Laplace invariants of hyperbolic equations linearizable by a differential substitution

The boundness of the order of generalized Laplace invariants of a scalar hyperbolic equation is a necessary condition for the existence of a differential substitution transforming solutions of the equation into those of a linear hyperbolic equation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 2, pp. 237–247, August, 1999.     

Continuous Galerkin finite element methods for a forward‐backward heat equation

A space‐time finite element method is introduced to solve a model forward‐backward heat equation. The scheme uses the continuous Galerkin method for the time discretization. An error analysis for the method is presented. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 257–265, 1999     

Pathwise uniqueness for reflecting Brownian motion in Euclidean domains

For a bounded C ^1,α domain in ℝ ^d we show that there exists a strong solution to the multidimensional Skorokhod equation and that weak uniqueness holds for this equation. These results imply that pathwise uniqueness and strong uniqueness hold for the Skorokhod equation. Received: 3 February 1999 / Revised version: 2 September 1999 /?Published online: 11 April 2000     

Geometry and multidimensional soliton equations

The connection between the differential geometry of curves and (2+1)-dimensional integrable systems is established. The Zakharov equation, the modified Veselov-Novikov equation, the modified Kortewegde Vries equation, etc., are equivalent in the Lakshmanan sense to (2+1)-dimensional spin systems. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 3, pp. 441–451, March, 1999.     

Direct computation of multivalued phase space solutions for Hamilton‐Jacobi equations

A method is presented for the complete resolution of Hamiltonian systems using only the corresponding Hamilton‐Jacobi equation. © 1999 John Wiley & Sons, Inc.     

Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case

We establish global wellposedness and scattering for the -critical defocusing NLS in 3D

assuming radial data , . In particular, it proves global existence of classical solutions in the radial case. The same result is obtained in 4D for the equation

     

Some results on a class of nonliner Schrödinger equations

By using a Liapunov-Schmidt reduction we prove an existence result for the nonlinear Schr?dinger equation in where satisfies suitable assumptions. We also provide a necessary condition for the existence of solutions. Received June 7, 1999 / in final form November 10, 1999 / Published online July 20, 2000     

A finite difference scheme for solving the heat transport equation at the microscale

Heat transport at the microscale is of vital importance in microtechnology applications. In this study, we develop a finite difference scheme of the Crank‐Nicholson type by introducing an intermediate function for the heat transport equation at the microscale. It is shown by the discrete energy method that the scheme is unconditionally stable. Numerical results show that the solution is accurate. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 697–708, 1999     

A free boundary problem in an absorbing porous material with saturation dependent permeability

We prove a well posedness result for a free boundary problem describing the filtration of an incompressible viscous fluid in a porous medium containing water absorbing granules.?The location of the wetting front (the free boundary) is described by a Stefan like problem for a parabolic equation which is coupled with an hyperbolic equation describing the absorption kinetic of the granules. Received December 1999     

Explicit and approximate solutions of second‐order evolution differential equations in Hilbert space

The explicit closed‐form solutions for a second‐order differential equation with a constant self‐adjoint positive definite operator coefficient A (the hyperbolic case) and for the abstract Euler–Poisson–Darboux equation in a Hilbert space are presented. On the basis of these representations, we propose approximate solutions and give error estimates. The accuracy of the approximation automatically depends on the smoothness of the initial data. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 111–131, 1999     

Stationary layered solutions in {\mathbb{R}}^2 for a class of non autonomous Allen-Cahn equations

We consider a class of non autonomous Allen-Cahn equations where is a multiple-well potential and is a periodic, positive, non-constant function. We look for solutions to (0.1) having uniform limits as corresponding to minima of W. We show, via variational methods, that under a nondegeneracy condition on the set of heteroclinic solutions of the associated ordinary differential equation the equation (0.1) has solutions which depends on both the variables x andy. In contrast, when a is constant such nondegeneracy condition is not satisfied and all such solutions are known to depend only on x. Received April 16, 1999 / Accepted October 1, 1999 / Published online June 28, 2000     

Edge asymptotics for the radiosity equation over polyhedral boundaries

In the present paper we consider the radiosity equation over the boundary of a polyhedral domain. Similarly to corresponding results on the double‐layer potential equation, the solution of the second kind integral equation with non‐compact integral operator is piecewise continuous. The partial derivatives, however, are not bounded. In the present paper we derive the first term in the asymptotic expansion of the solution in the vicinity of an edge. Note that, knowing this term, optimal mesh gradings can be designed for the numerical solution of this equation. Copyright © 1999 John Wiley & Sons, Ltd.     

