On nonlinear wave equations with degenerate damping and source terms

In this article we focus on the global well-posedness of the differential equation , where is a sub-differential of a continuous convex function . Under some conditions on and the parameters in the equations, we obtain several results on the existence of global solutions, uniqueness, nonexistence and propagation of regularity. Under nominal assumptions on the parameters we establish the existence of global generalized solutions. With further restrictions on the parameters we prove the existence and uniqueness of a global weak solution. In addition, we obtain a result on the nonexistence of global weak solutions to the equation whenever the exponent is greater than the critical value , and the initial energy is negative. We also address the issue of propagation of regularity. Specifically, under some restriction on the parameters, we prove that solutions that correspond to any regular initial data such that , are indeed strong solutions.

     

On local properties of functions and singular integrals in terms of the mean oscillation

This paper is devoted to research on local properties of functions and multidimensional singular integrals in terms of their mean oscillation. The conditions guaranteeing existence of a derivative in the L ^p -sense at a given point are found. Spaces which remain invariant under singular integral operators are considered.      

On anisotropic caginalp phase-field type models with singular nonlinear terms

Our aim in this paper is to study the well-posedness and the existence of the global attractor of anisotropic Caginalp phase-field type models with singular nonlinear terms. The main difficulty is to prove, in one and two space dimensions, that the order parameter remains in the physically relevant range and this is achieved by deriving proper a priori estimates.     

A deficient spline function approximation to fourth-order differential equations

An approximate spline solution is developed for the initial value problem of a fourth-order ordinary differential equation. The approximation is based on deficient spline polynomials of degree M = 8 and deficiency 4. The existence and uniqueness of the solution, which satisfies a Lipschitz condition, are proved. The consistency, stability, and consequently convergence of the solution are established. Furthermore, the method is proved to be of order 9, and the errors are limited by the relation S⁽ⁱ⁾(x) − y⁽ⁱ⁾(x) = O(h^{9 − i}), I = 0(1)8.     

Regular approximation of singular self-adjoint differential operators

Given a singular self-adjoint differential operator of order 2n with real coefficients we constructtwo sequences of regular self-adjoint differential expressions^r which converge to ina generalized sense of resolvent convergence. The first constructionis suitable when no information about the real resolvent setof is available. The second is suitablewhen we know a real point of the resolvent set of .The main application of this construction is in numerical solutionof singular differential equations.     

Jacobi interpolation approximations and their applications to singular differential equations

Jacobi–Gauss-type interpolations are considered. Some approximation results in certain Hilbert spaces are established. They are used for numerical solutions of singular differential equations and other related problems. The numerical results are illustrated.     

On the numerical solution of diffusion–reaction equations with singular source terms

A numerical study is presented of reaction–diffusion problems having singular reaction source terms, singular in the sense that within the spatial domain the source is defined by a Dirac delta function expression on a lower dimensional surface. A consequence is that solutions will be continuous, but not continuously differentiable. This lack of smoothness and the lower dimensional surface form an obstacle for numerical discretization, including amongst others order reduction. In this paper the standard finite volume approach is studied for which reduction from order two to order one occurs. A local grid refinement technique is discussed which overcomes the reduction.     

A basis function for approximation and the solutions of partial differential equations

In this article, we introduce a type of basis functions to approximate a set of scattered data. Each of the basis functions is in the form of a truncated series over some orthogonal system of eigenfunctions. In particular, the trigonometric eigenfunctions are used. We test our basis functions on recovering the well‐known Franke's and Peaks functions given by scattered data, and on the extension of a singular function from an irregular domain onto a square. These basis functions are further used in Kansa's method for solving Helmholtz‐type equations on arbitrary domains. Proper one level and two level approximation techniques are discussed. A comparison of numerical with analytic solutions is given. The numerical results show that our approach is accurate and efficient. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

A novel discriminant approximation of periodic differential equations

A new approximation of the discriminant of a second order periodic differential equation is presented as a recursive summation of the evaluation of its excitation function at different values of time. The new approximation is obtained, at first, by means of Walsh functions and then, by using some algebraic properties the dependence on the Walsh functions is eliminated. This new approximation is then used to calculate the boundaries of stability. We prove that by letting the summation elements number approach to infinite, the discriminant approximation can be rewritten as a summation of definite integrals. Finally we prove that the definite integrals summation is equivalent to the discriminant approximation made by A. Lyapunov which consists in an alternating series of coefficients defined by multiple definite integrals, that is, a series of the form $A=A_0?A_1+\dots +{(?1)}^n{A}_n$, where each coefficient $A_n$ is defined as an n-multiple definite integral.     

Convergence of a singular Euler-Poisson approximation of the incompressible Navier-Stokes equations

In this note, we rigorously justify a singular approximation of the incompressible Navier-Stokes equations. Our approximation combines two classical approximations of the incompressible Euler equations: a standard relaxation approximation, but with a diffusive scaling, and the Euler-Poisson equations in the quasineutral regime.

     

On the approximation of stochastic differential equations

The stability properties of stochastic differential equations with respetct to the perturbation of the coefficients and of the driving processes are investigated in the topology of uniform convergence in probability     

On numerical approximation using differential equations with piecewise-constant arguments

In this paper we give a brief overview of the application of delay differential equations with piecewise constant arguments (EPCAs) for obtaining numerical approximation of delay differential equations, and we show that this method can be used for numerical approximation in a class of age-dependent population models. We also formulate an open problem for stability and oscillation of a class of linear delay equations with continuous and piecewise constant arguments. This research was partially supported by Hungarian NFSR Grant No. T046929.     

Series solutions of coupled differential equations with one regular singular point

We consider two linear second-order ordinary differential equations. r=0 is a regular singular point of these equations. Applying the classical Method of Frobenius, we do not obtain any indicial equation and therefore no solution, because the differential equations are coupled.

In this paper, we present an extended Method of Frobenius on a coupled system of two ordinary differential equations. These equations come from the micropolar theory, which is one of the three kinds of the new 3M physics.     

Gradient estimate for solutions of nonlinear singular elliptic equations below the duality exponent

In this paper, we study the homogeneous Dirichlet problem for an elliptic equation whose simplest model is where , N≥3 is an open bounded set, θ∈]0,1[, and f belongs to a suitable Morrey space. We will show that the Morrey property of the datum is transmitted to the gradient of a solution.     

一类具有不连续源的奇摄动半线性微分方程组边值问题

讨论了一类具有不连续源的奇摄动半线性微分方程组边值问题,构造了形式渐近解.利用Hartman-Nagumo不等式证明了奇摄动半线性微分方程组的解的存在性与唯一性,利用Aumann介值定理,得到了该方程组解的光滑性,并且得到了一致有效估计.     

On the approximation of singular integral equations by equations with smooth kernels

Singular integral equations with Cauchy kernel and piecewise-continuous matrix coefficients on open and closed smooth curves are replaced by integral equations with smooth kernels of the form(t–)[(t–) ²–n ² (t) ²]^–1,0, wheren(t), t , is a continuous field of unit vectors non-tangential to . we give necessary and sufficient conditions under which the approximating equations have unique solutions and these solutions converge to the solution of the original equation. For the scalar case and the spaceL ₂() these conditions coincide with the strong ellipticity of the given equation.This work was fulfilled during the first author's visit to the Weierstrass Institute for Applied Analysis and Stochastics, Berlin in October 1993.