Taylor‐Galerkin B‐spline finite element method for the one‐dimensional advection‐diffusion equation

The advection‐diffusion equation has a long history as a benchmark for numerical methods. Taylor‐Galerkin methods are used together with the type of splines known as B‐splines to construct the approximation functions over the finite elements for the solution of time‐dependent advection‐diffusion problems. If advection dominates over diffusion, the numerical solution is difficult especially if boundary layers are to be resolved. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show the behavior of the method with emphasis on treatment of boundary conditions. Taylor‐Galerkin methods have been constructed by using both linear and quadratic B‐spline shape functions. Results shown by the method are found to be in good agreement with the exact solution. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Degenerate Dirichlet Problems Related to the Invariant Measure of Elasto-Plastic Oscillators

A stochastic variational inequality is proposed to model a white noise excited elasto-plastic oscillator. The solution of this inequality is essentially a continuous diffusion process for which a governing diffusion equation is obtained to study the evolution in time of its probability distribution. The diffusion equation is degenerate, but using the fact that the degeneracy occurs on a bounded region we are able to show the existence of a unique solution satisfying the desired properties. We prove the ergodic properties of the process and characterize the invariant measure. Our approach relies on extending Khasminskii’s method (Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, 1980), which in the present context leads to the study of degenerate Dirichlet problems with nonlocal boundary conditions. This research was partially supported by a grant from CEA, Commissariat à l’énergie atomique and by the National Science Foundation under grant DMS-0705247.     

Convergence of an operator splitting method on a bounded domain for a convection-diffusion-reaction system

We solve a convection-diffusion-sorption (reaction) system on a bounded domain with dominant convection using an operator splitting method. The model arises in contaminant transport in groundwater induced by a dual-well, or in controlled laboratory experiments. The operator splitting transforms the original problem to three subproblems: nonlinear convection, nonlinear diffusion, and a reaction problem, each with its own boundary conditions. The transport equation is solved by a Riemann solver, the diffusion one by a finite volume method, and the reaction equation by an approximation of an integral equation. This approach has proved to be very successful in solving the problem, but the convergence properties where not fully known. We show how the boundary conditions must be taken into account, and prove convergence in L1,loc of the fully discrete splitting procedure to the very weak solution of the original system based on compactness arguments via total variation estimates. Generally, this is the best convergence obtained for this type of approximation. The derivation indicates limitations of the approach, being able to consider only some types of boundary conditions. A sample numerical experiment of a problem with an analytical solution is given, showing the stated efficiency of the method.     

Using generating functions to convert an implicit (3,3) finite difference method to an explicit form on diffusion equation with different boundary conditions

In this article, our main goal is to develop an idea to convert an implicit (3,3) ??-scheme finite difference method to an explicit form for both linear and nonlinear diffusion equations and also for nonlinear advection-diffusion equation with different boundary conditions. Accordingly, we assist power series generating functions which are a routine method in discrete mathematics. Also, the stability analysis of ??–scheme to implement in nonlinear advection–diffusion equation has been investigated. Finally, the new approach has been implemented for Fisher, reaction–diffusion, Burgers and coupled Burgers equations as test problems to verify the ability and efficiency of the method proposed in this paper.     

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.     

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     

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.     

On the Adomian Decomp osition Metho d for Solving PDEs

In this paper, we explore some issues related to adopting the Adomian decomposition method (ADM) to solve partial differential equations (PDEs), par-ticularly linear diffusion equations. Through a proposition, we show that extending the ADM from ODEs to PDEs poses some strong requirements on the initial and boundary conditions, which quite often are violated for problems encountered in en-gineering, physics and applied mathematics. We then propose a modified approach, based on combining the ADM with the Fourier series decomposition, to provide solu-tions for those problems when these conditions are not met. In passing, we shall also present an argument that would address a long-term standing“pitfall”of the original ADM and make this powerful approach much more rigorous in its setup. Numeri-cal examples are provided to show that our modified approach can be used to solve any linear diffusion equation (homogeneous or non-homogeneous), with reasonable smoothness of the initial and boundary data.     

Attractor for a conserved phase‐field system with hyperbolic heat conduction

We consider a conserved phase‐field system of Caginalp type, characterized by the assumption that both the internal energy and the heat flux depend on the past history of the temperature and its gradient, respectively. The latter dependence is a law of Gurtin–Pipkin type, so that the equation ruling the temperature evolution is hyperbolic. Thus, the system consists of a hyperbolic integrodifferential equation coupled with a fourth‐order evolution equation for the phase‐field. This model, endowed with suitable boundary conditions, has already been analysed within the theory of dissipative dynamical systems, and the existence of an absorbing set has been obtained. Here we prove the existence of the universal attractor. Copyright © 2004 John Wiley & Sons, Ltd.     

Two‐dimensional fractal Burgers equation with step‐like initial conditions

We consider the two‐dimensional convection–diffusion equation with a fractional Laplacian, supplemented with step‐like initial conditions. We show that the large time behavior of solutions to this IVP is described either by rarefaction waves, or diffusion waves, or suitable self‐similar solutions, depending on the order of the fractional dissipation and on a direction of a convective nonlinearity. Copyright © 2014 John Wiley & Sons, Ltd.     

