more conditions than those required by the governing partial differential equations. These additional conditions for the finite-difference equations are often called numerical conditions. The conditions cannot be imposed arbitrarily but are determined, in general, using interior information, for example, by extrapolation or uncentered approximations. In this paper, any procedure used to provide a condition will be called a boundary scheme/' Whatever schemes are used for the conditions, it is a common practice to assume that the scheme has a local effect and will not affect the solution globally. During the early 1970s, Kreiss,1'2 Osher,3 Gustafsson et al.,4 Varah,5 and Gustafsson6 published a series of papers establishing methods for checking the stability and accuracy of difference approximations with schemes included. Since then, further progress has been made in the theory of linear difference approximations for initial boundary-value problems of the hyperbolic and parabolic type.7'12 Because improper treatment of the conditions can lead to instability and inaccuracy, even though we start with a stable interior scheme (i.e., scheme for the interior points), it is appropriate to adopt an approach that includes the stability and accuracy of the combined interior and schemes. Surveys of recent developments and extensive bibliographies are included in papers by Coughran13 and Yee.14 The purpose of this paper is to examine the applicability to practical calculations (for nonlinear gasdynamic problems) of recent theoretical stability analyses of implicit difference approximations for initial boundary-value problems of the hyperbolic type. As computations have progressed, the use of the conservative form of the gasdynamic equations has gained popularity. For physical reasons it is sometimes desirable to specify conditions in the nonconservative variables and to compute with conservative variables in the interior. We will consider the additional complications introduced by this procedure.