The wave equation for a hypoelliptic operator with symplectic characteristics of codimension two

1984 ◽  
Vol 44 (1) ◽  
pp. 134-182 ◽  
Author(s):  
Richard B. Melrose
Keyword(s):  
Wave Equation ◽  
Hypoelliptic Operator ◽  
Codimension Two

Rough estimates on the scalar heat kernel

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Wave Equation ◽  
Heat Equation ◽  
Heat Kernel ◽  
Heat Kernels ◽  
Hypoelliptic Operator ◽  
Uniform Bounds ◽  
Standard Heat ◽  
Probabilistic Construction

This chapter establishes rough estimates on the heat kernel rb,tX for the scalar hypoelliptic operator AbX on X defined in the preceding chapter. By rough estimates, this chapter refers to just the uniform bounds on the heat kernel. The chapter also obtains corresponding bounds for the heat kernels associated with operators AbX and another AbX over ̂X. Moreover, it gives a probabilistic construction of the heat kernels. This chapter also explains the relation of the heat equation for the hypoelliptic Laplacian on X to the wave equation on X and proves that as b → 0, the heat kernel rb,tX converges to the standard heat kernel of X.

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

DeWitt geometry and the wave equation in hyper-volume

2020 ◽  
Author(s):  
Vitaly Kuyukov
Keyword(s):  
Wave Equation

DeWitt geometry and the wave equation in hyper-volume

Classical and generalized of a mixed problem– solutions for a non-homogeneous wave equation

2019 ◽  
Vol 484 (1) ◽  
pp. 18-20
Author(s):  
A. P. Khromov ◽  
V. V. Kornev
Keyword(s):  
Wave Equation ◽  
Fourier Series ◽  
Mixed Problem ◽  
Homogeneous Equation ◽  
Generalized Solution ◽  
Problem Solution ◽  
Homogeneous Wave ◽  
Homogeneous Wave Equation ◽  
Explicit Expressions

This study follows A.N. Krylov’s recommendations on accelerating the convergence of the Fourier series, to obtain explicit expressions of the classical mixed problem–solution for a non-homogeneous equation and explicit expressions of the generalized solution in the case of arbitrary summable functions q(x), ϕ(x), y(x), f(x, t).

Sign in / Sign up

Export Citation Format

Share Document