ON THE EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS FOR A CLASSOF DEGENERATE ELLIPTIC SYSTEMS

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Monotone iterations for numerical solutions of reaction-diffusion-convection equations with time delay

In this article we use the monotone method for the computation of numerical solutions of a nonlinear reaction-diffusion-convection problem with time delay. Three monotone iteration processes for a suitably formulated finite-difference system of the problem are presented. It is shown that the sequence of iteration from each of these iterative schemes converges from either above or below to a unique solution of the finite-difference system without any monotone condition on the nonlinear reaction function. An analytical comparison result among the three processes of iterations is given. Also given is the application of the iterative schemes to some model problems in population dynamics, including numerical results of a model problem with known analytical solution. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 339–351, 1998     

Positive solutions of quasilinear parabolic systems with nonlinear boundary conditions

The aim of this paper is to investigate the existence, uniqueness, and asymptotic behavior of solutions for a coupled system of quasilinear parabolic equations under nonlinear boundary conditions, including a system of quasilinear parabolic and ordinary differential equations. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system as well as the uniqueness of a positive steady-state solution. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients Di(ui) may have the property Di(0)=0 for some or all i. Our approach to the problem is by the method of upper and lower solutions and its associated monotone iterations. It is shown that the time-dependent solution converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a porous medium type of problem, a heat-transfer problem, and a two-component competition model in ecology. These applications illustrate some very interesting distinctive behavior of the time-dependent solutions between density-independent and density-dependent diffusions.     

A quasilinearization method for elliptic problems with a nonlinear boundary condition

We study a nonlinear elliptic second order problem with a nonlinear boundary condition. Assuming the existence of an ordered couple of a supersolution and a subsolution, we develop a quasilinearization method in order to construct an iterative scheme that converges to a solution. Furthermore, under an extra assumption we prove that the convergence is quadratic.     

Iterative method for solving a nonlinear boundary value problem

In this paper, a boundary value problem for a nonlinear second-order ordinary differential equation is studied. By means of the maximum principle we established the existence and the uniqueness of a solution of the problem. Then for finding the solution an iterative method is proposed. It is proved that this method converges much faster than the Picar successive approximations and in a particular case it gives two-sided monotone approximations to the exact solution of the problem. Finally, some illustrative examples are considered to confirm the efficiency of the method.     

Evaluating American put options on zero-coupon bonds by a penalty method

In this paper, American put options on zero-coupon bonds are priced under a single factor model of short-term rate. The linear complementarity problem of the option value is solved numerically by a penalty method, by which the problem is transformed into a nonlinear PDE by adding a power penalty term. The solution of the penalized problem converges to that of the original problem. A numerical scheme is established by using the finite volume method and the corresponding stability and convergence are discussed. Numerical results are presented to show the usefulness of the method.     

A block monotone domain decomposition algorithm for a semilinear convection–diffusion problem

This paper deals with discrete monotone iterative algorithms for solving a nonlinear singularly perturbed convection–diffusion problem. A block monotone domain decomposition algorithm based on a Schwarz alternating method and on block iterative scheme is constructed. This monotone algorithm solves only linear discrete systems at each iterative step of the iterative process and converges monotonically to the exact solution of the nonlinear problem. The rate of convergence of the block monotone domain decomposition algorithm is estimated. Numerical experiments are presented.     

Error bounds for euler approximation of a state and control constrained optimal control problem 1

We examine convergence of the Euler approximation to a nonlinear optimal control problem subject to mixed state-control and pure state constraints. We prove that under smoothness, independence, controllability and coercivity conditions at a reference solution of the continuous problem, there exists a locally unique solution to the Euler approximation, for sufficiently fine discretization, which converges to the reference solution with rate proportional to the mesh size.     

A power penalty approach to a mixed quasilinear elliptic complementarity problem

In this paper, a power penalty approximation method is proposed for solving a mixed quasilinear elliptic complementarity problem. The mixed complementarity problem is first reformulated as a double obstacle quasilinear elliptic variational inequality problem. A nonlinear elliptic partial differential equation is then defined to approximate the resulting variational inequality by using a power penalty approach. The existence and uniqueness of the solution to the partial differential penalty equation are proved. It is shown that, under some mild assumptions, the sequence of solutions to the penalty equations converges to the unique solution of the variational inequality problem as the penalty parameter tends to infinity. The error estimates of the convergence of this penalty approach are also derived. At last, numerical experimental results are presented to show that the power penalty approximation method is efficient and robust.

     

Viscous limits for piecewise smooth solutions of the p-system

We study the zero-dissipation problem for a one-dimensional model system for the isentropic flow of a compressible viscous gas, the so-called p-system with viscosity. When the solution of the inviscid problem is piecewise smooth and having finitely many noninteracting shocks satisfying the entropy condition, there exists unique solution to the viscous problem which converges to the given inviscid solution away from shock discontinuities at a rate of order ε as the viscosity coefficient ε goes to zero. The proof is given by a matched asymptotic analysis and an elementary energy method. And we do not need the smallness condition on the shock strength.     

GLOBAL EXISTENCE, UNIQUENESS, AND STABILITY FOR A NONLINEAR HYPERBOLIC-PARABOLIC PROBLEM IN PULSE COMBUSTION

A global existence theorem is established for an initial-boundary value problem,with time-dependent boundary data,arising in a lumped parameter model of pulse combustion; the model in question gives ri...     

