Stability estimates of solutions to extremal problems for a nonlinear convection-diffusion-reaction equation

We consider an identification problem for a stationary nonlinear convection–diffusion–reaction equation in which the reaction coefficient depends nonlinearly on the concentration of the substance. This problem is reduced to an inverse extremal problem by an optimization method. The solvability of the boundary value problem and the extremal problem is proved. In the case that the reaction coefficient is quadratic, when the equation acquires cubic nonlinearity, we deduce an optimality system. Analyzing it, we establish some estimates of the local stability of solutions to the extremal problem under small perturbations both of the quality functional and the given velocity vector which occurs multiplicatively in the convection–diffusion–reaction equation.     

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.     

平面二维水沙模型的特征有限元数值模拟

1引言水流泥沙数学模型是定量预测水沙运动及河床演变的重要手段,常用于模拟三角洲的形成与发展、淤积区的扩展、淤泥和泥沙输运和沉积而引起的河道的变迁等水利水沙问题.这一模型的数值研究已引起了水动力学工程师们的极大关注,对此问题的细致的研     

Approximation of Stochastic Nonlinear Equations of Schrödinger Type by the Splitting Method

In this article, we construct a splitting method for nonlinear stochastic equations of Schrödinger type. We approximate the solution of our problem by the sequence of solutions of two types of equations: one without stochastic integral term, but containing the Laplace operator and the other one containing only the stochastic integral term. The two types of equations are connected to each other by their initial values. We prove that the solutions of these equations both converge strongly to the solution of the Schrödinger type equation.     

Numerical study of Sivashinsky equation using a splitting scheme based on Crank‐Nicolson method

The mathematical modeling of a planar solid‐liquid interface in the solidification of a dilute binary alloy is formulating by one of nonintegrable, nonlinear evolution equation known as Sivashinsky equation. In the first part of this paper, the mathematical modeling of Sivashinsky equation is briefly discussed. Since, the exact solutions of this equation is yet unknown, obtaining its numerical solution plays an important role to simulate its behavior. Therefore, in the second part, a second‐order splitting finite difference scheme, based on Crank‐Nicolson method, is investigated to approximate the solution of the Sivashinsky equation with homogeneous boundary conditions. We prove the solvability of the present scheme and establish the error estimate of the numerical scheme.     

The vanishing viscosity method for the sensitivity analysis of an optimal control problem of conservation laws in the presence of shocks

In this article we study, by the vanishing viscosity method, the sensitivity analysis of an optimal control problem for 1-D scalar conservation laws in the presence of shocks. It is reduced to investigate the vanishing viscosity limit for the nonlinear conservation law, the corresponding linearized equation and its adjoint equation, respectively. We employ the method of matched asymptotic expansions to construct approximate solutions to those equations. It is then proved that the approximate solutions, respectively, satisfy those viscous equations in the asymptotic sense, and converge to the solutions of the corresponding inviscid problems with certain convergent rates. A new equation for the variation of shock positions is derived. It is also discussed how to identify descent directions to find the minimizer of the viscous optimal control problem in the quasi-shock case.     

Convergence analysis of a DDFV scheme for a system describing miscible fluid flows in porous media

In this article, we prove the convergence of a discrete duality finite volume scheme for a system of partial differential equations describing miscible displacement in porous media. This system is made of two coupled equations: an anisotropic diffusion equation on the pressure and a convection‐diffusion‐dispersion equation on the concentration. We first establish some a priori estimates satisfied by the sequences of approximate solutions. Then, it yields the compactness of these sequences. Passing to the limit in the numerical scheme, we finally obtain that the limit of the sequence of approximate solutions is a weak solution to the problem under study. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 723–760, 2015     

Convergence of the finite element method for an elliptic equation with strong degeneration

Under consideration are the questions of the numerical solution by the finite element method (FEM) of the first boundary value problem for an elliptic equation with degeneration on a part of the boundary. The weak and strong variational statements are posed in the function spaces with the coordinated weights that correspond to the problem. Using the method of the multiplicative extraction of singularities for the finite element method that utilizes piecewise linear elements, we prove that the convergence of the approximate solutions to the exact solution in the weighted norm is not worse than in the case of an elliptic equation without degeneration.     

