Multidimensional Diffusion Equation Research Articles

On multidimensional exact solutions of the nonlinear diffusion equation of the pantograph type with variable delay

We consider a multidimensional pantograph-type nonlinear diffusion equation with a linearly increasing time delay and scaling with respect to spatial variables in the source (sink). It is proposed to construct exact solutions by the reduction method using two ansatzes with a quadratic dependence on spatial variables. The dependence of the solution on spatial variables is found from a system of algebraic equations, and the dependence on time is found from a system of ordinary differential equations with a linearly increasing delay of the argument. A number of examples of exact solutions are given, both radially symmetric and anisotropic with respect to spatial variables.

Hybrid technique for multi-dimensional fractional diffusion problems involving Caputo–Fabrizio derivative

In this study, a precise and analytical method, namely Shehu transform decomposition method (STDM), is applied to examine multi-dimensional fractional diffusion equations, which will describe density dynamics in a material undergoing diffusion. The fractional derivative is taken into account by the Caputo–Fabrizio (CF) derivative. The proposed approach combines the Shehu transformation (ST) with the Adomian decomposition method (ADM), employing Adomain polynomials to handle nonlinear terms. Fixed point theorem has been used to prove the uniqueness and existence of the solution of governing differential equation. To validate the accuracy of proposed method, we present the analytical solutions of three applications of multi-dimensional fractional diffusion equations. Furthermore, we compute the graphical and numerical results using MATLAB to demonstrate the close-form analytical solution in the comparison of the exact solution. The obtained findings are promising and suitable for the solution of multi-dimensional diffusion problems with time-fractional derivatives. The main advantage is that our developed scheme does not require assumptions or restrictions on variables that ruin the actual problem. This scheme plays a significant role in finding the solution and overcoming the restriction of variables that may cause difficulty in modeling the problem.

Asymptotic analysis of fundamental solution of multi-dimensional distributed-order time-fractional diffusion equation with unit density function

In this paper, we consider the multi-dimensional distributed-order time-fractional diffusion equation with the unit density function. We introduce the new Volterra–Bessel function and give the integral representations of fundamental solutions of equations in terms of this function in the whole- and half-space. The fractional moments of fundamental solutions are also provided in the higher dimensions using the Mellin transforms. We further apply steepest descent method to find the asymptotic behaviors of solutions using the Schläfli integral of the Volterra–Bessel function. In this respect, we study the asymptotic analysis of the Volterra–Bessel function with the large parameters, and subsequently obtain the asymptotic behaviors of fundamental solutions with a discussion on the large space variable, large time variable, higher dimensions and small diffusivity constant.

New Stability Results of the Modified Craig-Sneyd Scheme in a Multidimensional Diffusion Equation with Mixed Derivative Terms

The time-dependent multidimensional diffusion equations with mixed derivative terms have been widely used in mathematics. Due to the mixed derivative terms, it is difficult to solve this pattern of multidimensional diffusion equations. The Modified Craig-Sneyd scheme (MCS), which has been applied to value the power derivatives, is one of the most promising methods for solving such equations. Therefore, it is of great importance to study its stability. The unconditional stability problem of MCS for a multidimensional diffusion equation with mixed derivative terms is investigated by taking into account the sizes. A new sufficient condition, necessary condition, and equivalence condition on the parameter of the MCS scheme for unconditional stability are proposed in case of the two and three-dimensional diffusion equations.

