scholarly journals Explicit stencil computation schemes generated by Poisson's formula for the 2D wave equation

2020 ◽  
Vol 2 (5) ◽  
Author(s):  
Naum M. Khutoryansky
Keyword(s):  
Wave Equation ◽  
Stencil Computation

scholarly journals Explicit Stencil Computation Schemes Generated by Poisson's Formula for the 2D Wave Equation

2019 ◽  
Author(s):  
Naum Khutoryansky
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Integral Representation ◽  
Initial Conditions ◽  
Representation Formula ◽  
Time Step ◽  
Stencil Computation ◽  
Integral Representation Formula ◽  
Time Marching ◽  
First Time

An approach to building explicit time-marching stencil computation schemes for the transient 2D acoustic wave equation without using finite-difference approximations is proposed and implemented. It is based on using the integral representation formula (Poisson's formula) that provides the exact solution of the initial-value problem for the transient 2D scalar wave equation at any time point through the initial conditions. For the purpose of constructing a two-step time-marching algorithm, a modified integral representation formula involving three time levels is also employed. It is shown that integrals in the two representation formulas are exactly calculated if the initial conditions and the sought solution at each time level as functions of spatial coordinates are approximated by stencil interpolation polynomials in the neighborhood of any point in a 2D Cartesian grid. As a result, if a uniform time grid is chosen, the proposed time-marching algorithm consists of two numerical procedures: 1) the solution calculation at the first time-step through the initial conditions; 2) the solution calculation at the second and next time-steps using a generated two-step numerical scheme. Three particular explicit stencil schemes (with five, nine and 13 space points) are built using the proposed approach. Their stability regions are presented. The obtained stencil expressions are compared with the corresponding finite-difference schemes available in the literature. Their novelty features are discussed. Simulation results with new and conventional schemes are presented for two benchmark problems that have exact solutions. It is demonstrated that using the new first time-step calculation procedure instead of the conventional one can provide a significant improvement of accuracy even for later time steps.

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