Two explicit and implicit finite difference schemes for time fractional Riesz space diffusion equation

Finite difference solution of heat and mass transfer problems related to precooling of food

In physics and mathematics, heat equation is a special case of diffusion equation and is a partial differential equation (PDE). Partial differential equations are useful tools for mathematical modeling. A few problems can be solved analytically, whereas difficult boundary value problem can be solved by numerical methods easily. A very popular numerical method known as finite difference methods (explicit and implicit schemes) is applied expansively for solving heat equations successfully. Explicit schemes are Forward Time and Centre Space (FTCS), Dufort and Frankel methods, whereas implicit schemes are Laasonen and Crank-Nicolson methods. In this study, explicit and implicit finite difference schemes are applied for simple one-dimensional transient heat conduction equation with Dirichlet’s initial-boundary conditions. MATLAB code is used to solve the problem for each scheme in fine mesh grids. Comparing results with analytical results, Crank-Nicolson method gives the best approximate solution. FTCS scheme is conditionally stable, whereas other schemes are unconditionally stable. Convergence, stability and truncation error analysis are investigated. Transient temperature distribution plot and surface temperature plots for different time are presented. Also, unstable plot for FTCS method is represented.

For the first time, it is demonstrated that the methanol diffusion in the H-ZSM-5/Al 2 O 3 catalyst pellet is anomalous and is described by the time-fractional diffusion equation. The regime of the methanol transport is sub-diffusive. The aim of the present work is a description of the experimental data of the methanol transport through the pellet of the zeolite-containing catalyst for olefins synthesis form the methanol based on the solutions of the diffusion equation of the second Fick’s law of diffusion and the time-fractional diffusion equation. In this work, the mesoporous catalyst based on zeolite H-ZSM-5 and alumina with zeolite/alumina ratio 3/1 by mass is used. The methanol transport has been studied using the developed method of mass transfer process investigation in the porous solid media in flow regime. The method is based on the porous sample saturation by a pulse of the adsorbate. The porous sample is installed to the diffusion cell. Adsorbate quantity evolution versus time is chromatographically analyzed. The porous sample is installed to the diffusion cell so that half of its surface is impermeable for adsorbate which allows applying the von Neumann boundary conditions to sole the diffusion equation. The investigation resulted in the obtaining the relative concentration decay versus time on the boundary of the catalyst pellet. The obtained experimental dependences are analyzed on the whole temporal scale using the analytic solution of the diffusion equation based on the second Fick’s law of diffusion. However, the correspondence between the theoretical solution and the experimental data is very poor. The asymptotic analysis of the experimental data in the long-time range linearized in the logarithmic coordinates according to the solution of the standard diffusion equation has demonstrated that the equation for the second Fick’s law of diffusion is inapplicable to a description of the obtained experimental data because the slope of the experimental data is far from the theoretical one which is equal to unity. On the other, an hand analysis of the experimental data in the long-time range linearized in the logarithmic coordinates according to the solution of the time-fractional diffusion equation revealed the high correlation between the theoretical solution and the experimental data. The calculated values of the fractional order and the fractional diffusion coefficient are independent on the experimental conditions. This means that these characteristics are individual for each pair of the porous media and diffusate and may be associated with the methanol adsorption on the active sites on the surface of the catalyst. The fractional order value is lower than unity, which reveals the presence of the sub-diffusive regime of transport, which is slower comparing to the standard diffusion. Based on the analysis of the methanol mass transfer in the pellet of the zeolite-containing catalyst, it is found that the solution of the time-fractional diffusion equation gives good fit to the experimental data comparing to the solution of the standard diffusion equation. The values of the diffusion coefficients and the fractional orders calculated on the long times are equal to the values estimated for the whole temporal range. It is found that the methanol transfer in the catalyst pellet occurs in the slow sub-diffusive regime. The experimental evidence of the presence of an anomalous diffusion is fundamental for the theoretical understanding the mass transfer process and modeling as well as for application during solving the engineering problems.