Shear dispersion of multispecies electrolyte solutions in the channel domain

In multispecies electrolyte solutions, even in the absence of an external electric field, differences in ion diffusivities induce an electric potential and generate additional fluxes for each species. This electro-diffusion process is well-described by the advection Nernst–Planck equation. This study aims to analyse the long-time behaviour of the governing equation under electroneutrality and zero current conditions, and to investigate how the diffusion-induced electric potential and shear flow enhance the effective diffusion coefficients of each species in channel domains. The exact solutions of the effective equation with certain special parameters, as well as the asymptotic analyses for ions with large diffusivity discrepancies, are presented. Furthermore, there are several interesting properties of the effective equation. First, it is a generalization of the Taylor dispersion, with a nonlinear diffusion tensor replacing the scalar diffusion coefficient. Second, the effective equation exhibits a scaling relation, revealing that the system with a weak flow is equivalent to the system with a strong flow under scaled physical parameters. Third, in the case of injecting an electrolyte solution into a channel containing well-mixed buffer solutions or electrolyte solutions with the same ion species, if the concentration of the injected solution is lower than that of the pre-existing solution, then the effective equation simplifies to a multi-dimensional diffusion equation. However, when introducing the electrolyte solution into a channel filled with deionized water, the ion–electric interaction results in several phenomena not present in the advection–diffusion equation, including upstream migration of some species, spontaneous separation of ions, and non-monotonic dependence of the effective diffusivity on Péclet numbers. Finally, the dependence of effective diffusivity on concentration and ion diffusivity suggests a method to infer the concentration ratio of each component and ion diffusivity by measuring the effective diffusivity.

A coupled nonlinear finite element scheme for anisotropic diffusion equation with nonlinear capacity term

A new nonlinear finite element method is studied to solve multi-dimensional anisotropic nonlinear diffusion equation with nonlinear capacity term, especially to secure a vivid simulation in the case of problems with transient physical quantities. In the scheme design, fully implicit two-layer coupled discretization is applied to assure high time accuracy, and finite element discretization is applied to assure high space accuracy. By introducing Ritz projection and new inductive reasoning methods, we develop the discrete functional analysis technique from that for finite difference schemes, and establish a new general analysis framework for nonlinear finite element schemes. Consequently, we overcome the difficulties arising from the strong coupling of the nonlinear finite element discretizations for the capacity term and diffusion operator, and prove the existence and boundedness of the nonlinear finite element solution, as well as its second-order time and optimal order space convergence. Numerical experiments and comparisons confirm the theoretical analysis results and demonstrate its high performance.

A rational preconditioner for multi-dimensional Riesz fractional diffusion equations

We propose a rational preconditioner for an efficient numerical solution of linear systems arising from the discretization of multi-dimensional Riesz fractional diffusion equations. In particular, the discrete problem is obtained by employing finite difference or finite element methods to approximate the fractional derivatives of order α with α∈(1,2]. The proposed preconditioner is then defined as a rational approximation of the Riesz operator expressed as the integral of the standard heat diffusion semigroup. We show that, being the sum of k inverses of shifted Laplacian matrices, the resulting preconditioner belongs to the generalized locally Toeplitz class. As a consequence, we are able to provide the asymptotic description of the spectrum of the preconditioned matrices and we show that, despite the lack of clustering just as for the Laplacian, our preconditioner for α close to 1 and k≠1 reasonably small, provides better results than the Laplacian itself, while sharing the same computational complexity.

Numerical analysis of multi-dimensional time-fractional diffusion problems under the Atangana-Baleanu Caputo derivative.

This paper presents the Elzaki homotopy perturbation transform scheme (EHPTS) to analyze the approximate solution of the multi-dimensional fractional diffusion equation. The Atangana-Baleanu derivative is considered in the Caputo sense. First, we apply Elzaki transform (ET) to obtain a recurrence relation without any assumption or restrictive variable. Then, this relation becomes very easy to handle for the implementation of the homotopy perturbation scheme (HPS). We observe that HPS produces the iterations in the form of convergence series that approaches the precise solution. We provide the graphical representation in 2D plot distribution and 3D surface solution. The error analysis shows that the solution derived by EHPTS is very close to the exact solution. The obtained series shows that EHPTS is a very simple, straightforward, and efficient tool for other problems of fractional derivatives.

THE NEW ANALYSIS OF FRACTIONAL-ORDER MULTI-DIMENSIONAL DIFFUSION EQUATIONS BY ZZ TRANSFORM WITH MITTAG-LEFFLER KERNEL

In this paper, two well-known analytic approaches for solving diffusion equations are implemented. We propose a modified version of the homotopy perturbation method and Adomian decomposition methods utilizing the ZZ transform. In this sense, the fractional derivative of the Atangana–Baleanu derivative is provided. In addition, concrete examples are as well provided to demonstrate the precision of the proposed methodologies. The proposed solution is observed to have the desired rate of convergence towards the exact answer. The key advantage of the proposed method is the small amount of calculations. To demonstrate the validity of the suggested methods, we give graphical representation of analytical and exact solutions that are in close contact.

