scholarly journals Splitting the one-Dimensional Wave Equation, Part II: Additional Data are Given by an End Displacement Measurement

2021 ◽  
pp. 233-239
Author(s):  
Shilan Othman Hussein ◽  
Mohammed Sabah Hussein
Keyword(s):  
Wave Equation ◽  
Additional Data ◽  
Numerical Solutions ◽  
Separation Of Variables ◽  
Displacement Measurement ◽  
Tikhonov Regularization Method ◽  
Ill Posed ◽  
The One ◽  
Inverse Force ◽  
Force Problem

     In this research, an unknown space-dependent force function in the wave equation is studied. This is a natural continuation of [1] and chapter 2 of [2] and [3], where the finite difference method (FDM)/boundary element method (BEM), with the separation of variables method, were considered. Additional data are given by the one end displacement measurement. Moreover, it is a continuation of [3], with exchanging the boundary condition, where  are extra data, by the initial condition. This is an ill-posed inverse force problem for linear hyperbolic equation. Therefore, in order to stabilize the solution, a zeroth-order Tikhonov regularization method is provided. To assess the accuracy, the minimum error between exact and numerical solutions for the force is computed for various regularization parameters. Numerical results are presented and a good agreement was obtained for the exact and noisy data.

A hybrid scheme for absorbing edge reflections in numerical modeling of wave propagation

Geophysics ◽  
2010 ◽  
Vol 75 (2) ◽  
pp. A1-A6 ◽  
Author(s):  
Yang Liu ◽  
Mrinal K. Sen
Keyword(s):  
Numerical Modeling ◽  
Wave Equation ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Computational Domain ◽  
Perfect Absorption ◽  
Transition Area ◽  
Efficient Scheme ◽  
Grid Points ◽  
The One

We propose an efficient scheme to absorb reflections from the model boundaries in numerical solutions of wave equations. This scheme divides the computational domain into boundary, transition, and inner areas. The wavefields within the inner and boundary areas are computed by the wave equation and the one-way wave equation, respectively. The wavefields within the transition area are determined by a weighted combination of the wavefields computed by the wave equation and the one-way wave equation to obtain a smooth variation from the inner area to the boundary via the transition zone. The results from our finite-difference numerical modeling tests of the 2D acoustic wave equation show that the absorption enforced by this scheme gradually increases with increasing width of the transition area. We obtain equally good performance using pseudospectral and finite-element modeling with the same scheme. Our numerical experiments demonstrate that use of 10 grid points for absorbing edge reflections attains nearly perfect absorption.

scholarly journals Determination of a space-dependent force function in the one-dimensional wave equation

2014 ◽  
Vol 12 (1) ◽  
Author(s):  
S. O. Hussein ◽  
D. Lesnic
Keyword(s):  
Input Data ◽  
Regularization Parameter ◽  
Stable Solution ◽  
Force Function ◽  
Tikhonov Regularization Method ◽  
Vibrating String ◽  
Ill Posed ◽  
Curve Method ◽  
The One

<p class="p1">The determination of an unknown spacewice dependent force function acting on a vibrating string from over-specied Cauchy boundary data is investigated numerically using the boundary element method (BEM) combined with a regularized method of separating variables. This linear inverse problem is ill-posed since small errors in the input data cause large errors in the output force solution. Consequently, when the input data is contaminated with noise we use the Tikhonov regularization method in order to obtain a stable solution. The choice of the regularization parameter is based on the L-curve method. Numerical results show that the solution is accurate for exact data and stable for noisy data.</p>

Wave Mechanics & Elements of Mathematical Physics

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Wave Equation ◽  
Helmholtz Equation ◽  
Separation Of Variables ◽  
Three Dimensional ◽  
Mathematical Physics ◽  
Wave Mechanics ◽  
Molecular Physics ◽  
Classical Wave ◽  
The One ◽  
Dimensional Wave

