Dimensional Diffusion Equation Research Articles

The high dimensional Fisher-KPP nonlocal diffusion equation with free boundary and radial symmetry, part 2: Sharp estimates

This is the second part of a two-part series devoted to an in depth understanding of the dynamical behaviour of the radially symmetric Fisher-KPP nonlocal diffusion equation with free boundary |x|=h(t). In Part 1 [19], we have shown that the long-time dynamics of this problem is characterised by a spreading-vanishing dichotomy, and there exists a threshold condition on the diffusion kernel J(|x|) such that the spreading speed is ∞ when this condition is not satisfied, and when it is satisfied, the finite spreading speed c0 is determined by an associated semi-wave problem established in [15]. In Part 2 here, we obtain more precise description of the spreading profile by focusing on some natural classes of kernel functions, including those satisfying J(|x|)∼|x|−β for |x|≫1 in RN. Our results for such kernels reveal a striking difference of behaviour from the pattern exhibited in the one dimension case [18] when β crosses the value N+2. More precisely, (a) when β∈(N,N+1], we show that for t≫1, h(t)∼t1/(β−N) if β∈(N,N+1), and h(t)∼tln⁡t if β=N+1, which is of the same pattern as in dimension one, namely we recover the result in [18] by letting N=1 in the above statements; (b) when β∈(N+1,N+2], the front has a finite spreading speed c0=c0(β) in the sense that limt→∞⁡h(t)/t=c0, and our results here on the order of shift c0t−h(t) are again of the same pattern as in dimension one; (c) when β>N+2, the front still has a finite spreading speed c0, but a significant change happens to the order of shift c0t−h(t) between N≥2 and N=1: for t≫1, c0t−h(t)∼ln⁡t when N≥2, but c0t−h(t)∼1 when N=1.

Cost evaluation of finite dimensional and approximate null-controllability for the one dimensional half-heat equation

In this article we evaluate the cost of the finite dimensional and the approximate controllability of a one dimensional anomalous diffusion equation which is not null-controllable. The finite dimensional controllability property consists in driving to zero a finite number of frequencies of the corresponding solutions. To evaluate its cost, we construct an explicit biorthogonal sequence to a finite part of a family of real exponential functions. Since the exponents of these functions satisfy the so-called Müntz density condition, there is no biorthogonal sequence to the entire family which shows that the equation is not even spectrally controllable. For the approximate controllability problem, we combine the idea of finite dimensional controllability studied before and the intrinsic dissipative nature of the system. These arguments allow to show that the controlled solution can be made arbitrarily small after sufficiently large time and provide estimates for the corresponding approximate controls.

KERNEL DETERMINATION PROBLEM FOR ONE PARABOLIC EQUATION WITH MEMORY

This paper studies the inverse problem of determining a multidimensional kernel function of an integral term which depends on the time variable \(t\) and \((n-1)\)-dimensional space variable \(x'= \left(x_1,\ldots, x_ {n-1}\right)\) in the \(n\)-dimensional diffusion equation with a time-variable coefficient at the Laplacian of a direct problem solution. Given a known kernel function, a Cauchy problem is investigated as a direct problem. The integral term in the equation has convolution form: the kernel function is multiplied by a solution of the direct problem's elliptic operator. As an overdetermination condition, the result of the direct question on the hyperplane \(x_n = 0\) is used. An inverse question is replaced by an auxiliary one, which is more suitable for further investigation. After that, the last problem is reduced to an equivalent system of Volterra-type integral equations of the second order with respect to unknown functions. Applying the fixed point theorem to this system in Hölder spaces, we prove the main result of the paper, which is a local existence and uniqueness theorem.

Implicit Runge‐Kutta with spectral Galerkin methods for the fractional diffusion equation with spectral fractional Laplacian

AbstractAn efficient numerical method with high accuracy both in time and in space is proposed for solving ‐dimensional fractional diffusion equation with spectral fractional Laplacian. The main idea is discretizing the time by an ‐stage implicit Runge‐Kutta method and approximating the space by a spectral Galerkin method with Fourier‐like basis functions. In view of the orthogonality, the mass matrix of the spectral Galerkin method is an identity and the stiffness matrix is diagonal, which makes the proposed method is efficient for high‐dimensional problems. The proposed method is showed to be stable and convergent with at least order in time, when the implicit Runge‐Kutta method with classical order () is algebraically stable. As another important contribution of this paper, we derive the spatial error estimate with optimal convergence order which depends on the regularity of the exact solution but not on the fractional parameter . This improves the previous result which depends on the fractional parameter . Numerical experiments verify and complement our theoretical results.

