On a high-order iterative scheme for a nonlinear Love equation

Le Thi Phuong Ngoc; Nguyen Tuan Duy; Nguyen Thanh Long

Displaying similar documents to “On a high-order iterative scheme for a nonlinear Love equation”

New modification of Maheshwari’s method with optimal eighth order convergence for solving nonlinear equations

Somayeh Sharifi, Massimiliano Ferrara, Mehdi Salimi, Stefan Siegmund (2016)

Open Mathematics

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In this paper, we present a family of three-point with eight-order convergence methods for finding the simple roots of nonlinear equations by suitable approximations and weight function based on Maheshwari’s method. Per iteration this method requires three evaluations of the function and one evaluation of its first derivative. These class of methods have the efficiency index equal to [...] 814≈1.682 $8^{\frac{1}{4}} \approx 1.682$ . We describe the analysis of the proposed methods along with numerical experiments including...

Existence of positive solutions for a nonlinear fourth order boundary value problem

Ruyun Ma (2003)

Annales Polonici Mathematici

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We study the existence of positive solutions of the nonlinear fourth order problem $u^{(4)} (x) = λ a (x) f (u (x))$ , u(0) = u’(0) = u”(1) = u”’(1) = 0, where a: [0,1] → ℝ may change sign, f(0) < 0, and λ < 0 is sufficiently small. Our approach is based on the Leray-Schauder fixed point theorem.

Multiplicity of positive solutions for a nonlinear fourth order equation

D. R. Dunninger (2001)

Annales Polonici Mathematici

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We study the existence and multiplicity of positive solutions of the nonlinear fourth order problem ⎧ $u^{(4)} = λ f (u)$ in (0,1), ⎨ ⎩u(0) = a ≥ 0, u’(0) = a’ ≥ 0, u(1) = b ≥ 0, u(1) = -b’ ≤ 0 The methods employed are upper and lower solutions and degree theory arguments.

Some results on strongly nonlinear anisotropic differential equations

L. Bougoffa, A. El Khalil, S. El Manouni (2010)

Applicationes Mathematicae

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The paper concerns the existence of weak solutions to nonlinear elliptic equations of the form A(u) + g(x,u,∇u) = f, where A is an operator from an appropriate anisotropic function space to its dual and the right hand side term is in $L^{1 + m}$ with 0 < m < 1. We assume a sign condition on the nonlinear term g, but no growth restrictions on u.

Newton-type iterative methods for nonlinear ill-posed Hammerstein-type equations

Monnanda Erappa Shobha, Ioannis K. Argyros, Santhosh George (2014)

Applicationes Mathematicae

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We use a combination of modified Newton method and Tikhonov regularization to obtain a stable approximate solution for nonlinear ill-posed Hammerstein-type operator equations KF(x) = y. It is assumed that the available data is $y^{δ}$ with $| | y - y^{δ} | | \leq δ$ , K: Z → Y is a bounded linear operator and F: X → Z is a nonlinear operator where X,Y,Z are Hilbert spaces. Two cases of F are considered: where $F^{'} {(x ₀)}^{- 1}$ exists (F’(x₀) is the Fréchet derivative of F at an initial guess x₀) and where F is a monotone operator....

Analysis of the discontinuous Galerkin finite element method applied to a scalar nonlinear convection-diffusion equation

Hozman, Jiří, Dolejší, Vít

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We deal with a scalar nonstationary convection-diffusion equation with nonlinear convective as well as diffusive terms which represents a model problem for the solution of the system of the compressible Navier-Stokes equations describing a motion of viscous compressible fluids. We present a discretization of this model equation by the discontinuous Galerkin finite element method. Moreover, under some assumptions on the nonlinear terms, domain partitions and the regularity of the exact...

Some Remarks on Nonlinear Composition Operators in Spaces of Differentiable Functions

J. Appell, Z. Jesús, O. Mejía (2011)

Bollettino dell'Unione Matematica Italiana

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In this note we study the nonlinear composition operator $f\mapsto g\circ f$ in various spaces of differentiable functions over an interval. It turns out that this operator is always bounded in the corresponding norm, whenever it maps such a space into itself, but continuous only in exceptional cases.

A convergent nonlinear splitting via orthogonal projection

Jan Mandel (1984)

Aplikace matematiky

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We study the convergence of the iterations in a Hilbert space $V, x_{k + 1} = W (P) x_{k}, W (P) z = w = T (P w + (I - P) z)$ , where $T$ maps $V$ into itself and $P$ is a linear projection operator. The iterations converge to the unique fixed point of $T$ , if the operator $W (P)$ is continuous and the Lipschitz constant $∥(I - P) W (P)∥ < 1$ . If an operator $W (P_{1})$ satisfies these assumptions and $P_{2}$ is an orthogonal projection such that $P_{1} P_{2} = P_{2} P_{1} = P_{1}$ , then the operator $W (P_{2})$ is defined and continuous in $V$ and satisfies $∥(I - P_{2}) W (P_{2})∥ \leq ∥(I - P_{1}) W (P_{1})∥$ .

Blow-up of the solution to the initial-value problem in nonlinear three-dimensional hyperelasticity

J. A. Gawinecki, P. Kacprzyk (2008)

Applicationes Mathematicae

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We consider the initial value problem for the nonlinear partial differential equations describing the motion of an inhomogeneous and anisotropic hyperelastic medium. We assume that the stored energy function of the hyperelastic material is a function of the point x and the nonlinear Green-St. Venant strain tensor $e_{j k}$ . Moreover, we assume that the stored energy function is $C^{\infty}$ with respect to x and $e_{j k}$ . In our description we assume that Piola-Kirchhoff’s stress tensor $p_{j k}$ depends on the tensor...

An adaptive $h p$ -discontinuous Galerkin approach for nonlinear convection-diffusion problems

Dolejší, Vít

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We deal with a numerical solution of nonlinear convection-diffusion equations with the aid of the discontinuous Galerkin method (DGM). We propose a new $h p$ -adaptation technique, which is based on a combination of a residuum estimator and a regularity indicator. The residuum estimator as well as the regularity indicator are easily evaluated quantities without the necessity to solve any local problem and/or any reconstruction of the approximate solution. The performance of the proposed $h p$ -DGM...

On the principal eigencurve of the p-Laplacian related to the Sobolev trace embedding

Abdelouahed El Khalil, Mohammed Ouanan (2005)

Applicationes Mathematicae

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We prove that for any λ ∈ ℝ, there is an increasing sequence of eigenvalues μₙ(λ) for the nonlinear boundary value problem ⎧ $Δ ₚ u = {| u |}^{p - 2} u$ in Ω, ⎨ ⎩ ${| \nabla u |}^{p - 2} \partial u / \partial ν = {λ ϱ (x) | u |}^{p - 2} u + μ {| u |}^{p - 2} u$ on crtial ∂Ω and we show that the first one μ₁(λ) is simple and isolated; we also prove some results about variations of the density ϱ and the continuity with respect to the parameter λ.

On the number of positive solutions of singularly perturbed 1D nonlinear Schrödinger equations

Patricio Felmer, Salomé Martínez, Kazunaga Tanaka (2006)

Journal of the European Mathematical Society

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We study singularly perturbed 1D nonlinear Schrödinger equations (1.1). When $V (x)$ has multiple critical points, (1.1) has a wide variety of positive solutions for small $ε$ and the number of positive solutions increases to $\infty$ as $ε \to 0$ . We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of $V (x)$ . Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.