守恒型扩散方程非线性离散格式的性质分析和快速求解

对于守恒型扩散方程,研究其二阶时间精度非线性全隐有限差分离散格式的性质,证明了其解的存在唯一性.研究了二阶时间精度的Picard-Newton迭代格式,证明了迭代解对原问题真解的二阶时间和空间收敛性,以及对非线性离散解的二次收敛速度,实现了非线性问题的快速求解.本文中方法也适用于一阶时间精度格式的分析,并可推广至对流扩散问题.数值实验验证了二阶时间精度Picard-Newton迭代格式的高精度和高效率.     

Successive iterations for unique positive solution of a nonlinear fractional q-integral boundary value problem

In this paper, under certain nonlinear growth conditions, we investigate the existence and successive iterations for the unique positive solution of a nonlinear fractional $q$-integral boundary problem by employing hybrid monotone method, which is a novel approach to nonlinear fractional $q$-difference equation. This paper not only proves the existence of the unique positive solution, but also gives some computable explicit hybrid iterative sequences approximating to the unique positive solution.     

关于一类具阻尼项的非线性波方程的Cauchy问题

The paper concerns with the existence, uniqueness and nonexistence of global solution to the Cauchy problem for a class of nonlinear wave equations with damping term. It proves that under suitable assumptions on nonlinear the function and initial data the above-mentioned problem admits a unique global solution by Fourier transform method. The sufficient conditions of nonexistence of the global solution to the above-mentioned problem are given by the concavity method.     

Fully discrete T‐ψ finite element method to solve a nonlinear induction hardening problem

We study an induction hardening model described by Maxwell's equations coupled with a heat equation. The magnetic induction field is assumed a nonlinear constitutional relation and the electric conductivity is temperature‐dependent. The T ‐ψ method is to transform Maxwell's equations to the vector–scalar potential formulations and to solve the potentials by means of the finite element method. In this article, we present a fully discrete T ‐ψ finite element scheme for this nonlinear coupled problem and discuss its solvability. We prove that the discrete solution converges to a weak solution of the continuous problem. Finally, we conclude with two numerical experiments for the coupled system.     

Regularizing properties of a truncated newton-cg algorithm for nonlinear inverse problems

This paper develops truncated Newton methods as an appropriate tool for nonlinear inverse problems which are ill-posed in the sense of Hadamard. In each Newton step an approximate solution for the linearized problem is computed with the conjugate gradient method as an inner iteration. The conjugate gradient iteration is terminated when the residual has been reduced to a prescribed percentage. Under certain assumptions on the nonlinear operator it is shown that the algorithm converges and is stable if the discrepancy principle is used to terminate the outer iteration. These assumptions are fulfilled, e.g., for the inverse problem of identifying the diffusion coefficient in a parabolic differential equation from distributed data.     

Algorithms for solutions of extended general mixed variational inequalities and fixed points

In this paper, we introduce and study a new class of extended general nonlinear mixed variational inequalities and a new class of extended general resolvent equations and establish the equivalence between the extended general nonlinear mixed variational inequalities and implicit fixed point problems as well as the extended general resolvent equations. Then by using this equivalent formulation, we discuss the existence and uniqueness of solution of the problem of extended general nonlinear mixed variational inequalities. Applying the aforesaid equivalent alternative formulation and a nearly uniformly Lipschitzian mapping S, we construct some new resolvent iterative algorithms for finding an element of set of the fixed points of nearly uniformly Lipschitzian mapping S which is the unique solution of the problem of extended general nonlinear mixed variational inequalities. We study convergence analysis of the suggested iterative schemes under some suitable conditions. We also suggest and analyze a class of extended general resolvent dynamical systems associated with the extended general nonlinear mixed variational inequalities and show that the trajectory of the solution of the extended general resolvent dynamical system converges globally exponentially to the unique solution of the extended general nonlinear mixed variational inequalities. The results presented in this paper extend and improve some known results in the literature.     

A Variational Analysis for the Moving Finite Element Method for Gradient Flows

By using the Onsager principle as an approximation tool, we give a novel derivation for the moving finite element method for gradient flow equations. We show that the discretized problem has the same energy dissipation structure as the continuous one. This enables us to do numerical analysis for the stationary solution of a nonlinear reaction diffusion equation using the approximation theory of free-knot piecewise polynomials. We show that under certain conditions the solution obtained by the moving finite element method converges to a local minimizer of the total energy when time goes to infinity. The global minimizer, once it is detected by the discrete scheme, approximates the continuous stationary solution in optimal order. Numerical examples for a linear diffusion equation and a nonlinear Allen-Cahn equation are given to verify the analytical results.     

A block monotone iterative method for numerical solutions of nonlinear elliptic boundary value problems

The aim of this article is to develop a new block monotone iterative method for the numerical solutions of a nonlinear elliptic boundary value problem. The boundary value problem is discretized into a system of nonlinear algebraic equations, and a block monotone iterative method is established for the system using an upper solution or a lower solution as the initial iteration. The sequence of iterations can be computed in a parallel fashion and converge monotonically to a maximal solution or a minimal solution of the system. Three theoretical comparison results are given for the sequences from the proposed method and the block Jacobi monotone iterative method. The comparison results show that the sequence from the proposed method converges faster than the corresponding sequence given by the block Jacobi monotone iterative method. A simple and easily verified condition is obtained to guarantee a geometric convergence of the block monotone iterations. The numerical results demonstrate advantages of this new approach. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011