New strategy for the numerical solution of multi-dimensional diffusion equations

PurposeThe purpose of this paper is to introduce an innovative strategy for the approximate solution of the heat flow problems in two- and three-dimensional spaces. This new strategy is very easy to implement and handles the restrictive variable that may ruin the physical nature of the problem.Design/methodology/approachThis study combines Sawi transform (ST) and the homotopy perturbation method (HPM) to formulate the idea of Sawi homotopy perturbation transform method (SHPTM). First, this study implements ST to handle the recurrence relation and then incorporates HPM to derive the series solutions of this recurrence relation. ST has the advantage in that it does not require any assumptions or hypothesis for the evaluation of series solutions.FindingsThis strategy finds the results very accurate and close to the precise solution. The graphical observations and the surface solution demonstrate that SHPTM is a reliable and powerful scheme for finding the approximate solution of heat flow problems.Originality/valueThe study presents an original work. This study develops SHPTM for the approximate solution of two- and three-dimensional heat flow problems. The obtained results and graphical representation demonstrate that SHPTM is a very authentic and reliable approach.

Simultaneous inversion of a fractional order and a space source term in an anomalous diffusion model

Abstract This paper is devoted to recovering simultaneously the fractional order and the space-dependent source term from partial Cauchy’s boundary data in a multidimensional time-fractional diffusion equation. The uniqueness of the inverse problem is obtained by employing analytic continuation and the Laplace transform. Then a modified non-stationary iterative Tikhonov regularization method with a regularization parameter chosen by a sigmoid-type function is used to find a stable approximate solution for the source term and the fractional order. Numerical examples in one-dimensional and two-dimensional cases are provided to illustrate the efficiency of the proposed algorithm.

Spectral Analysis for Preconditioning of Multi-Dimensional Riesz Fractional Diffusion Equations

In this paper, we analyze the spectra of the preconditioned matrices arising from discretized multi-dimensional Riesz spatial fractional diffusion equations. The finite difference method is employed to approximate the multi-dimensional Riesz fractional derivatives, which will generate symmetric positive definite ill-conditioned multi-level Toeplitz matrices. The preconditioned conjugate gradient method with a preconditioner based on the sine transform is employed to solve the resulting linear system. Theoretically, we prove that the spectra of the preconditioned matrices are uniformly bounded in the open interval (1/2,3/2) and thus the preconditioned conjugate gradient method converges linearly. The proposed method can be extended to multi-level Toeplitz matrices generated by functions with zeros of fractional order. Our theoretical results fill in a vacancy in the literature. Numerical examples are presented to demonstrate our new theoretical results in the literature and show the convergence performance of the proposed preconditioner that is better than other existing preconditioners.

Higher order compact finite difference schemes for steady and transient solutions of space–time neutron diffusion model

One of the most difficult tasks in the numerical simulation of nuclear reactors is to find solution of a system arising from spatial discretization of multidimensional multigroup neutron diffusion equations by finite difference method. The big disadvantage is that it requires smaller mesh spacing. Using smaller mesh spacing produces reasonable results but it consumes huge computational effort. In this work, we overcome this flaw by constructing higher order compact finite difference schemes to approximate the neutron leakage terms. The technique is based on using concentrated points around a central point instead of minimizing the mesh spacing taking into account the location of the diffusion coefficients. Beside the accuracy of the proposed schemes, the dimension of the coefficient matrix and therefore the complexity of the system still the same. Five point stencils with equally distance h in each direction are used to produce more than one of standard central difference formulas to be combined producing the fourth order finite difference scheme with order of accuracy O(h4). Also, seven point stencils are used to get the sixth order scheme with accuracy O(h6). The proposed fourth and sixth order schemes are highlighted through two theories. The maximum error norms and the convergence rates of the proposed schemes are discussed. For the time discretization, multi-step differential transform method MDTM is employed to solve the neutron diffusion model. To illustrate the applicability and accuracy of the proposed schemes, results are tested on some multidimensional homogenous and heterogeneous benchmark problems. Our static normalized assembly power densities that are obtained by dividing the whole core into just few dozens or hundreds of fuel assemblies are compared with the Improved SIDK, FLUENT and BRBFCM methods that divide the core into many thousands of fuel assemblies. It is proved that the obtained results are very close to those methods.