Numerical Solution of Two-Dimensional Fractional Order Diffusion Equation by using Elzaki Transform with Residual Power Series Method

The Elzaki transform with RPS method is an efficient and reliable approach for solution of non-integer order linear as well as non-linear differential equations. The major endeavourof current work is to find the solution of two dimensional non-integer order diffusion equation by Elzaki transform with RPS method. Firstly, Elzaki is applied on this equation. Secondly, inverse Elzaki is taken on this equation for finding the expressionof solution. Thirdly, assumed approximate solution is substituted on this equation. Then unidentified coefficient functions are obtained by using residual function is equal to zero as well as combining its initial circumstances. At last, coefficient functions are substituted in power series form for finding finite approximate logical solutions. The judgment between exact solution and approximate analytic solutions with different number of terms of this equation are determined and compared for reliability. This method reduces the size of computational works of solution of non-integer order non-linear diffusion equation

A fast ADI based matrix splitting preconditioning method for the high dimensional space fractional diffusion equations in conservative form

In this paper, we consider the high-dimensional two-sided space fractional diffusion equations with monotonic variable diffusion coefficients, which are derived from the fractional Fick's law. We apply the implicit Euler scheme to discretize the temporal derivative and certain difference operator to discretize the space fractional derivatives. For the discretized linear systems, we apply the alternating direction implicit (ADI) scheme to split the linear systems into two sub-systems of linear equations. For both coefficient matrices of the sub-systems of linear equations, we propose the lopsided scaled diagonal and Toeplitz splitting preconditioner. The generalized minimal residual (GMRES) method combined with the proposed preconditioner is applied to solve both sub-systems of linear equations. The spectral distributions of the preconditioned matrices are analyzed, and the theoretical results are given. Numerical results demonstrate that the proposed preconditioner is efficient in accelerating the convergence rate of the GMRES method for solving the discretized system of linear equations.

Upper and lower bounds for the speed of fronts of the reaction diffusion equation with Stefan boundary conditions

We establish two integral variational principles for the spreading speed of the one dimensional reaction diffusion equation with Stefan boundary conditions. The first principle is valid for monostable reaction terms and the second principle is valid for arbitrary reaction terms. These principles allow to obtain several upper and lower bounds for the speed. In particular, we construct a generalized Zeldovich–Frank–Kamenetskii type lower bound for the speed and upper bounds in terms of the speed of the standard reaction diffusion problem. We construct asymptotically exact lower bounds previously obtained by perturbation theory.

Numerical Simulation of One-Dimensional Advection Diffusion Equation by New Hybrid Explicit Finite Difference Schemes

In this paper, the aim is to develop two Hybrid Explicit Schemes based on the Finite Difference method for a one- dimensional Advection Diffusion Equation. Moreover, the study considered the advection-diffusion equation as an initial boundary value problem (IBVP) for numerical solutions obtained from various second-order explicit methods along with the solution by proposed methods. Von-Neumann stability analysis is used to analyze the stability of the developed schemes graphically. In the numerical analysis of errors, the L 2 has been computed to compare proposed methods with existing methods in literature and has been carried out in different conditions and step sizes. Proposed methods are robust explicit methods for the purpose of solution of the 1-D advection-diffusion equation.

Thermophoresis and Brownian motion effects on 3D flow of Casson nanofluid consisting microorganisms over a Riga plate using PSO: A numerical study

The objective of this research work is to analyze the nature of heat and mass transfer on 3D flow of Casson nanofluid that comprise of gyrotactic microorganisms over a Riga plate with modified magnetic field, thermophoresis and Brownian motion. The three dimensional flow, energy, diffusion and microorganisms equations are solved by numerical integration scheme along with PSO “Particle Swarm Optimization”. In this overall analysis, the patterns of temperature, velocities, concentration and gyrotactic microorganisms for Casson nanofluid are depicted due to existing parameters. From this study, it has been noted that due to increase in Casson factor and Brownian motion parameter values, mass transfer rate of working fluid is depreciated. Moreover, motile gyrotactic microorganisms profiles declined with increase in values of Peclet number, Hartmann number, Brownian and bioconvection parameters.

