The method of fundamental solutions with eigenfunction expansion method for nonhomogeneous diffusion equation

In this article we describe a numerical method to solve a nonhomogeneous diffusion equation with arbitrary geometry by combining the method of fundamental solutions (MFS), the method of particular solutions (MPS), and the eigenfunction expansion method (EEM). This forms a meshless numerical scheme of the MFS‐MPS‐EEM model to solve nonhomogeneous diffusion equations with time‐independent source terms and boundary conditions for any time and any shape. Nonhomogeneous diffusion equation with complex domain can be separated into a Poisson equation and a homogeneous diffusion equation using this model. The Poisson equation is solved by the MFS‐MPS model, in which the compactly supported radial basis functions are adopted for the MPS. On the other hand, utilizing the EEM the diffusion equation is first translated to a Helmholtz equation, which is then solved by the MFS together with the technique of the singular value decomposition (SVD). Since the present meshless method does not need mesh generation, nodal connectivity, or numerical integration, the computational effort and memory storage required are minimal as compared with other numerical schemes. Test results for two 2D diffusion problems show good comparability with the analytical solutions. The proposed algorithm is then extended to solve a problem with irregular domain and the results compare very well with solutions of a finite element scheme. Therefore, the present scheme has been proved to be very promising as a meshfree numerical method to solve nonhomogeneous diffusion equations with time‐independent source terms of any time frame, and for any arbitrary geometry. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

The effect of a changing diffusion coefficient in super-Case II polymer-penetrant systems

In certain polymer-penetrant systems, nonlinear viscoelasticeffects dominate those of Ficlcian diffusion. By introducinga dependence of the chemical potential on concentration history,this behaviour can be modelled by a memory integral. The mathematicalframework presented is a moving boundary-value problem wherethe boundary separates the polymer into two distinct states:glassy and rubbery. In each region, a different operator holdsat leading order. The problem which results is not solvableby similarity solutions, but can be solved by perturbation andintegral equation techniques. By introducing a new model wherethe diffusion coefficient changes with phase, asymptotic solutionsare obtained where sharp fronts initially move like t³/2. This‘super-Case II’ behaviour is found in various non-Fickianpolymer-penetrant systems.     

Numerical solution of a heat diffusion problem by boundary element methods using the Laplace transform

This paper is concerned with a heat diffusion problem in a half-space which is motivated by the detection of material defects using thermal measurements. This problem is solved by inverting the Laplace transform with respect to time on a contour in the complex plane using an exponentially convergent quadrature rule. This leads to a finite number of time-independent problems, which can be solved in parallel using boundary integral equation methods. We provide a full numerical analysis of this scheme on compact time intervals. Our results are formulated in a way that they can easily be used for other diffusion problems in exterior or interior domains.     

Generalized eigenvalue decomposition of the field autocorrelation in correlation diffusion of photons in turbid media

We propose a novel numerical method based on a generalized eigenvalue decomposition for solving the diffusion equation governing the correlation diffusion of photons in turbid media. Medical imaging modalities such as diffuse correlation tomography and ultrasound‐modulated optical tomography have the (elliptic) diffusion equation parameterized by a time variable as the forward model. Hitherto, for the computation of the correlation function, the diffusion equation is solved repeatedly over the time parameter. We show that the use of a certain time‐independent generalized eigenfunction basis results in the decoupling of the spatial and time dependence of the correlation function, thus allowing greater computational efficiency in arriving at the forward solution. Besides presenting the mathematical analysis of the generalized eigenvalue problem on the basis of spectral theory, we put forth the numerical results that compare the proposed numerical method with the standard technique for solving the diffusion equation. Copyright © 2012 John Wiley & Sons, Ltd.     