Space-Fractional Diffusion with Variable Order and Diffusivity: Discretization and Direct Solution Strategies

We consider the multidimensional space-fractional diffusion equations with spatially varying diffusivity and fractional order. Significant computational challenges are encountered when solving these equations due to the kernel singularity in the fractional integral operator and the resulting dense discretized operators, which quickly become prohibitively expensive to handle because of their memory and arithmetic complexities. In this work, we present a singularity-aware discretization scheme that regularizes the singular integrals through a singularity subtraction technique adapted to the spatial variability of diffusivity and fractional order. This regularization strategy is conveniently formulated as a sparse matrix correction that is added to the dense operator, and is applicable to different formulations of fractional diffusion equations. We also present a block low rank representation to handle the dense matrix representations, by exploiting the ability to approximate blocks of the resulting formally dense matrix by low rank factorizations. A Cholesky factorization solver operates directly on this representation using the low rank blocks as its atomic computational tiles, and achieves high performance on multicore hardware. Numerical results show that the singularity treatment is robust, substantially reduces discretization errors, and attains the first-order convergence rate allowed by the regularity of the solutions. They also show that considerable savings are obtained in storage (\(O(N^{1.5})\)) and computational cost (\(O(N^2)\)) compared to dense factorizations. This translates to orders-of-magnitude savings in memory and time on multidimensional problems, and shows that the proposed methods offer practical tools for tackling large nonlocal fractional diffusion simulations.

ADI Methods for Three-dimensional Fractional Diffusions

ADI methods can be generalized to solve numerically multidimensional fractional diffusion equations, which describe fluid flows through porous media better than classical diffusion equations. A new, unconditionally stable, second-order and well balanced in space, third-order in time ADI scheme has been constructed and its convergence accelerated by an extrapolation technique coupled with the PageRank algorithm.

High-order bicompact schemes for the quasilinear multidimensional diffusion equation

High-order bicompact schemes are designed for the quasilinear multidimensional diffusion equation. They are constructed using the method of lines, the finite volume method, the bicubic (or tricubic) Hermite interpolation. Implicit-explicit and diagonally-implicit Runge-Kutta methods are applied to time integration. The resulting schemes are stable for any ratios between grid steps, are conservative, have approximation of the fourth order in space and the third order in time. To implement the implicit-explicit schemes, an iteration method is proposed based on the approximate factorization of their multidimensional difference operators. This method is modified to implement the schemes with diagonally-implicit Runge-Kutta time stepping. High-order grid convergence and implementation efficiency of the new bicompact schemes are demonstrated on numerical examples.

Asymptotic behavior of solutions to the multidimensional semidiscrete diffusion equation

We study the asymptotic behavior of solutions to the multidimensional diffusion (heat) equation with continuous time and discrete space. We focus on initial-value problems with bounded initial data, and provide sufficient conditions for the existence of pointwise and uniform limits of solutions.

Efficient spatial second-/fourth-order finite difference ADI methods for multi-dimensional variable-order time-fractional diffusion equations

Efficient spatial second-/fourth-order finite difference ADI methods for multi-dimensional variable-order time-fractional diffusion equations

Fundamental solution of a multi-dimensional distributed order fractional diffusion equation

In this paper, we consider the multi-dimensional distributed order fractional diffusion equation in the whole space, the half space and the corridor space. We use the Titchmarsh theorem to show that the fundamental solutions are expressed in terms of the Laplace-type integrals and can be applied for the Gauss–Laguerre quadrature. The mathematical identities and their relationships for the fundamental solutions in different dimensions are also discussed, and the alternative representation is given in terms of the Mellin–Barnes integrals for the fundamental solution in the whole space.

Multi-dimensional diffusion process of allele frequencies in population genetics.

One of the ultimate goals of population genetics is to theoretically describe the behavior of allele frequency. Diffusion theory has been commonly used for this purpose mainly in one-locus one-population models, although it is not easy to handle diffusion theory in models with multiple loci or with multiple populations. This review introduces several successful cases, where multi-dimensional diffusion equations contributed to addressing evolutionary questions, thereby demonstrating its strong potential in population genetics.