Nonlinear Differential Operator Research Articles

A priori error estimates of finite element solutions of parametrized strongly nonlinear boundary value problems

Nonlinear boundary value problems with parameters are called parametrized nonlinear boundary problems. This paper studies a priori error estimates of finite element solutions of second-order parametrized strongly nonlinear boundary value problems in divergence form on one-dimensional bounded intervals. The Banach space W 0 1, ∞ is chosen in formulation of the error analysis so that the nonlinear differential operators defined by the differential equations are nonlinear Fredholm operators of index 1. Finite element solutions are defined in a natural way, and several a priori estimates are proved on regular branches and on branches around turning points. In the proofs the extended implicit function theorem due to Brezzi et al. (1980) plays an essential role.

The study on the nonlinear computations of the DQ and DC methods

This article points out that the differential quadrature (DQ) and differential cubature (DC) methods, due to their global domain property, are more efficient for nonlinear problems than the traditional numerical techniques such as finite element and finite difference methods. By introducing the Hadamard product of matrices, we obtain an explicit matrix formulation for the DQ and DC solutions of nonlinear differential and integro-differential equations. Due to its simplicity and flexibility, the present Hadamard product approach makes the DQ and DC methods much easier to use. Many studies on the Hadamard product can be fully exploited for the DQ and DC nonlinear computations. Furthermore, we first present the SJT product of matrixandvectortocomputeaccuratelyandefficientlytheFrechetderivativematrixintheNewton{Raphson method for the solution of the nonlinear formulations. We also propose a simple approach to simplify the DQ or DC formulations for some nonlinear differential operators and, thus, the computational efficiency of these methods is significantly improved. We give the matrix multiplication formulas to efficiently compute the weighting coefficient matrices of the DC method. The spherical harmonics are suggested as the test functionsintheDCmethodtohandlethenonlineardifferentialequationsoccurringinglobalandhemispheric weather forecasting problems. Some examples are analyzed to demonstrate the simplicity and efficiency of the presented techniques. It is emphasized that the innovations presented are applicable to the nonlinear computations of the other numerical methods as well. c 1997 John Wiley & Sons, Inc.

The study on the nonlinear computations of the DQ and DC methods

This paper points out that the differential quadrature (DQ) and differential cubature (DC) methods due to their global domain property are more efficient for nonlinear problems than the traditional numerical techniques such as finite element and finite difference methods. By introducing the Hadamard product of matrices, we obtain an explicit matrix formulation for the DQ and DC solutions of nonlinear differential and integro-differential equations. Due to its simplicity and flexibility, the present Hadamard product approach makes the DQ and DC methods much easier to be used. Many studies on the Hadamard product can be fully exploited for the DQ and DC nonlinear computations. Furthermore, we first present SJT product of matrix and vector to compute accurately and efficiently the Frechet derivative matrix in the Newton-Raphson method for the solution of the nonlinear formulations. We also propose a simple approach to simplify the DQ or DC formulations for some nonlinear differential operators and thus the computational efficiency of these methods is improved significantly. We give the matrix multiplication formulas to compute efficiently the weighting coefficient matrices of the DC method. The spherical harmonics are suggested as the test functions in the DC method to handle the nonlinear differential equations occurring in global and hemispheric weather forecasting problems. Some examples are analyzed to demonstrate the simplicity and efficiency of the presented techniques. It is emphasized that innovations presented are applicable to the nonlinear computations of the other numerical methods as well.

Energy demodulation of two-component AM-FM signal mixtures

An algorithm for the separation and energy-based demodulation of two-component mixtures of AM-FM signals is presented. The proposed algorithm is based on the generating differential or difference equation of the mixture signal and nonlinear differential energy operators.

HIGH-ORDER BEM FORMULATIONS FOR STRONGLY NON-LINEAR PROBLEMS GOVERNED BY QUITE GENERAL NON-LINEAR DIFFERENTIAL OPERATORS

SUMMARY In this paper the basic idea of homotopy in topology is applied to give a kind of high-order BEM formulation for general non-linear problems governed by non-linear differential operators which need not contain any linear operators at all. As a result, the traditional REM for non-linear problems is just a special case of the proposed method. Three simple examples are used to show the effectiveness of the proposed quite general BEM for nonlinear problems.