Anomalous diffusion on the percolating networks

According to the standard diffusion equation, by introducing reasonably into a anomalous diffusion coefficient, the generalized diffusion equation, which describes anomalous diffusion on the percolating networks with a power-law distribution of waiting times, is derived in this paper. The solution of the generalized diffusion equation is obtained by using the method, which is used by Barta. The problems of anomalous diffusion on percolating networks with a power-law distribution of waiting times, which aren't solved by Barta, are resolved.     

Global dynamics of a spatial heterogeneous viral infection model with intracellular delay and nonlocal diffusion

In this paper, we propose a spatial heterogeneous viral infection model, where heterogeneous parameters, the intracellular delay and nonlocal diffusion of free virions are considered. The global well-posedness, compactness and asymptotic smoothness of the semiflow generated by the system are established. It is shown that the principal eigenvalue problem of a perturbation of the nonlocal diffusion operator has a principal eigenvalue associated with a positive eigenfunction. The principal eigenvalue plays the same role as the basic reproduction number being defined as the spectral radius of the next generation operator. The existence of the unique chronic-infection steady state is established by the super-sub solution method. Furthermore, the uniform persistence of the model is investigated by using the persistence theory of infinite dimensional dynamical systems. By setting the eigenfunction as the integral kernel of Lyapunov functionals, the global threshold dynamics of the system is established. More precisely, the infection-free steady state is globally asymptotically stable if the basic reproduction number is less than one; while the chronic-infection steady state is globally asymptotically stable if the basic reproduction number is larger than one. Numerical simulations are carried out to illustrate the effects of intracellular delay and diffusion rate on the final concentrations of infected cells and free virions, respectively.     

Wavelet estimation of the diffusion coefficient in time dependent diffusion models

The estimation problem for diffusion coefficients in diffusion processes has been studied in many papers,where the diffusion coefficient function is assumed to be a 1-dimensional bounded Lipschitzian function of the state or the time only.There is no previous work for the nonparametric estimation of time-dependent diffusion models where the diffusion coefficient depends on both the state and the time.This paper introduces and studies a wavelet estimation of the time-dependent diffusion coefficient under a more general assumption that the diffusion coefficient is a linear growth Lipschitz function.Using the properties of martingale,we translate the problems in diffusion into the nonparametric regression setting and give the L~r convergence rate.A strong consistency of the estimate is established.With this result one can estimate the time-dependent diffusion coefficient using the same structure of the wavelet estimators under any equivalent probability measure.For example, in finance,the wavelet estimator is strongly consistent under the market probability measure as well as the risk neutral probability measure.     

On a nonlinear renewal equation with diffusion

In this paper, we consider a nonlinear age structured McKendrick–von Foerster population model with diffusion term. Here we prove existence and uniqueness of the solution of the equation. We consider a particular type of nonlinearity in the renewal term and prove Generalized Relative Entropy type inequality. Longtime behavior of the solution has been addressed for both linear and nonlinear versions of the equation. In linear case, we prove that the solution converges to the first eigenfunction with an exponential rate. In nonlinear case, we have considered a particular type of nonlinearity that is present in the mortality term in which we can predict the longtime behavior. Copyright © 2015 John Wiley & Sons, Ltd.     

Quenching phenomenon of a time‐fractional diffusion equation with singular source term

In this paper, a time‐fractional diffusion equation with singular source term is considered. The Caputo fractional derivative with order 0<α ?1 is applied to the temporal variable. Under specific initial and boundary conditions, we find that the time‐fractional diffusion equation presents quenching solution that is not globally well‐defined as time goes to infinity. The quenching time is estimated by using the eigenfunction of linear fractional diffusion equation. Moreover, by implementing a finite difference scheme, we give some numerical simulations to demonstrate the theoretical analysis. Copyright © 2017 John Wiley & Sons, Ltd.     

