scholarly journals Error estimates for a multidimensional meshfree Galerkin method with diffuse derivatives and stabilization

2013 ◽  
Vol 9 (17) ◽  
pp. 53-76
Author(s):  
Mauricio Osorio ◽  
Donald French
Keyword(s):  
Differential Equation ◽  
Convergence Rate ◽  
Error Analysis ◽  
Boundary Conditions ◽  
Galerkin Method ◽  
Approximation Error ◽  
Neumann Boundary ◽  
Meshfree Method ◽  
General Elliptic ◽  
Better Than

A meshfree method with diffuse derivatives and a penalty stabilization is developed. An error analysis for the approximation of the solution of a general elliptic differential equation, in several dimensions, with Neumann boundary conditions is provided. Theoretical and numerical results show that the approximation error and the convergence rate are better than the diffuse element method.

scholarly journals Uniqueness of Iterative Positive Solution to Nonlinear Fractional Differential Equations with Negatively Perturbed Term

2020 ◽  
Vol 2020 ◽  
pp. 1-6
Author(s):  
Jingjing Tan ◽  
Meixia Li ◽  
Aixia Pan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Convergence Rate ◽  
Boundary Value Problem ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Fractional Differential Equations ◽  
Iterative Scheme ◽  
Approximation Error ◽  
Fractional Differential

We prove that there are unique positive solutions for a new kind of fractional differential equation with a negatively perturbed term boundary value problem. Our methods rely on an iterative algorithm which requires constructing an iterative scheme to approximate the solution. This allows us to calculate the estimation of the convergence rate and the approximation error.

Ambrosetti–Prodi Periodic Problem Under Local Coercivity Conditions

2018 ◽  
Vol 18 (1) ◽  
pp. 169-182 ◽  
Author(s):  
Elisa Sovrano ◽  
Fabio Zanolin
Keyword(s):  
Differential Equation ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Periodic Boundary ◽  
Periodic Problem ◽  
Neumann Boundary ◽  
Continuous Functions ◽  
Periodic Boundary Value Problem ◽  
Multiplicity Result ◽  
Coercivity Conditions

AbstractIn this paper we focus on the periodic boundary value problem associated with the Liénard differential equation{x^{\prime\prime}+f(x)x^{\prime}+g(t,x)=s}, wheresis a real parameter,fandgare continuous functions andgisT-periodic in the variablet. The classical framework of Fabry, Mawhin and Nkashama, related to the Ambrosetti–Prodi periodic problem, is modified to include conditions without uniformity, in order to achieve the same multiplicity result under local coercivity conditions ong. Analogous results are also obtained for Neumann boundary conditions.

scholarly journals High Order Difference Schemes for a Time Fractional Differential Equation with Neumann Boundary Conditions

2014 ◽  
Vol 4 (3) ◽  
pp. 222-241 ◽  
Author(s):  
Seakweng Vong ◽  
Zhibo Wang
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Fractional Differential Equation ◽  
Neumann Boundary ◽  
High Order ◽  
Neumann Boundary Conditions ◽  
Stability And Convergence ◽  
Alternating Direction ◽  
Fractional Differential ◽  
The Stability

AbstractA compact finite difference scheme is derived for a time fractional differential equation subject to Neumann boundary conditions. The proposed scheme is second-order accurate in time and fourth-order accurate in space. In addition, a high order alternating direction implicit (ADI) scheme is also constructed for the two-dimensional case. The stability and convergence of the schemes are analysed using their matrix forms.

scholarly journals An Iterative Algorithm for Solving n -Order Fractional Differential Equation with Mixed Integral and Multipoint Boundary Conditions

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Jingjing Tan ◽  
Xinguang Zhang ◽  
Lishan Liu ◽  
Yonghong Wu
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Positive Solution ◽  
Fractional Differential Equation ◽  
Iterative Algorithm ◽  
Iterative Scheme ◽  
Approximation Error ◽  
Multipoint Boundary ◽  
Multipoint Boundary Conditions ◽  
Fractional Differential

In this paper, we consider the iterative algorithm for a boundary value problem of n -order fractional differential equation with mixed integral and multipoint boundary conditions. Using an iterative technique, we derive an existence result of the uniqueness of the positive solution, then construct the iterative scheme to approximate the positive solution of the equation, and further establish some numerical results on the estimation of the convergence rate and the approximation error.

On the Treatment of Neumann Boundary Conditions in Solving Poisson Equation Using Meshless Galerkin Method

2013 ◽  
Vol 816-817 ◽  
pp. 734-738
Author(s):  
Xiang Dong Zhang ◽  
Lei Wang
Keyword(s):  
Boundary Conditions ◽  
Galerkin Method ◽  
Poisson Equation ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Proper Method ◽  
Meshless Galerkin Method

This paper studies the treatment of Neumann boundary conditions when solving Poisson equation using meshless Galerkin method. We find that Neumann boundary conditions can be implemented more accurately by adopting proper method. Advantages of doing this are also shown.

scholarly journals Analysis of Dirichlet–Robin Iterations for Solving the Cauchy Problem for Elliptic Equations

2020 ◽  
Author(s):  
Pauline Achieng ◽  
Fredrik Berntsson ◽  
Jennifer Chepkorir ◽  
Vladimir Kozlov
Keyword(s):  
Cauchy Problem ◽  
Boundary Conditions ◽  
Helmholtz Equation ◽  
Iterative Algorithm ◽  
Elliptic Equations ◽  
Numerical Experiments ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
General Elliptic ◽  
The Cauchy Problem

Abstract The Cauchy problem for general elliptic equations of second order is considered. In a previous paper (Berntsson et al. in Inverse Probl Sci Eng 26(7):1062–1078, 2018), it was suggested that the alternating iterative algorithm suggested by Kozlov and Maz’ya can be convergent, even for large wavenumbers $$k^2$$ k 2 , in the Helmholtz equation, if the Neumann boundary conditions are replaced by Robin conditions. In this paper, we provide a proof that shows that the Dirichlet–Robin alternating algorithm is indeed convergent for general elliptic operators provided that the parameters in the Robin conditions are chosen appropriately. We also give numerical experiments intended to investigate the precise behaviour of the algorithm for different values of $$k^2$$ k 2 in the Helmholtz equation. In particular, we show how the speed of the convergence depends on the choice of Robin parameters.

scholarly journals The irrotational solution of an elliptic differential equation with an unknown coefficient

1963 ◽  
Vol 59 (3) ◽  
pp. 680-682 ◽  
Author(s):  
J. R. cannon ◽  
J. H. Halton
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Laplace Equation ◽  
Dimensional Space ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Unknown Coefficient ◽  
Bounded Region ◽  
Image Position ◽  
Normal Differentiation

Let G be a bounded region in k-dimensional space, with boundary Γ, such that the Laplace equation,is uniquely soluble (to within an added constant) under the Neumann boundary conditionswhere ∂/∂n denotes outward normal differentiation on Γ, and it is assumed that h is a function in G ∪ ∂, and thus that g is a function on ∂. In what follows, we shall assume certain properties of the solution h: these are all well known (see, for example, Osgood(l) or Courant(2)).

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