Bifurcation for nonlinear elliptic boundary value problems. III.

This paper is devoted to global static bifurcation theory for a class of degenerateboundary value problems for nonlinear second-order elliptic differential operators which includes as particular cases the Dirichlet and Neumann problems. In the previous paper [13] we treated the asymptotic linear case, for example, such nonlinear terms as $u^p$, $p > 1$, near $u = 0$ but $u + 1/u$ near $u = +\infty$. The purpose of this paper is to study the asymptotic nonlinear case, for example, such nonlinear terms as $u^p$ also near $u = +\infty$. First we prove a general existence and uniqueness theorem of positive solutions for our nonlinear boundary value problems, by using the super-subsolution method, and then we study in great detail the asymptotic nonlinear case.

Nonlinear differential operators of first and second order possessing invariant linear spaces of maximal dimension

In connection with the approach to the construction of explicit solutions to nonlinear partial differential equations proposed by S. S. Titov and V. A. Galaktionov, the problem of describing nonlinear differential operators F[y(x)] possessing finite-dimensional invariant linear spaces arises. It has previously been proved that for the m-th order operators, the dimension of an invariant space cannot exceed2m+1. In the present paper, we consider the cases where this value is achieved. The first- and second-order operators are studied and it is shown that they are quadratic in y. Full descriptions of the first-order operators and the second-order quadratic operators with constant coefficients are obtained.

A new approach to minimum and comparison principles for nonlinear ordinary differential operators of second order

A new approach to minimum and comparison principles for nonlinear ordinary differential operators of second order

Generalized Swan’s theorem and its application

Swan’s theorem verifies the equivalence between finitely generated projective modules over function algebras and smooth vector bundles. We define A ( r ) {A^{(r)}} -maps that correspond to usual non-linear differential operators of degree r under the equivalence of Swan’s theorem and thus generalize Swan’s theorem to include non-linear differential operators as morphisms. An A ( r ) {A^{(r)}} -manifold structure is introduced on the space of sections of a fiber bundle through charts with A ( r ) {A^{(r)}} -maps as transition homeomorphisms. A characterization for all the smooth maps between the spaces of sections of vector bundles, whose kth derivatives are linear differential operators of degree r in each variable, is given in terms of A ( r ) {A^{(r)}} -maps.

On the Gelfand–Dickey algebra GD(SLn) and the W n-symmetry. I. The bosonic case

The algebra Ξ of nonlinear (local and nonlocal) differential operators, acting on the ring of analytic functions, is studied. It is shown in particular that this space splits into 3×2 special subalgebras ∑jr, j=0,±1, r=±1. Each subalgebra is completely specified by quantum numbers s and (p,q) describing the conformal spin, and the lowest and the highest degrees, respectively. The algebra ∑++ (and its dual ∑−−) of local (pure nonlocal) differential operators is used to calculate the general expression of the Gelfand–Dickey bracket and the Wn-symmetry Poisson, one in terms of a set of spin j canonical fields uj, 2≤j≤n and a nonlinear u-cubic dependent differential operator D(n,i,j;z,u). The explicit form of this operator is worked out. Other remarkable features are also discussed.

Induced subbundles and Nash's implicit function theorem

Given two manifolds X and Y and smooth subbundles S ⊂ T( X) and T ⊂ T( Y), we study C ∞-maps ƒ : X → Y which induce S from T. Our study relies on Nash–Gromov implicit function theorem. We show that, under certain non-degeneracy conditions, the (nonlinear differential) operator D T : ƒ → S = ƒ ∗( T) is an open map. Moreover, under some additional topological restriction on X, we establish the existence of maps ƒ inducing S from T.

A penalty method for the identification of nonlinear elliptic differential operator

A penalty approximation sheme for identification of nonlinear elliptic operators along with numerical results axe given.

