New applications of the existence of solutions for equilibrium equations with Neumann type boundary condition

We consider the following high order periodic-type boundary value problem \[ \lefteqn{(PBVP)} \left\{\begin{array}{lll} (E)~u^{(n)}(t)= f(t,u(t),u^{(1)}(t), \cdots, u^{(n-2)}(t), u^{(n-1)}(t))~\mbox{for}~t\in (0,T) \\ (PBC)~\left\{\begin{array}{lll} u^{(i)}(0)=0,~0\leq i\leq n-3,\\ u^{(n-2)}(0)= u^{(n-2)}(T),\\ u^{(n-1)}(0)= u^{(n-1)}(T), \end{array}\right. \end{array}\right. \] where $f\in C([0,T]\times\mathbb{R}^n,\mathbb{R})$, $n\geq 2$ and satisfies the so-called Nagumo's condition. In this article, we will use a general upper and lower solution method to establish an existence theorem for solutions of $(PBVP)$.

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Inverse Problem of Finding the Coecient of the Lowest Term in Two-dimensional Heat Equation with Ionkin-type Boundary Condition

Журнал вычислительной математики и математической физики ◽

10.1134/s0044466919050168 ◽

2019 ◽

Vol 59 (5) ◽

pp. 859-859

Author(s):

M. I. Ismailov ◽

S. Erkovan

Keyword(s):

Boundary Condition ◽

Inverse Problem ◽

Heat Equation ◽

Two Dimensional ◽

Type Boundary ◽

Type Boundary Condition

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An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2019089 ◽

2020 ◽

Vol 54 (4) ◽

pp. 1373-1413 ◽

Cited By ~ 1

Author(s):

Huaiqian You ◽

XinYang Lu ◽

Nathaniel Task ◽

Yue Yu

Keyword(s):

Boundary Condition ◽

Boundary Conditions ◽

Nonlocal Boundary ◽

Nonlocal Diffusion ◽

Order Convergence ◽

Flux Boundary Condition ◽

Nonlocal Interactions ◽

Local Case ◽

Neumann Type ◽

Type Boundary

In this paper we consider 2D nonlocal diffusion models with a finite nonlocal horizon parameter δ characterizing the range of nonlocal interactions, and consider the treatment of Neumann-like boundary conditions that have proven challenging for discretizations of nonlocal models. We propose a new generalization of classical local Neumann conditions by converting the local flux to a correction term in the nonlocal model, which provides an estimate for the nonlocal interactions of each point with points outside the domain. While existing 2D nonlocal flux boundary conditions have been shown to exhibit at most first order convergence to the local counter part as δ → 0, the proposed Neumann-type boundary formulation recovers the local case as O(δ2) in the L∞ (Ω) norm, which is optimal considering the O(δ2) convergence of the nonlocal equation to its local limit away from the boundary. We analyze the application of this new boundary treatment to the nonlocal diffusion problem, and present conditions under which the solution of the nonlocal boundary value problem converges to the solution of the corresponding local Neumann problem as the horizon is reduced. To demonstrate the applicability of this nonlocal flux boundary condition to more complicated scenarios, we extend the approach to less regular domains, numerically verifying that we preserve second-order convergence for non-convex domains with corners. Based on the new formulation for nonlocal boundary condition, we develop an asymptotically compatible meshfree discretization, obtaining a solution to the nonlocal diffusion equation with mixed boundary conditions that converges with O(δ2) convergence.

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