Stabilization by Slow Diffusion in a Real Ginzburg-Landau System

The Ginzburg-Landau equation is essential for understanding the dynamics of patterns in a wide variety of physical contexts. It governs the evolution of small amplitude instabilities near criticality. It is well known that the (cubic) Ginzburg-Landau equation has various unstable solitary pulse solutions. However, such localized patterns have been observed in systems in which there are two competing instability mechanisms. In such systems, the evolution of instabilities is described by a Ginzburg-Landau equation coupled to a diffusion equation. In this article we study the influence of this additional diffusion equation on the pulse solutions of the Ginzburg-Landau equation in light of recently developed insights into the effects of slow diffusion on the stability of pulses. Therefore, we consider the limit case of slow diffusion, i.e., the situation in which the additional diffusion equation acts on a long spatial scale. We show that the solitary pulse solution of the Ginzburg-Landau equation persists under this coupling. We use the Evans function method to analyze the effect of the slow diffusion and to show that it acts as a control mechanism that influences the (in)stability of the pulse. We establish that this control mechanism can indeed stabilize a pulse when higher order nonlinearities are taken into account.     

具有阶段结构的Lotka-Volterra合作系统的稳定性和行波解

文建立并研究了一个两物种成年个体相互合作的时滞反应扩散模型.利用线性化稳定性方法和Redlinger上、下解方法证明了该模型具有简单的动力学行为,即零平衡点和边界平衡点是不稳定的,而唯一的正平衡点是全局渐近稳定的.同时, 利用Wang, Li 和Ruan建立的具有非局部时滞的反应扩散系统的波前解的存在性,证明了该模型连接零平衡点与唯一正平衡点的波前解的存在性.     

Modelling and forecasting vehicle stocks using the trends of stochastic Gompertz diffusion models: The case of Spain

In the present study, we treat the stochastic homogeneous Gompertz diffusion process (SHGDP) by the approach of the Kolmogorov equation. Firstly, using a transformation in diffusion processes, we show that the probability transition density function of this process has a lognormal time‐dependent distribution, from which the trend and conditional trend functions and the stationary distribution are obtained. Second, the maximum likelihood approach is adapted to the problem of parameters estimation in the drift and the diffusion coefficient using discrete sampling of the process, then the approximated asymptotic confidence intervals of the parameter are obtained. Later, we obtain the corresponding inference of the stochastic homogeneous lognormal diffusion process as limit from the inference of SHGDP when the deceleration factor tends to zero. A statistical methodology, based on the above results, is proposed for trend analysis. Such a methodology is applied to modelling and forecasting vehicle stocks. Finally, an application is given to illustrate the methodology presented using real data, concretely the total vehicle stocks in Spain. Copyright © 2008 John Wiley & Sons, Ltd.     

Traveling wavefronts for a two-species predator–prey system with diffusion terms and stage structure

In this paper, we considered an important model describing a two-species predator–prey system with diffusion terms and stage structure. By using the linearized method, we investigated the locally asymptotical stability of the nonnegative equilibria of the system and obtained the locally stable conditions. And by using the approach introduced by Canosa [J. Canosa, On a nonlinear diffusion equation describing population growth, IBM J. Res. Dev. 17 (1973) 307–313] and the method of upper and lower solutions, we studied the existence of traveling wavefronts, connecting the zero solution with the positive equilibrium of the system. Our results show that the traveling wavefronts exist and appear to be monotone. Finally, we given a conclusion to summarize the overall achievements of the work presented in the paper.     

Nonlinear diffusion and viral spread through the leaf of a plant

The spread of a virus through the leaf of a plant is both spatially and temporally causal in that the present status depends on the past and the spatial spread is compactly supported and progresses outwards. Such spatial spread is known to occur for certain nonlinear diffusion processes. The first compactly supported solution for nonlinear diffusion equations appears to be that of Pattle published in 1959. In that paper, no explanation is given as to how the solution was derived. Here, we show how the solution can be derived using Lie symmetry analysis. This lays a foundation for exploring the behavior of other choices for nonlinear diffusion and exploring the addition of reaction terms which do not eliminate the compactly supported structure. The implications associated with using the reaction–diffusion equation to model the spatial–temporal spread of a virus through the leaf of a plant are discussed.     