A new fractional finite volume method for solving the fractional diffusion equation

The inherent heterogeneities of many geophysical systems often gives rise to fast and slow pathways to water and chemical movement. One approach to model solute transport through such media is by fractional diffusion equations with a space–time dependent variable coefficient. In this paper, a two-sided space fractional diffusion model with a space–time dependent variable coefficient and a nonlinear source term subject to zero Dirichlet boundary conditions is considered.Some finite volume methods to solve a fractional differential equation with a constant dispersion coefficient have been proposed. The spatial discretisation employs fractionally-shifted Grünwald formulas to discretise the Riemann–Liouville fractional derivatives at control volume faces in terms of function values at the nodes. However, these finite volume methods have not been extended to two-dimensional and three-dimensional problems in a natural manner. In this paper, a new weighted fractional finite volume method with a nonlocal operator (using nodal basis functions) for solving this two-sided space fractional diffusion equation is proposed. Some numerical results for the Crank–Nicholson fractional finite volume method are given to show the stability, consistency and convergence of our computational approach. This novel simulation technique provides excellent tools for practical problems even when a complex transition zone is involved. This technique can be extend to two-dimensional and three-dimensional problems with complex regions.     

Group theoretic approach for solving the problem of diffusion of a drug through a thin membrane

The transformation group theoretic approach is applied to study the diffusion process of a drug through a skin-like membrane which tends to partially absorb the drug. Two cases are considered for the diffusion coefficient. The application of one parameter group reduces the number of independent variables by one, and consequently the partial differential equation governing the diffusion process with the boundary and initial conditions is transformed into an ordinary differential equation with the corresponding conditions. The obtained differential equation is solved numerically using the shooting method, and the results are illustrated graphically and in tables.     

A new inverse problem for the diffusion operator

In this work, we have estimated nodal points and nodal lengths for the diffusion operator. Furthermore, by using these new spectral parameters, we have shown that the potential function of the diffusion operator can be established uniquely. An analogous inverse problem was solved for the Sturm–Liouville problem in recent years.     

Numerical study of an influenza epidemic model with diffusion

A diffusive epidemic model is investigated with a view to describe the transmission of influenza as an epidemic. The equations are solved numerically using the splitting method under different initial distribution of population density. It is shown that the initial population distribution and diffusion play an important role for spread of disease. It is also shown that interventions (medical and nonmedical) significantly slow down the spread of disease. Stability of equilibria of the numerical solutions are also established.     

Simultaneous identification of diffusion coefficient,spacewise dependent source and initial value for one‐dimensional heat equation

This paper deals with an inverse problem of determining the diffusion coefficient, spacewise dependent source term, and the initial value simultaneously for a one‐dimensional heat equation based on the boundary control, boundary measurement, and temperature distribution at a given single instant in time. By a Dirichlet series representation for the boundary observation, the identification of the diffusion coefficient and initial value can be transformed into a spectral estimation problem of an exponential series with measurement error, which is solved by the matrix pencil method. For the identification of the source term, a finite difference approximation method in conjunction with the truncated singular value decomposition is adopted, where the regularization parameter is determined by the generalized cross‐validation criterion. Numerical simulations are performed to verify the result of the proposed algorithm. Copyright © 2016 John Wiley & Sons, Ltd.     

The influence of a metasolution on the behaviour of the logistic equation with nonlocal diffusion coefficient

In this paper we use the bifurcation method and fixed point arguments to study a logistic equation with nonlocal diffusion coefficient. We prove the existence of an unbounded continuum of positive solutions that bifurcates from the trivial solution. The global behaviour of this continuum depends strongly on the value of the nonlocal diffusion coefficient at infinity as well as the relative position between the refuge of the species and the weight of the diffusion coefficient. Moreover, we show the complexity of the structure of the set of positive solutions using fixed point arguments.     

Tikhonov regularization method for a backward problem for the time-fractional diffusion equation

This paper is devoted to solve a backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain by the Tikhonov regularization method. Based on the eigenfunction expansion of the solution, the backward problem for searching the initial data is changed to solve a Fredholm integral equation of the first kind. The conditional stability for the backward problem is obtained. We use the Tikhonov regularization method to deal with the integral equation and obtain the series expression of solution. Furthermore, the convergence rates for the Tikhonov regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Two numerical examples in one-dimensional and two-dimensional cases respectively are investigated. Numerical results show that the proposed method is effective and stable.     

