scholarly journals On the Operator ⊕k,m Related to the Wave Equation and Laplacian

2021 ◽  
Vol 14 (3) ◽  
pp. 881-894
Author(s):  
Sudprathai Bupasiri
Keyword(s):  
Wave Equation ◽  
Euclidean Space ◽  
Fundamental Solution ◽  
Real Number ◽  
Nonnegative Integer ◽  
Unknown Function ◽  
Nonnegative Real Number ◽  
Generalized Function

In this article, we study the fundamental solution of the operator $\oplus _{m}^{k}$, iterated $k$-times and is defined by$$\oplus _{m}^{k} = \left[\left(\sum_{r=1}^{p} \frac{\partial^2} {\partial x_r^2}+m^{2}\right)^4 - \left( \sum_{j=p+1}^{p+q} \frac{\partial^2}{\partial x_{j}^2} \right)^4 \right ]^k,$$ where $m$ is a nonnegative real number, $p+q=n$ is the dimension of the Euclidean space $\mathbb{R}^n$,$x=(x_1,x_2,\ldots,x_n)\in\mathbb{R}^n$, $k$ is a nonnegative integer. At first we study the fundamental solution of the operator $\oplus _{m}^{k}$ and after that, we apply such the fundamental solution to solve for the solution of the equation $\oplus _{m}^{k}u(x)= f(x)$, where $f(x)$ is generalized function and $u(x)$ is unknown function for $ x\in \mathbb{R}^{n}$.

scholarly journals On the Elementary Solution for the Partial Differential Operator $\circledcirc_c^{k}$ Related to the Wave Equation

2018 ◽  
Vol 11 (2) ◽  
pp. 390-399 ◽  
Author(s):  
Sudprathai Bupasiri
Keyword(s):  
Wave Equation ◽  
Differential Operator ◽  
Euclidean Space ◽  
Real Number ◽  
Nonnegative Integer ◽  
Positive Real Number ◽  
Partial Differential Operator ◽  
Nonnegative Real Number ◽  
Dimensional Euclidean Space ◽  
Positive Real

In this article, we defined the operator $\diamondsuit _{m,c}^{k}$ which is iterated $k$-times and is defined by$$\diamondsuit _{m,c}^{k}=\left[\left(\frac{1}{c^2}\sum_{i=1}^{p}\frac{\partial ^{2}}{\partial x_{i}^{2}} +\frac{m^{2}}{2}\right)^{2} - \left(\sum_{j=p+1}^{p+q}\frac{\partial ^{2}}{\partial x_{j}^{2}} - \frac{m^{2}}{2}\right)^{2}\right]^{k},$$where $m$ is a nonnegative real number, $c$ is a positive real number and $p+q=n$ is the dimension of the $n$-dimensional Euclidean space $\mathbb{R}^{n}$, $x=(x_{1},\ldots x_{n})\in\mathbb{R}^{n}$ and $k$ is a nonnegative integer. We obtain a causal and anticausal solutionof the operator $\diamondsuit _{m,c}^{k}$, iterated $k$-times.

scholarly journals On the Solutionn-Dimensional of the Product⊗kOperator and Diamond Bessel Operator

2010 ◽  
Vol 2010 ◽  
pp. 1-20
Author(s):  
Wanchak Satsanit ◽  
Amnuay Kananthai
Keyword(s):  
Wave Equation ◽  
Positive Integer ◽  
Nonlinear Equation ◽  
Unknown Function ◽  
Bessel Operator ◽  
Generalized Function ◽  
Inhomogeneous Wave ◽  
Inhomogeneous Wave Equation ◽  
Unknown Unknown

Firstly, we studied the solution of the equation⊗k◊Bku(x)=f(x)whereu(x)is an unknown unknown function forx=(x1,x2,…,xn)∈ℝn,f(x)is the generalized function,kis a positive integer. Finally, we have studied the solution of the nonlinear equation⊗k◊Bku(x)=f(x,□k−1LkΔBk□Bku(x)). It was found that the existence of the solutionu(x)of such an equation depends on the condition offand□k−1LkΔBk□Bku(x). Moreover such solutionu(x)is related to the inhomogeneous wave equation depending on the conditions ofp,q, andk.