Identification of an inverse source problem for time‐fractional diffusion equation with random noise

In this paper, we consider an inverse source problem for a time fractional diffusion equation. In general, this problem is ill posed, therefore we shall construct a regularized solution using the filter regularization method in the random noise case. We will provide appropriate conditions to guarantee the convergence of the approximate solution to the exact solution. Then, we provide examples of filters in order to obtain error estimates for their approximate solutions. Finally, we present a numerical example to show efficiency of the method.     

Exact travelling wave solutions for the dissipative (2 + 1)-dimensional AKNS equation

This paper employs the theory of planar dynamical systems and undetermined coefficient method to study travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation. By qualitative analysis, global phase portraits of the dynamic system corresponding to the equation are obtained under different parameter conditions. Furthermore, the relations between the properties of travelling wave solutions and the dissipation coefficient r of the equation are investigated. In addition, the possible bell profile solitary wave solution, kink profile solitary wave solutions and approximate damped oscillatory solutions of the equation are obtained by using undetermined coefficient method. Error estimates indicate that the approximate solutions are meaningful. Based on above studies, a main contribution in this paper is to reveal the dissipation effect on travelling wave solutions of the dissipative (2 + 1)-dimensional AKNS equation.     

Finite element approach to the Stokes problem

We approximate the Stokes problem by using a finite element method. This method utilizes the approach of Kleiser–Schumann, in which a boundary condition for the pressure is implicitly defined by a condition on the velocity. We consider a suitable splitting of the unknowns that allows one to reduce the Stokes problem to a cascade of classical Dirichlet problems and to a boundary integral equation.     

Approximation of Solution Components for Ill-Posed Problems by the Tikhonov Method with Total Variation

An ill-posed problem in the form of a linear operator equation given on a pair of Banach spaces is considered. Its solution is representable as a sum of a smooth and a discontinuous component. A stable approximation of the solution is obtained using a modified Tikhonov method in which the stabilizer is constructed as a sum of the Lebesgue norm and total variation. Each of the functionals involved in the stabilizer depends only on one component and takes into account its properties. Theorems on the componentwise convergence of the regularization method are stated, and a general scheme for the finite-difference approximation of the regularized family of approximate solutions is substantiated in the n-dimensional case.     

Identification of Parameters through the Approximate Periodic Solutions of a Parabolic System

This work is concerned with the identification problem for the perturbation term or error term in a parabolic partial differential equation through its approximate periodic solutions. The observation is made over a subregion of the physical domain. The existence and uniqueness problem of the approximate periodic solutions is studied in the first part of the paper. A solution to the identification problem is given in the second part of the paper. The main ingredients to be used include the classical Galerkin method and the unique continuation property for a parabolic system. This work was supported by the National Natural Science Foundation of China Grant 10571161.     

On the approximation of nonlinear singular self-adjoint second order boundary value problems

We investigate the approximation of the solutions of a class of nonlinear second order singular boundary value problems with a self-adjoint linear part. Our strategy involves two ingredients. First, we take advantage of certain boundary condition functions to obtain well behaved functions of the solutions. Second, we integrate the problem over an interval that avoids the singularity. We are able to prove a uniform convergence result for the approximate solutions. We describe how the approximation is constructed for the various values of the deficiency index associated with the differential equation. The solution of the nonlinear problem is obtained by a globally convergent iterative method.     

On the background of limit pass for Korteweg-de Vries equation as the dispersion vanishes

We present the new approach to the background of approximate methods of convergence based on the theory of functional solutions and solutions in the mean one for conservation laws. The applications to the Cauchy problem to KdV equation, when dispersion tends to zero are considered. Also the Galerkin method for a periodic problem for the KdV equation is considered.     