A new coupled model for alloy solidification

A new coupled model in the binary alloy solidification has been developed. The model is based on the cellular automaton (CA) technique to calculate the evolution of the interface governed by temperature, solute diffusion and Gibbs-Thomson effect. The diffusion equation of temperature with the release of latent heat on the solid/liquid (S/L) interface is valid in the entire domain. The temperature diffusion without the release of latent heat and solute diffusion are solved in the entire domain. In the interface cells, the energy and solute conservation, thermodynamic and chemical potential equilibrium are adopted to calculate the temperature, solid concentration, liquid concentration and the increment of solid fraction. Compared with other models where the release of latent heat is solved in implicit or explicit form according to the solid/liquid (S/L) interface velocity, the energy diffusion and the release of latent heat in this model are solved at different scales, i.e. the macro-scale and micro-scale. The variation of solid fraction in this model is solved using several algebraic relations coming from the chemical potential equilibrium and thermodynamic equilibrium which can be cheaply solved instead of the calculation of S/L interface velocity. With the assumption of the solute conservation and energy conservation, the solid fraction can be directly obtained according to the thermodynamic data. This model is natural to be applied to multiple (< 2) spatial dimension case and multiple (< 2) component alloy. The morphologies of equiaxed dendrite are obtained in numerical experiments.     

Inverse problem for a space‐time fractional diffusion equation: Application of fractional Sturm‐Liouville operator

An inverse problem of determining a time‐dependent source term from the total energy measurement of the system (the over‐specified condition) for a space‐time fractional diffusion equation is considered. The space‐time fractional diffusion equation is obtained from classical diffusion equation by replacing time derivative with fractional‐order time derivative and Sturm‐Liouville operator by fractional‐order Sturm‐Liouville operator. The existence and uniqueness results are proved by using eigenfunction expansion method. Several special cases are discussed, and particular examples are provided.     

Bifurcation analysis of a predator–prey model with memory-based diffusion

In this paper, we investigate the predator–prey model equipped with Fickian diffusion and memory-based diffusion of predators. The stability and bifurcation analysis explores the impacts of the memory-based diffusion and the averaged memory period on the dynamics near the positive steady state. Specifically, when the memory-based diffusion coefficient is less than a critical value, we show that the stability of the positive steady state can be destabilized as the average memory period increases, which leads to the occurrence of Hopf bifurcations. Moreover, we also analyze the bifurcation properties using the central manifold theorem and normal form theory. This allows us to prove the existence of stable spatially inhomogeneous periodic solutions arising from Hopf bifurcation. In addition, the sufficient and necessary conditions for the occurrence of stability switches are also provided.     

Kronecker product approximation preconditioners for convection–diffusion model problems

We consider the iterative solution of linear systems arising from four convection–diffusion model problems: scalar convection–diffusion problem, Stokes problem, Oseen problem and Navier–Stokes problem. We design preconditioners for these model problems that are based on Kronecker product approximations (KPAs). For this we first identify explicit Kronecker product structure of the coefficient matrices, in particular for the convection term. For the latter three model cases, the coefficient matrices have a 2 × 2 block structure, where each block is a Kronecker product or a summation of several Kronecker products. We then use this structure to design a block diagonal preconditioner, a block triangular preconditioner and a constraint preconditioner. Numerical experiments show the efficiency of the three KPA preconditioners, and in particular of the constraint preconditioner that usually outperforms the other two. This can be explained by the relationship that exists between these three preconditioners: the constraint preconditioner can be regarded as a modification of the block triangular preconditioner, which at its turn is a modification of the block diagonal preconditioner based on the cell Reynolds number. Copyright © 2009 John Wiley & Sons, Ltd.