scholarly journals On the Operator ⨁BkRelated to Bessel Heat Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-12
Author(s):  
Wanchak Satsanit
Keyword(s):  
Heat Equation ◽  
Heat Kernel ◽  
Positive Constant ◽  
Nonnegative Integer ◽  
Unknown Function ◽  
Generalized Function ◽  
Initial Condition

We study the equation(∂/∂t)u(x,t)=c2⊕Bku(x,t)with the initial conditionu(x,0)=f(x)forx∈Rn+.The operator⊕Bkis the operator iterated k-times and is defined by⊕Bk=((∑i=1pBxi)4-(∑j=p+1p+qBxi)4)k, wherep+q=nis the dimension of theRn+,Bxi=∂2/∂xi2+(2vi/xi)(∂/∂xi),2vi=2αi+1,αi>-1/2,i=1,2,3,…,n, andkis a nonnegative integer,u(x,t)is an unknown function for(x,t)=(x1,x2,…,xn,t)∈Rn+×(0,∞),f(x)is a given generalized function, andcis a positive constant. We obtain the solution of such equation, which is related to the spectrum and the kernel, which is so called Bessel heat kernel. Moreover, such Bessel heat kernel has interesting properties and also related to the kernel of an extension of the heat equation.

A review: The Laplacian and the d’Alembertian

2017 ◽  
Author(s):  
Christopher D. Sogge
Keyword(s):  
Wave Equation ◽  
Euclidean Space ◽  
Fundamental Solution ◽  
Minkowski Space ◽  
Riemannian Manifolds ◽  
Wave Operator ◽  
Wave Equations ◽  
Fundamental Solutions ◽  
Specific Function ◽  
Solutions Of Wave Equations

This chapter reviews the Laplacian and the d'Alembertian. It begins with a brief discussion on the solution of wave equation both in Euclidean space and on manifolds and how this knowledge can be used to derive properties of eigenfunctions on Riemannian manifolds. A key step in understanding properties of solutions of wave equations on manifolds is to compute the types of distributions that include the fundamental solution of the wave operator in Minkowski space (d'Alembertian), with a specific function for the Euclidean Laplacian on Rn. The chapter also reviews another equation involving the Laplacian, before discussing the fundamental solutions of the d'Alembertian in R1+n.

A Modulus for the 3-Dimensional Wave Equation With Noise: Dealing With a Singular Kernel

1993 ◽  
Vol 45 (6) ◽  
pp. 1263-1275
Author(s):  
C. Mueller
Keyword(s):  
Wave Equation ◽  
Fundamental Solution ◽  
Modulus Of Continuity ◽  
Gaussian Noise ◽  
Singular Kernel ◽  
3 Dimensional ◽  
Noise Term ◽  
Dimensional Wave

AbstractWe give a modulus of continuity for solutions of the wave equation with a noise term:utt = Δu + a(u) + b(u)G, x ∈ ℝ3where G is a Gaussian noise. This case is more difficult than in lower dimensions because the fundamental solution of the wave equation is singular.

scholarly journals On the convolution product of the distributional kernelKα,β,γ,ν

2003 ◽  
Vol 2003 (3) ◽  
pp. 153-158
Author(s):  
A. Kananthai ◽  
S. Suantai
Keyword(s):  
Euclidean Space ◽  
Nonnegative Integer ◽  
Dimensional Euclidean Space ◽  
Convolution Product ◽  
Gamma Prime ◽  
Alpha Beta

We introduce a distributional kernelKα,β,γ,νwhich is related to the operator⊕ kiteratedktimes and defined by⊕ k=[(∑r=1p∂2/∂xr2)4−(∑j=p+1p+q∂2/∂xj2)4] k, wherep+q=nis the dimension of the spaceℝ nof then-dimensional Euclidean space,x=(x1,x2,…,xn)∈ℝ n,kis a nonnegative integer, andα,β,γ, andνare complex parameters. It is found that the existence of the convolutionKα,β,γ,ν∗Kα′,β′,γ′,ν′is depending on the conditions ofpandq.

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