Capturing complexities with composite operator and differential operators with non-singular kernel

2019 ◽  
Vol 29 (2) ◽  
pp. 023103 ◽  
Author(s):  
Abdon Atangana ◽  
Toufik Mekkaoui
Keyword(s):  
Differential Operators ◽  
Composite Operator ◽  
Singular Kernel

Existence of Solutions for Schrödinger–Kirchhoff Type Problems Involving Nonlocal Elliptic Operators

2016 ◽  
Vol 17 (7-8) ◽  
pp. 369-379
Author(s):  
Mingqi Xiang ◽  
Zhengquan Yang
Keyword(s):  
Differential Operator ◽  
Measurable Function ◽  
Differential Operators ◽  
Existence Of Solutions ◽  
Continuous Functions ◽  
Singular Kernel ◽  
Kirchhoff Type ◽  
Nonnegative Solutions ◽  
Carathéodory Function ◽  
Kirchhoff Type Problems

Abstract:The aim of this paper is to establish the existence of nonnegative solutions for a class of Schrödinger–Kirchhoff type problems driven by nonlocal integro-differential operators, that is, $$\begin{align*}&M\left(\mathop{\iint_{\mathbb{R}^{2N}}}|u(x)-u(y)|^pK(x-y)dxdy,\int_{\mathbb{R}^N}V(x)|u|^pdx\right)\\&\kern10pt \left(\mathcal{L}_Ku+V(x)|u|^{p-2}u\right)\\&=G\left(\mathop{\iint_{\mathbb{R}^{2N}}}|u(x)-u(y)|^pK(x-y)dxdy,\int_{\mathbb{R}^N}V(x)|u|^pdx\right)\nonumber\\&\kern11pt f(x,u)+h(x)\ \ \ \ {\rm in}\ \mathbb{R}^N,\end{align*}$$where $\mathcal{L}_K$ is a nonlocal integro-differential operator with singular kernel $K:\mathbb{R}^N\,\backslash\,\{0\}\rightarrow(0,\infty)$, $M,G$ are two nonnegative continuous functions on $(0,\infty)\times(0,\infty)$, $V\in C(\mathbb{R}^N,\mathbb{R}^+)$, $h:\mathbb{R}^N\rightarrow (0,\infty)$ is a measurable function and $f:\mathbb{R}^N\times\mathbb{R}\rightarrow\mathbb{R}$ is a Carathéodory function. Employing several nonvariational techniques, we prove various results of existence of nonnegative solutions. The main feature of this paper is that the Kirchhoff function $M$ can be zero at zero and the problem is not variational in nature.

The Calder´on-Zygmund Theory I: Ellipticity

2017 ◽  
Author(s):  
Brian Street
Keyword(s):  
Classical Theory ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differential Operators ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Single Parameter ◽  
Partial Differential ◽  
Translation Invariant ◽  
Partial Differential Operators

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

Spectral Methods

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Simple Model ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Linear Algebra ◽  
Spectral Methods ◽  
Differential Operators ◽  
Higher Dimensions ◽  
Standard Linear ◽  
Sommerfeld Equation ◽  
Spectral Discretization

Thenumerical solution of ordinary differential equations (ODEs)with boundary conditions is studied here. Functions are approximated by polynomials in a Chebychev basis. Sections then cover spectral discretization, sampling, interpolation, differentiation, integration, and the basic ODE. Following Trefethen et al., differential operators are approximated as rectangular matrices. Boundary conditions add additional rows that turn them into square matrices. These can then be diagonalized using standard linear algebra methods. After studying various simple model problems, this method is applied to the Orr–Sommerfeld equation, deriving results originally due to Orszag. The difficulties of pushing spectral methods to higher dimensions are outlined.

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