Invariants on projective space

This article is concerned with the construction of invariant nonlinear differential operators for projective space, pn := P(R n+1) There is no loss in working at first on the projective n-sphere Sn, that is the space of rays in Rn+l. The consequences for pn will be explained at the end of the article. Working on Sn avoids notational difficulties arising from the nonorientablity of pn when n is even. We shall use the general notion of a homogeneous bundle as in [2]. Suppose E and F are homogeneous vector bundles on X, a homogeneous space for some Lie group G. The polynomials of degree at most d on a vector space V themselves form a vector space, namely EhdO 0h V* (where o is the symmetric tensor product). Therefore, the G-invariant differential operators E --+ F of order k and which are polynomial in the jets of E are precisely the G-invariant homomorphisms d C)h Jk E --* F

An inexact Newton method for nonlinear two-point boundary-value problems

The method of quasilinearization for nonlinear two-point boundary-value problems is an application of Newton's method to a nonlinear differential operator equation. Since the linear boundary-value problem to be solved at each iteration must be discretized, it is natural to consider quasilinearization in the framework of an inexact Newton method. More importantly, each linear problem is only a local model of the nonlinear problem, and so it is inefficient to try to solve the linear problems to full accuracy. Conditions on size of the relative residual of the linear differential equation can then be specified to guarantee rapid local convergence to the solution of the nonlinear continuous problem. If initial-value techniques are used to solve the linear boundary-value problems, then an integration step selection scheme is proposed so that the residual criteria are satisfied by the approximate solutions. Numerical results are presented that demonstrate substantial computational savings by this type of economizing on the intermediate problems.

On a class of nonlinear differential operators acting on polynomials

On a class of nonlinear differential operators acting on polynomials

A model trust-region modification of Newton's method for nonlinear two-point boundary-value problems

The method of quasilinearization for nonlinear two-point boundary-value problems is Newton's method for a nonlinear differential operator equation. A model trust-region approach to globalizing the quasilinearization algorithm is presented. A double-dogleg implementation yields a globally convergent algorithm that is robust in solving difficult problems.

Rondo: Rank-order based nonlinear differential operator

A class of new differential filter that is a generalized version of the weighted median filter is proposed. The weighted median filter is a generalization of the median filter that has weight coefficients. In this paper, we propose an extended version of the weighted median filter that has negative weight coefficients. We call it the rank-order based nonlinear differential operator (RONDO). We show that RONDO acts as an effective differential operator. It detects edges in images degraded by impulsive noise effectively. It also has clear selectivity of edge orientation.

Optimization of cascaded threshold logic filters for grayscale image processing

Threshold logic filter is a broad class of the nonlinear filters. A designing method of cascaded threshold logic filters is proposed. The designing method is based on the learning rule for the multilayer neural network. The results of the optimization of the cascaded threshold logic filters are presented. Optimization includes those of the weighted median filter and the rank-order based nonlinear differential operator (RONDO) for specific applications.

A Nonlinear Differential Operator Series That Commutes with Any Function

A natural differential operator series is one that commutes with every function. The only linear examples are the formal series operators $e^{\alpha zD} $ representing translations. This paper discusses a surprising natural nonlinear “normally ordered” differential operator series, arising from the Lagrange inversion formula. The operator provides a wide range of new higher-order derivative identities and identities among Bell polynomials. These identities specialize to a large variety of interesting identities among binomial coefficients and classical orthogonal polynomials, a number of which are new.

Singular Limits in Free Boundary Problems

We analyze the following class of nonlinear eigenvalue problems: find (uµ) є B x R satisfying (1) Du + µH(a.u - 1)f(u) = 0 in Ω ⊆ R^N, (2) u = O on ∂Ω. Here H (X) is the Heaviside step-function defined by H(X) =0, X ≤ 0 H (X) = 1 X > 0. B is some Banach space appropriate to the problem. D is taken to be a (possibly nonlinear) differential operator with the property than, when µ = 0, equations (1,2) have the unique solution u