The High Dimensional Fisher-KPP Nonlocal Diffusion Equation with Free Boundary and Radial Symmetry, Part 1

This is Part 1 of a two-part series, where we study the radially symmetric high dimensional Fisher-KPP nonlocal diffusion equation with free boundary, and obtain a rather complete description of its long-time dynamical behavior, which reveals both similarities and fundamental differences to its one dimensional version considered in [J. Cao, Y. Du, F. Li, and W. T. Li, J. Funct. Anal., 277 (2019), pp. 2772--2814; Y. Du, F. Li, and M. Zhou, J. Math. Pure Appl., 154 (2021), pp. 30--66; Y. Du and W. Ni, Rate of Propagation for the Fisher-KPP Equation with Nonlocal Diffusion, and Free Boundaries, preprint, 2021] recently. Our goals on the long-time dynamics of the model include (a) find the threshold condition on the kernel function that governs the onset of accelerated spreading, (b) determine the spreading speed when it is finite, (c) obtain sharp estimates of the spreading profile for finite speed spreading as well as for accelerated spreading. The tasks in (a) and (b) are carried out here in Part 1, while (c) is done in a separate Part 2. This high dimensional problem poses considerable technical difficulties when the rate of spreading is considered, and we overcome that by introducing an intermediate kernel function which plays a crucial role both in determining the onset condition for accelerated spreading and in obtaining the spreading speed when it is finite.

A new FV scheme and fast cell-centered multigrid solver for 3D anisotropic diffusion equations with discontinuous coefficients

In this paper, an efficient cell-centered extrapolation cascadic multigrid (CEXCMG) method is proposed for solving large linear system of equations resulting from finite volume (FV) discretizations of three dimensional (3D) anisotropic diffusion equations with discontinuous coefficients. For cell-centered FV schemes, the values at vertex need to be approximated often by a linear combination of neighboring cell-centered values. In the literature, the weighted coefficients are obtained by solving local linear system of equations which is costly in 3D. One of the novelties of this paper is a new approach for obtaining vertex values by interpolating the cell-centered ones, which avoids solving local linear system of equations even with arbitrary diffusion tensors. Another main novelty of this paper is a new cascadic multigrid solver based on a prolongation operator, the newly developed explicit gradient transfer method, and a splitting extrapolation operator for solving 3D anisotropic diffusion equations with discontinuous coefficients. Numerical experiments are presented to demonstrate the efficiency and robustness of the CEXCMG method in terms of the mesh size and the contrast in the coefficients of the anisotropic diffusion tensor.

Collocation method for one dimensional nonlocal diffusion equations

In this paper, the collocation method for solving one dimensional steady state and time dependent nonlocal diffusion equations is analyzed. The difficulty of applying collocation method to nonlocal diffusion equations comes from the singularity of the kernel. If the kernel is weakly singular, however, if the kernel is not integrable in Riemann sense. So that the Hadamard finite part integral is introduced to overcome this difficulty. For analysis and performance, a “balance” term is added to discretize the nonlocal operator. Numerical results validate the theorems.

A new numerical scheme for solving the two dimensional fractional diffusion equation

A new numerical scheme for solving the two dimensional fractional diffusion equation

The Explicit Formula for Solution of Anomalous Diffusion Equation in the Multi-Dimensional Space

This paper intends on obtaining the explicit solution of $$n$$ -dimensional anomalous diffusion equation in the infinite domain with non-zero initial condition and vanishing condition at infinity. It is shown that this equation can be derived from the parabolic integro-differential equation with memory in which the kernel is $$t^{-\alpha}E_{1-\alpha,1-\alpha}(-t^{1-\alpha})$$ , $$\alpha\in(0,1),$$ where $$E_{\alpha,\beta}$$ is the Mittag-Liffler function. Based on Laplace and Fourier transforms the properties of the Fox H-function and convolution theorem, explicit solution for anomalous diffusion equation is obtained.

Monte Carlo Method to Solve Diffusion Equation and Error Analysis

Three different mathematical approaches for the evolution of diffusion equation are presented. The evolution process of the diffusion equation is explained by principle of conservation law, probability distribution procedure, and finally though stochastic differential equation (SDE) driven by Brownian motion. The Monte Carlo method is discussed to solve the diffusion equation by generating the normally distributed random numbers and the root mean square error is derived for the Monte Carlo method. The numerical solutions are computed for 1-dimensional diffusion equation and results are compared with exact solution. Finally, theoretical root mean square error is compared with the maximum error and the L2-error by increasing the number of simulated points.

Finite volume approximation with ADI scheme and low-rank solver for high dimensional spatial distributed-order fractional diffusion equations

High dimensional conservative spatial distributed-order fractional diffusion equation is discretized by midpoint quadrature rule, Crank–Nicolson method, and a finite volume approximation, with alternating direction implicit scheme. The resulting scheme is shown to be consistent and unconditionally stable, hence convergent with order 3−α, where α is the maximum of the involving fractional orders. Moreover, if the initial condition and source term possess Tensor-Train format (TT-format) with low TT-ranks, the scheme can be solved in TT-format, such that higher dimensional cases can be considered. Perturbation analysis ensures that the accumulated errors due to data recompression do not affect the overall convergence order. Numerical examples with low TT-ranks initial conditions and source terms, and with dimensions up to 20 are tested.