The documentation of a computer code for the numerical solution of the linear diffusion equation in one or two dimensions in Cartesian or cylindrical coordinates is presented. Applications of the program include molecular diffusion, heat conduction, and fluid flow in confined systems. The flow media may be anisotropic and heterogeneous. The model is formulated by replacing the continuous linear diffusion equation by discrete finite-difference approximations at each node in a block-centered grid. The resulting matrix equation is solved by the method of preconditioned conjugate gradients. The conjugate gradient method does not require the estimation of iteration parameters and is guaranteed convergent in the absence of rounding error. The matrixes are preconditioned to decrease the steps to convergence. The model allows the specification of any number of boundary conditions for any number of stress periods, and the output of a summary table for selected nodes showing flux and the concentration of the flux quantity for each time step. The model is written in a modular format for ease of modification. The model was verified by comparison of numerical and analytical solutions for cases of molecular diffusion, two-dimensional heat transfer, and axisymmetric radial saturated fluid flow. Application of the model to a hypothetical two-dimensional field situation of gas diffusion in the unsaturated zone is demonstrated. The input and output files are included as a check on program installation. The definition of variables, input requirements, flow chart, and program listing are included in the attachments. (USGS)

In this paper, the solution of a fractional diffusion equation with a Hilfer-generalized Riemann–Liouville time fractional derivative is obtained in terms of Mittag–Leffler-type functions and Fox's H-function. The considered equation represents a quite general extension of the classical diffusion (heat conduction) equation. The methods of separation of variables, Laplace transform, and analysis of the Sturm–Liouville problem are used to solve the fractional diffusion equation defined in a bounded domain. By using the Fourier–Laplace transform method, it is shown that the fundamental solution of the fractional diffusion equation with a generalized Riemann–Liouville time fractional derivative defined in the infinite domain can be expressed via Fox's H-function. It is shown that the corresponding solutions of the diffusion equations with time fractional derivative in the Caputo and Riemann–Liouville sense are special cases of those diffusion equations with the Hilfer-generalized Riemann–Liouville time fractional derivative. The asymptotic behaviour of the solutions are found for large values of the spatial variable. The fractional moments of the fundamental solution of the fractional diffusion equation are obtained. The obtained results are relevant in the context of glass relaxation and aquifer problems.

Over the last two decades, anomalous diffusion processes in which the mean squares variance grows slower or faster than that in a Gaussian process have found many applications. At a macroscopic level, these processes are adequately described by fractional differential equations, which involves fractional derivatives in time or/and space. The fractional derivatives describe either history mechanism or long range interactions of particle motions at a microscopic level. The new physics can change dramatically the behavior of the forward problems. For example, the solution operator of the time fractional diffusion diffusion equation has only limited smoothing property, whereas the solution for the space fractional diffusion equation may contain weak singularity. Naturally one expects that the new physics will impact related inverse problems in terms of uniqueness, stability, and degree of ill-posedness. The last aspect is especially important from a practical point of view, i.e., stably reconstructing the quantities of interest. In this paper, we employ a formal analytic and numerical way, especially the two-parameter Mittag-Leffler function and singular value decomposition, to examine the degree of ill-posedness of several ‘classical’ inverse problems for fractional differential equations involving a Djrbashian–Caputo fractional derivative in either time or space, which represent the fractional analogues of that for classical integral order differential equations. We discuss four inverse problems, i.e., backward fractional diffusion, sideways problem, inverse source problem and inverse potential problem for time fractional diffusion, and inverse Sturm–Liouville problem, Cauchy problem, backward fractional diffusion and sideways problem for space fractional diffusion. It is found that contrary to the wide belief, the influence of anomalous diffusion on the degree of ill-posedness is not definitive: it can either significantly improve or worsen the conditioning of related inverse problems, depending crucially on the specific type of given data and quantity of interest. Further, the study exhibits distinct new features of ‘fractional’ inverse problems, and a partial list of surprising observations is given below. (a) Classical backward diffusion is exponentially ill-posed, whereas time fractional backward diffusion is only mildly ill-posed in the sense of norms on the domain and range spaces. However, this does not imply that the latter always allows a more effective reconstruction. (b) Theoretically, the time fractional sideways problem is severely ill-posed like its classical counterpart, but numerically can be nearly well-posed. (c) The classical Sturm–Liouville problem requires two pieces of spectral data to uniquely determine a general potential, but in the fractional case, one single Dirichlet spectrum may suffice. (d) The space fractional sideways problem can be far more or far less ill-posed than the classical counterpart, depending on the location of the lateral Cauchy data. In many cases, the precise mechanism of these surprising observations is unclear, and awaits further analytical and numerical exploration, which requires new mathematical tools and ingenuities. Further, our findings indicate fractional diffusion inverse problems also provide an excellent case study in the differences between theoretical ill-conditioning involving domain and range norms and the numerical analysis of a finite-dimensional reconstruction procedure. Throughout we will also describe known analytical and numerical results in the literature.