High-order approximation of Pearson diffusion processes

This paper focuses on Pearson diffusions and the spectral high-order approximation of their related Fokker–Planck equations. The Pearson diffusions is a class of diffusions defined by linear drift and quadratic squared diffusion coefficient. They are widely used in the physical and chemical sciences, engineering, rheology, environmental sciences and financial mathematics. In recent years diffusion models have been studied analytically and numerically primarily through the solution of stochastic differential equations. Analytical solutions have been derived for some of the Pearson diffusions, including the Ornstein–Uhlenbeck, Cox–Ingersoll–Ross and Jacobi processes. However, analytical investigations and computations for diffusions with so-called heavy-tailed ergodic distributions are more difficult to perform. The novelty of this research is the development of an accurate and efficient numerical method to solve the Fokker–Planck equations associated with Pearson diffusions with different boundary conditions. Comparisons between the numerical predictions and available time-dependent and equilibrium analytical solutions are made. The solution of the Fokker–Planck equation is approximated using a reduced basis spectral method. The advantage of this approach is that many models for pricing options in financial mathematics cannot be expressed in terms of a stochastic partial differential equation and therefore one has to resort to solving Fokker–Planck type equations.     

Finite integral transform-based analytical solutions of dual phase lag bio-heat transfer equation

A finite integral transform (FIT)-based analytical solution to the dual phase lag (DPL) bio-heat transfer equation has been developed. One of the potential applications of this analytical approach is in the field of photo-thermal therapy, wherein the interest lies in determining the thermal response of laser-irradiated biological samples. In order to demonstrate the applicability of the generalized analytical solutions, three problems have been formulated: (1) time independent boundary conditions (constant surface temperature heating), (2) time dependent boundary conditions (medium subjected to sinusoidal surface heating), and (3) biological tissue phantoms subjected to short-pulse laser irradiation. In the context of the case study involving biological tissue phantoms, the FIT-based analytical solutions of Fourier, as well as non-Fourier, heat conduction equations have been coupled with a numerical solution of the transient form of the radiative transfer equation (RTE) to determine the resultant temperature distribution. Performance of the FIT-based approach has been assessed by comparing the results of the present study with those reported in the literature. A comparison of DPL-based analytical solutions with those obtained using the conventional Fourier and hyperbolic heat conduction models has been presented. The relative influence of relaxation times associated with the temperature gradients (τ_T) and heat flux (τ_q) on the resultant thermal profiles has also been discussed. To the best of the knowledge of the authors, the present study is the first successful attempt at developing complete FIT-based analytical solution(s) of non-Fourier heat conduction equation(s), which have subsequently been coupled with numerical solutions of the transient form of the RTE. The work finds its importance in a range of areas such as material processing, photo-thermal therapy, etc.     

Unifying pricing formula for several stochastic volatility models with jumps

In this paper, we introduce a unifying approach to option pricing under continuous‐time stochastic volatility models with jumps. For European style options, a new semi‐closed pricing formula is derived using the generalized complex Fourier transform of the corresponding partial integro‐differential equation. This approach is successfully applied to models with different volatility diffusion and jump processes. We also discuss how to price options with different payoff functions in a similar way. In particular, we focus on a log‐normal and a log‐uniform jump diffusion stochastic volatility model, originally introduced by Bates and Yan and Hanson, respectively. The comparison of existing and newly proposed option pricing formulas with respect to time efficiency and precision is discussed. We also derive a representation of an option price under a new approximative fractional jump diffusion model that differs from the aforementioned models, especially for the out‐of‐the money contracts. Copyright © 2017 John Wiley & Sons, Ltd.     

A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude

The Helmholtz equation arises when modeling wave propagation in the frequency domain. The equation is discretized as an indefinite linear system, which is difficult to solve at high wave numbers. In many applications, the solution of the Helmholtz equation is required for a point source. In this case, it is possible to reformulate the equation as two separate equations: one for the travel time of the wave and one for its amplitude. The travel time is obtained by a solution of the factored eikonal equation, and the amplitude is obtained by solving a complex‐valued advection–diffusion–reaction equation. The reformulated equation is equivalent to the original Helmholtz equation, and the differences between the numerical solutions of these equations arise only from discretization errors. We develop an efficient multigrid solver for obtaining the amplitude given the travel time, which can be efficiently computed. This approach is advantageous because the amplitude is typically smooth in this case and, hence, more suitable for multigrid solvers than the standard Helmholtz discretization. We demonstrate that our second‐order advection–diffusion–reaction discretization is more accurate than the standard second‐order discretization at high wave numbers, as long as there are no reflections or caustics. Moreover, we show that using our approach, the problem can be solved more efficiently than using the common shifted Laplacian multigrid approach.     

Lattice Boltzmann model for the bimolecular autocatalytic reaction–diffusion equation

A lattice Boltzmann model for the bimolecular autocatalytic reaction–diffusion equation is proposed. By using multi-scale technique and the Chapman–Enskog expansion on complex lattice Boltzmann equation, we obtain a series of complex partial differential equations, complex equilibrium distribution function and its complex moments. Then, the complex reaction–diffusion equation is recovered with higher-order accuracy of the truncation error. This equation can be used to describe the bimolecular autocatalytic reaction–diffusion systems, in which a rich variety of behaviors have been observed. Based on this model, the Fitzhugh–Nagumo model and the Gray–Scott model are simulated. The comparisons between the LBM results and the Alternative Direction Implicit results are given in detail. The numerical examples show that assumptions of source term can be used to raise the accuracy of the truncation error of the lattice Boltzmann scheme for the complex reaction–diffusion equation.