Spin diffusion transfer difference (SDTD) NMR: An advanced method for the characterisation of water structuration within particle networks

HypothesisThe classical STD NMR protocol to monitor solvent interactions in gels is strongly dependent on gelator and solvent concentrations and does not report on the degree of structuration of the solvent at the particle/solvent interface. We hypothesised that, for suspensions of large gelator particles, solvent structuration could be characterised by STD NMR when taking into account the particle-to-solvent 1H–1H spin diffusion transfer using the 1D diffusion equation. ExperimentsWe have carried out a systematic study on effect of gelator and solvent concentrations, and gelator surface charge, affecting the behaviour of the classical STD NMR build-up curves. To do so, we have characterised solvent interactions in dispersions of starch and cellulose-like particles prepared in deuterated water and alcohol/D2O mixtures. FindingsThe Spin Diffusion Transfer Difference (SDTD) NMR protocol is independent of the gelator and solvent concentrations, hence allowing the estimation of the degree of solvent structuration within different particle networks. In addition, the simulation of SDTD build-up curves using the general one–dimensional diffusion equation allows the determination of minimum distances (r) and spin diffusion rates (D) at the particle/solvent interface. This novel NMR protocol can be readily extended to characterise the solvent(s) organisation in any type of colloidal systems constituted by large particles.

SOME NEW APPLICATIONS OF LAPLACE-WEIERSTRASS TRANSFORM

This paper is devoted to some new applications of Laplace-Weierstrass transform (i.e., for solving two dimensional diffusion equations). Solution of Cauchy’s linear differential equation is also given. Some results are also given which are required for solving Cauchy’s linear differential equation.

Fast direct solver for CN-ADI-FV scheme to two-dimensional Riesz space-fractional diffusion equations

Based on the Crank-Nicolson alternating direction implicit finite volume (CN-ADI-FV) discretization of two dimensional Riesz space-fractional diffusion equations (2 D RSFDEs) proposed in [1], a fast direct solver is proposed to obtain the numerical solutions. Linear systems result from the CN-ADI-FV discretization can be treated as two matrix equations with symmetric positive definite (SPD) Toeplitz coefficient matrices. Then numerical solutions, expressed as multiple matrix-vector multiplications involving the inverse of SPD Toeplitz matrices, can be effectively obtained by using Gohberg-Semencul formula (GSF), where the preconditioned conjugate gradient (PCG) method is applied as an acceleration technique. Moreover, under certain assumptions, the spectral of the preconditioned matrices of the linear system associated with the GSF are theoretically proved to be clustered around 1. Then the PCG method is verified to converge superlinearly and can be implemented with O(NlogN) operations, where N is the number of spatial unknowns in both x- and y- directions. Besides, the PCG method is used only twice in the whole solution process, and the complete solution process can be finished with O(MN2logN) operations and O(N) memory, where M is the number of temporal unknowns. Numerical experiments are given to demonstrate the efficiency of our approach.

The numerical solution of high dimensional variable-order time fractional diffusion equation via the singular boundary method

IntroductionThis study describes a novel meshless technique for solving one of common problem within cell biology, computer graphics, image processing and fluid flow. The diffusion mechanism has extremely depended on the properties of the structure. ObjectivesThe present paper studies why diffusion processes not following integer-order differential equations, and present novel meshless method for solving. diffusion problem on surface numerically. MethodsThe variable- order time fractional diffusion equation (VO-TFDE) is developed along with sense of the Caputo derivative for (0<α(t)<1). An efficient and accurate meshfree method based on the singular boundary method (SBM) and dual reciprocity method (DRM) in concomitant with finite difference scheme is proposed on three-dimensional arbitrary geometry. To discrete of the temporal term, the finite diffract method (FDM) is utilized. In the spatial variation domain; the proposal method is constructed two part. To evaluating first part, fundamental solution of (VO-TFDE) is transformed into inhomogeneous Helmholtz-type to implement the SBM approximation and other part the DRM is utilized to compute the particular solution. ResultsThe stability and convergent of the proposed method is numerically investigated on high dimensional domain. To verified the reliability and the accuracy of the present approach on complex geometry several examples are investigated. ConclusionsThe result of study provides a rapid and practical scheme to capture the behavior of diffusion process.