The article discusses different schemes for the numerical solution of the fractional Riccati equation with variable coefficients and variable memory, where the fractional derivative is understood in the sense of Gerasimov-Caputo. For a nonlinear fractional equation, in the general case, theorems of approximation, stability, and convergence of a nonlocal implicit finite difference scheme (IFDS) are proved. For IFDS, it is shown that the scheme converges with the order corresponding to the estimate for approximating the Gerasimov-Caputo fractional operator. The IFDS scheme is solved by the modified Newton’s method (MNM), for which it is shown that the method is locally stable and converges with the first order of accuracy. In the case of the fractional Riccati equation, approximation, stability, and convergence theorems are proved for a nonlocal explicit finite difference scheme (EFDS). It is shown that EFDS conditionally converges with the first order of accuracy. On specific test examples, the computational accuracy of numerical methods was estimated according to Runge’s rule and compared with the exact solution. It is shown that the order of computational accuracy of numerical methods tends to the theoretical order of accuracy with increasing nodes of the computational grid.

This paper presents a method for the solution of parabolic PDEs on parallel computers, which is a combination of implicit and explicit finite difference schemes based on a domain decomposition (DD) strategy. Moreover, this method is asynchronous (i.e., no explicit synchronization is required among processors). We determine the values at subdomains' boundaries by our new high-order asynchronous explicit schemes. Then, any known high-order implicit finite difference scheme can be applied within each subdomain. We present a technique for derivation of appropriate asynchronous-explicit schemes based on Green's functions. Synchronous versions of these schemes are obtained as special cases. The applicability of this method is also demonstrated for a family of nonlinear problems. Our new explicit schemes are of high order and yet stable for a large time step, as established in our analysis of their numerical properties. Moreover, these schemes provide attractive properties for parallel implementation. Being asynchronous, they allow local time stepping, thus eliminating the need for a global synchronized time step. Moreover, our asynchronous computation is time stabilizing, in the sense that the calculation implicitly prevents a growing time gap between neighboring subdomains. The locality property, due to the exponential decay of Green's functions, implies that communication is needed only between neighboring processors. Hence, this method which is designed to minimize the overhead associated with the synchronization of the multiple processors is specifically suitable for parallel computers having a high synchronization cost or highly varying load, even in cases in which some processors have persistent speed differences. Furthermore, the implementation of different resolution in each subdomain (e.g., irregular or unstructured grid) makes it valuable as an adaptive algorithm. The above schemes were implemented and tested on the shared-memory multi-user Cray J90 and Sequent Balance machines. These implementations prove high accuracy and high degree of parallelism. This work is complementary to our previous work on asynchronous schemes [Comput. Math. Appl., 24 (1992), pp. 33--53; Appl. Numer. Math., 12 (1993), pp. 27--45; Numer. Algorithms, 6 (1994), pp. 275--296; Numer. Algorithms, 12 (1996), pp. 159--192].