On Taylor-series expansion methods for the second kind integral equations

In this paper, we comment on the recent papers by Yuhe Ren et al. (1999) [1] and Maleknejad et al. (2006) [7] concerning the use of the Taylor series to approximate a solution of the Fredholm integral equation of the second kind as well as a solution of a system of Fredholm equations. The technique presented in Yuhe Ren et al. (1999) [1] takes advantage of a rapidly decaying convolution kernel k(|s−t|) as |s−t| increases. However, it does not apply to equations having other types of kernels. We present in this paper a more general Taylor expansion method which can be applied to approximate a solution of the Fredholm equation having a smooth kernel. Also, it is shown that when the new method is applied to the Fredholm equation with a rapidly decaying kernel, it provides more accurate results than the method in Yuhe Ren et al. (1999) [1]. We also discuss an application of the new Taylor-series method to a system of Fredholm integral equations of the second kind.     

Superconvergence of the orthogonal spline collocation solution of Poisson's equation

Superconvergence phenomena have been observed numerically in the piecewise Hermite bicubic orthogonal spline collocation solution of Poisson's equation on a rectangle. The purpose of this article is to demonstrate theoretically the superconvergent fourth‐order accuracy in the first‐order partial derivatives of the collocation solution at the partition nodes. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 285–303, 1999     

Asymptotic Behavior for Petrovsky Equation with Localized Damping

In this paper, we investigate asymptotic behavior for the solution of the Petrovsky equation with locally distributed damping. Without growth condition on the damping at the origin, we extend the energy decay result in Martinez (Rev. Math. Complut. Madr. 12(1):251–283, 1999) for the single wave equation to the Petrovsky equation. The explicit energy decay rate is established by using piecewise multiplier techniques and weighted nonlinear integral inequalities.     

The size of minimum 3‐trees: Cases 3 and 4 mod 6

A 3‐uniform hypergraph is called a minimum 3‐tree, if for any 3‐coloring of its vertex set there is a heterochromatic edge and the hypergraph has the minimum possible number of edges. Here we show that the number of edges in such 3‐tree is for any number of vertices n ≡ 3, 4 (mod 6). © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 157‐166, 1999     

Asymptotically self-similar global solutions of a general semilinear heat equation

We consider the general nonlinear heat equation on where and g satisfies certain growth conditions. We prove the existence of global solutions for small initial data with respect to a norm which is related to the structure of the equation. We also prove that some of those global solutions are asymptotic for large time to self-similar solutions of the single power nonlinear heat equation, i.e. with Received: 23 July 1999 / Accepted: 14 December 2000 / Published online: 23 July 2001     

Algebraic-geometric solutions of the Krichever-Novikov equation

A zero-curvature representation with constant poles on an elliptic curve is obtained for the Krichever-Novikov equation. Algebraic-geometric solutions of this equation are constructed. The consideration is based on reducing the theta function of a two-sheet covering of an elliptic curve to the Prym theta functions of codimension one. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 3, pp. 367–373, December, 1999.     

Sharp existence results for self-similar solutions of semilinear wave equations

The existence of self-similar and asymptotically self-similar solutions of the nonlinear wave equation with or in R ³×R ⁺ for small Cauchy data is proven if . A counterexample is given which shows that the lower bound on α is sharp. Received April 1999 – Accepted September 1999     

The Helmholtz equation in a quadrant with Robin's conditions

The boundary value problem for the Helmholtz equation in a quadrant with different Robin's or impedance‐type conditions is considered by the use of variational methods and boundary pseudodifferential equations. The coerciveness of the underlying sesquilinear form in dependence of the impedance parameters is proved. The equivalence between the boundary value problem and a scalar uniquely solvable Wiener–Hopf–Hankel equation is obtained. Copyright © 1999 John Wiley & Sons, Ltd.