Block‐centered upwind multistep difference method and convergence analysis for numerical simulation of oil reservoir

The three‐dimensional displacement of two‐phase flow in porous media is a preliminary problem of numerical simulation of energy science and mathematics. The mathematical model is formulated by a nonlinear system of partial differential equations to describe incompressible miscible case. The pressure is defined by an elliptic equation, and the concentration is defined by a convection‐dominated diffusion equation. The pressure generates Darcy velocity and controls the dynamic change of concentration. We adopt a conservative block‐centered scheme to approximate the pressure and Darcy velocity, and the accuracy of Darcy velocity is improved one order. We use a block‐centered upwind multistep method to solve the concentration, where the time derivative is approximated by multistep method, and the diffusion term and convection term are treated by a block‐centered scheme and an upwind scheme, respectively. The composite algorithm is effective to solve such a convection‐dominated problem, since numerical oscillation and dispersion are avoided and computational accuracy is improved. Block‐centered method is conservative, and the concentration and the adjoint function are computed simultaneously. This physical nature is important in numerical simulation of seepage fluid. Using the convergence theory and techniques of priori estimates, we derive optimal estimate error. Numerical experiments and data show the support and consistency of theoretical result. The argument in the present paper shows a powerful tool to solve the well‐known model problem.     

Exact and traveling-wave solutions for convection-diffusion-reaction equation with power-law nonlinearity

Based on the simplest equation method, we propose exact and traveling-wave solutions for a nonlinear convection-diffusion-reaction equation with power law nonlinearity. Such equation can be considered as a generalization of the Fisher equation and other well-known convection-diffusion-reaction equations. Two important cases are considered. The case of density-independent diffusion and the case of density-dependent diffusion. When the parameters of the equation are constant, the Bernoulli equation is used as the simplest equation. This leads to new traveling-wave solutions. Moreover, some wavefront solutions can be derived from the traveling-wave ones. The case of time-dependent velocity in the convection term is studied also. We derive exact solutions of the equations by using the Riccati equation as simplest equation. The exact and traveling-wave solutions presented in this paper can be used to explain many biological and physical phenomena.     

Almost Periodically Distributed Solutions for Diffusion Equations in Duals of Nuclear Spaces

We discuss the problem of the existence of almost periodic in distribution solutions of nuclear space-valued diffusion equations with almost periodic coefficients. Under a dissipativity condition we prove that the translation of the unique mean square bounded solution is almost periodically distributed. Similar results hold in the affine case under mean square stability of the linear part of the equation if the nuclear space is a component of a special compatible family. Accepted 19 December 1996     

Splitting problem for WKB asymptotics in a nonresonant case and the reduction method for linear systems

As applied to the problem of asymptotic integration of linear systems of ordinary differential equations, we propose a reduction of order method that allows one to effectively construct solutions indistinguishable in the growth/decrease rate at infinity. In the case of a third-order equation, we use the developed approach to answer Bellman’s problem on splitting WKB asymptotics of subdominant solutions that decrease at the same rate. For a family of Wigner–von Neumann type potentials, the method allows one to formulate a selection rule for nonresonance values of the parameters (for which the corresponding second-order equation has a Jost solution).     

A numerical method for solving the time fractional Schrödinger equation

In this article, we proposed a new numerical method to obtain the approximation solution for the time-fractional Schrödinger equation based on reproducing kernel theory and collocation method. In order to overcome the weak singularity of typical solutions, we apply the integral operator to both sides of differential equation and yield a integral equation. We divided the solution of this kind equation into two parts: imaginary part and real part, and then derived the approximate solutions of the two parts in the form of series with easily computable terms in the reproducing kernel space. New bases of reproducing kernel spaces are constructed and the existence of approximate solution is proved. Numerical examples are given to show the accuracy and effectiveness of our approach.