Fuzzy fractional diffusion equations are used to model certain phenomena in physics, hydrology biology and amongst others. In this paper, an implicit finite difference scheme is developed, analysed and applied to numerically solve a fuzzy time fractional diffusion equation. For our case, the fuzziness is in the coefficients as well as initial and boundary conditions. The time fractional derivative is defined using the Caputo formula. The stability of the implicit finite difference scheme is analysed by means of the Von Neumann method. A numerical example has been given to check the feasibility of the approach and to examine certain related aspects. It was found that the results obtained are in good agreement with the proposed theory. Hence, the proposed scheme is suitable for solving fuzzy time fractional diffusion equations.

Efficient, high-order accurate methods for the numerical solution of the standard (narrow-angle) parabolic equation for underwater sound propagation are developed. Explicit and implicit numerical schemes, which are second- or higher-order accurate in time-like marching and fourth-order accurate in the space-like direction are presented. The explicit schemes have severe stability limitations and some of the proposed high-order accurate implicit methods were found conditionally stable. The efficiency and accuracy of various numerical methods are evaluated for Cartesian-type meshes. The standard parabolic equation is transformed to body fitted curvilinear coordinates. An unconditionally stable, implicit finite-difference scheme is used for numerical solutions in complex domains with deformed meshes. Simple boundary conditions are used and the accuracy of the numerical solutions is evaluated by comparing with an exact solution. Numerical solutions in complex domains obtained with a finite element method show excellent agreement with results obtained with the proposed finite difference methods.

Various scholars have lately employed a wide range of strategies to resolve specific types of symmetrical fractional differential equations. This paper introduces a new implicit finite difference method with variable-order time-fractional Caputo derivative to solve semi-linear initial boundary value problems. Despite its extensive use in other areas, fractional calculus has only recently been applied to physics. This paper aims to find a solution for the fractional diffusion equation using an implicit finite difference scheme, and the results are displayed graphically using MATLAB and the Fourier technique to assess stability. The findings show the unconditional stability of the implicit time-fractional finite difference method. This method employs a variable-order fractional derivative of time, enabling greater flexibility and the ability to tackle more complicated problems.

Construction of explicit and implicit dynamic finite difference schemes and application to the large-eddy simulation of the Taylor–Green vortex

In level pool routing, which is the simplest hydrological routing method, the downstream discharge may be expressed explicitly in terms of the inflow and the channel or reservoir characteristics. The level pool routing equation can also be used to estimate the inflow hydrograph given the outflow hydrograph and the water level in the reservoir. Unfortunately, use of the traditional level pool routing method, which is based on the implicit finite difference scheme, for reverse routing has been unsuccessful, despite the simplicity of the problem. If a simple explicit centered differencing scheme is used instead for simulating the inflow hydrograph, the problems associated with traditional schemes, which require the application of filtering techniques, are bypassed. This is demonstrated using a realistic hypothetical example and a case study. The explicit scheme results are comparable in accuracy with results from the implicit scheme without resorting to the use of filtering techniques. The simple explicit scheme produces a direct solution to the problem and should be the preferred method for both level pool and reverse level pool routing.

A finite-difference method is used to transform the initial/boundary-value problem associated with the nonlinear Kadomtsev-Petviashvili equation, into an explicit scheme. The numerical method is developed by replacing the time and space derivatives by central-difference approximants. The resulting finite-difference method is analysed for local truncation error, stability and convergence. The results of a number of numerical experiments are given.

A new technique for the development of finite difference schemes for diffusion equations is presented. The model equations are the one space variable advection diffusion equation and the two space variable diffusion equation, each with Dirichlet boundary conditions. A two-step hybrid technique, which combines perturbation methods, based on the parameter ρ = Δt / (Δx)2, with the Galerkin method, provides a systematic way to develop new finite difference methods, referred to as hybrid equations. The main contributions of this paper include: 1) recovery of classical explicit or implicit finite difference schemes using only the perturbation terms; 2) development of new finite difference schemes, referred to as hybrid equations, which have better stability properties than the classical finite difference equations, permitting the use of larger values of the parameter ρ; and 3) higher order accurate methods, with either O((Δx)4) or O((Δx)6) truncation error, formed by convex linear combinations of the classical and hybrid equations. The solution of the hybrid finite difference equations requires only a tridiagonal equation solver and, hence, does not lead to excessive computational effort.