This chapter presents an in-depth look at classical wave mechanics and mathematical physics, containing key examples directly relevant to molecular physics. The separation of variables is used to construct the Helmholtz equation from the one-dimensional wave equation before considering the three-dimensional wave equation. From this, equations for the temporal, radial, azimuth and angular components are developed and solutions using the Bessel equations and Legendre polynomials are found. Boundary conditions are explained and the Rayleigh plane wave expansion as the general solution to the Helmholtz equation is reconstructed. Both the Hermite equation and the Legendre equation are derived using the series solution method, and the Laplace equation is discussed.

scholarly journals The One-dimensional Coulomb-force Problem in Classical Radiation Reaction Theories

1977 ◽  
Vol 32 (7) ◽  
pp. 685-691
Author(s):  
W. Heudorfer ◽  
M. Sorg
Keyword(s):  
Coulomb Potential ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Radiation Reaction ◽  
Coulomb Force ◽  
Stationary States ◽  
One Dimensional ◽  
The One ◽  
Force Problem ◽  
Classical Radiation

Abstract Numerical solutions of the recently proposed equations of motion for the classically radiating electron are obtained for the case where the particle moves in a one-dimensional Coulomb potential (both attractive and repulsive). The solutions are discussed and found to be meaningful also in that case, where the well-known Lorentz-Dirac equation fails (attractive Coulomb force). Discrete, stationary states are found in a non-singular version of the Coulomb potential. During the transition between those stationary states the particle looses energy by emission of radiation, which results in a smaller amplitude of the stationary oscillations.

scholarly journals Splitting the One-Dimensional Wave Equation. Part I: Solving by Finite-Difference Method and Separation Variables

2020 ◽  
Vol 17 (2(SI)) ◽  
pp. 0675
Author(s):  
Shilan Othman Hussein
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Boundary Data ◽  
Zeroth Order ◽  
Output Force ◽  
Ill Posed ◽  
The Finite Difference Method ◽  
The One

In this study, an unknown force function dependent on the space in the wave equation is investigated. Numerically wave equation splitting in two parts, part one using the finite-difference method (FDM). Part two using separating variables method. This is the continuation and changing technique for solving inverse problem part in (1,2). Instead, the boundary element method (BEM) in (1,2), the finite-difference method (FDM) has applied. Boundary data are in the role of overdetermination data. The second part of the problem is inverse and ill-posed, since small errors in the extra boundary data cause errors in the force solution. Zeroth order of Tikhonov regularization, and several parameters of regularization are employed to decrease errors for output force solution. It is obvious from figures how error affects the results and zeroth order stables the solution.

scholarly journals Modelling of Flood Wave Propagation with Wet-dry Front by One-dimensional Diffusive Wave Equation

2014 ◽  
Vol 61 (3-4) ◽  
pp. 111-125 ◽  
Author(s):  
Dariusz Gąsiorowski
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Numerical Solutions ◽  
Numerical Algorithms ◽  
Finite Volume Methods ◽  
Flood Wave ◽  
One Dimensional ◽  
Benchmark Solution ◽  
The One ◽  
Water Depths

Abstract A full dynamic model in the form of the shallow water equations (SWE) is often useful for reproducing the unsteady flow in open channels, as well as over a floodplain. However, most of the numerical algorithms applied to the solution of the SWE fail when flood wave propagation over an initially dry area is simulated. The main problems are related to the very small or negative values of water depths occurring in the vicinity of a moving wet-dry front, which lead to instability in numerical solutions. To overcome these difficulties, a simplified model in the form of a non-linear diffusive wave equation (DWE) can be used. The diffusive wave approach requires numerical algorithms that are much simpler, and consequently, the computational process is more effective than in the case of the SWE. In this paper, the numerical solution of the one-dimensional DWE based on the modified finite element method is verified in terms of accuracy. The resulting solutions of the DWE are compared with the corresponding benchmark solution of the one-dimensional SWE obtained by means of the finite volume methods. The results of numerical experiments show that the algorithm applied is capable of reproducing the reference solution with satisfactory accuracy even for a rapidly varied wave over a dry bottom.

scholarly journals Analytical and Numerical Modelling of One-Dimensional Consolidation of Stabilized Peat

2019 ◽  
Vol 5 (2) ◽  
pp. 398 ◽  
Author(s):  
Leong Sing Wong ◽  
Shamini Somanathan
Keyword(s):  
Finite Element Method ◽  
Organic Matter ◽  
Numerical Solutions ◽  
Separation Of Variables ◽  
Treatment Method ◽  
The Finite Element Method ◽  
Close Approximation ◽  
One Dimensional ◽  
Plant Decomposition ◽  
The One

The objective of the paper is to compare and evaluate analytical and numerical solutions of one-dimensional consolidation of stabilized peat. The type of analytical method used to solve the problem is exact method by separation of variables and utilization of Fourier series. Plaxis 2D 8.2 Professional version software was used to find numerical solution to the problem by employing the finite element method. One-dimensional consolidation problem of stabilized peat was solved numerically and validated with the one solved analytically based on laboratory experimental results. From the results, it was discovered that the consolidation characteristics of stabilized peat evaluated numerically were found to have close approximation to those evaluated analytically. There is a novel value in developing an accurate numerical prediction for the vertical consolidation of stabilized peat considering the complexity of the soil treatment method. It must be noted that peat is highly problematic because it is produced from plant decomposition with extremely high organic matter.

On Numerical Solutions to the One-Dimensional Wave Equation

1954 ◽  
Vol 61 (2) ◽  
pp. 115 ◽  
Author(s):  
W. P. Reid
Keyword(s):  
Wave Equation ◽  
Numerical Solutions ◽  
One Dimensional ◽  
The One ◽  
Dimensional Wave

Preface to the special issue, CMAM 2011, no. 3.

2011 ◽  
Vol 11 (3) ◽  
pp. 272
Author(s):  
Ivan Gavrilyuk ◽  
Boris Khoromskij ◽  
Eugene Tyrtyshnikov
Keyword(s):  
Separation Of Variables ◽  
Numerical Algorithms ◽  
Multilevel Methods ◽  
High Dimensional ◽  
Special Issue ◽  
High Dimensions ◽  
Guiding Principle ◽  
Tensor Methods ◽  
Closed World ◽  
The One

Abstract In the recent years, multidimensional numerical simulations with tensor-structured data formats have been recognized as the basic concept for breaking the "curse of dimensionality". Modern applications of tensor methods include the challenging high-dimensional problems of material sciences, bio-science, stochastic modeling, signal processing, machine learning, and data mining, financial mathematics, etc. The guiding principle of the tensor methods is an approximation of multivariate functions and operators with some separation of variables to keep the computational process in a low parametric tensor-structured manifold. Tensors structures had been wildly used as models of data and discussed in the contexts of differential geometry, mechanics, algebraic geometry, data analysis etc. before tensor methods recently have penetrated into numerical computations. On the one hand, the existing tensor representation formats remained to be of a limited use in many high-dimensional problems because of lack of sufficiently reliable and fast software. On the other hand, for moderate dimensional problems (e.g. in "ab-initio" quantum chemistry) as well as for selected model problems of very high dimensions, the application of traditional canonical and Tucker formats in combination with the ideas of multilevel methods has led to the new efficient algorithms. The recent progress in tensor numerical methods is achieved with new representation formats now known as "tensor-train representations" and "hierarchical Tucker representations". Note that the formats themselves could have been picked up earlier in the literature on the modeling of quantum systems. Until 2009 they lived in a closed world of those quantum theory publications and never trespassed the territory of numerical analysis. The tremendous progress during the very recent years shows the new tensor tools in various applications and in the development of these tools and study of their approximation and algebraic properties. This special issue treats tensors as a base for efficient numerical algorithms in various modern applications and with special emphases on the new representation formats.

scholarly journals A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation

2020 ◽  
Vol 18 (1) ◽  
pp. 1685-1697
Author(s):  
Zhenyu Zhao ◽  
Lei You ◽  
Zehong Meng
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Solution Process ◽  
Tikhonov Regularization Method ◽  
Hermite Expansion ◽  
The Cauchy Problem ◽  
Ill Posed

Abstract In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

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