Linear Waves in Bubbly Liquids via the Kramers-Kronig Relations

1991 ◽  
Vol 113 (3) ◽  
pp. 417-419
Author(s):  
N. Brauner ◽  
A. I. Beltzer
Keyword(s):  
Linear System ◽  
Closed Form ◽  
Acoustic Waves ◽  
Analytic Solution ◽  
Numerical Solutions ◽  
Bubbly Liquids ◽  
Linear Waves ◽  
Causal Approach ◽  
Effective Wave ◽  
Arbitrary Frequency

A simple closed form analytic solution, as well as numerical solutions, are derived for acoustic waves of an arbitrary frequency, propagating in dilute bubbly liquids. The applied method treats the effective wave as a causal response of a linear system. The results are compared with the prediction of Foldy’s theory, which in this particular case is shown to be consistent with the Kramers-Kronig relations. These results thus complement recently reported investigations concerning the causal approach as well as the validity of Foldy’s theory.

scholarly journals $L^{p}-L^{q}$-Maximal regularity of the Van Wijngaarden–Eringen equation in a cylindrical domain

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Carlos Lizama ◽  
Marina Murillo-Arcila
Keyword(s):  
Acoustic Waves ◽  
Maximal Regularity ◽  
Bubble Radius ◽  
Fourier Multipliers ◽  
Cylindrical Domain ◽  
Lebesgue Spaces ◽  
Regularity Problem ◽  
Bubbly Liquids

Abstract We consider the maximal regularity problem for a PDE of linear acoustics, named the Van Wijngaarden–Eringen equation, that models the propagation of linear acoustic waves in isothermal bubbly liquids, wherein the bubbles are of uniform radius. If the dimensionless bubble radius is greater than one, we prove that the inhomogeneous version of the Van Wijngaarden–Eringen equation, in a cylindrical domain, admits maximal regularity in Lebesgue spaces. Our methods are based on the theory of operator-valued Fourier multipliers.

An Improved Assembling Algorithm in Boundary Elements With Galerkin Weighting Applied to Three-Dimensional Stokes Flows

2017 ◽  
Vol 140 (1) ◽  
Author(s):  
Sofia Sarraf ◽  
Ezequiel López ◽  
Laura Battaglia ◽  
Gustavo Ríos Rodríguez ◽  
Jorge D'Elía
Keyword(s):  
Boundary Element Method ◽  
Linear System ◽  
Boundary Element ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Computational Time ◽  
Creeping Flows ◽  
Order Of Magnitude ◽  
Element Method

In the boundary element method (BEM), the Galerkin weighting technique allows to obtain numerical solutions of a boundary integral equation (BIE), giving the Galerkin boundary element method (GBEM). In three-dimensional (3D) spatial domains, the nested double surface integration of GBEM leads to a significantly larger computational time for assembling the linear system than with the standard collocation method. In practice, the computational time is roughly an order of magnitude larger, thus limiting the use of GBEM in 3D engineering problems. The standard approach for reducing the computational time of the linear system assembling is to skip integrations whenever possible. In this work, a modified assembling algorithm for the element matrices in GBEM is proposed for solving integral kernels that depend on the exterior unit normal. This algorithm is based on kernels symmetries at the element level and not on the flow nor in the mesh. It is applied to a BIE that models external creeping flows around 3D closed bodies using second-order kernels, and it is implemented using OpenMP. For these BIEs, the modified algorithm is on average 32% faster than the original one.

Parametric Deflection Approximations for End-Loaded, Large-Deflection Beams in Compliant Mechanisms

1995 ◽  
Vol 117 (1) ◽  
pp. 156-165 ◽  
Author(s):  
L. L. Howell ◽  
A. Midha
Keyword(s):  
Closed Form ◽  
Large Deflection ◽  
Numerical Solutions ◽  
Compliant Mechanisms ◽  
Elliptic Integral ◽  
Cantilever Beams ◽  
Simple Method ◽  
Body Angle ◽  
Analysis And Design ◽  
Integral Solutions

Geometric nonlinearities often complicate the analysis of systems containing large-deflection members. The time and resources required to develop closed-form or numerical solutions have inspired the development of a simple method of approximating the deflection path of end-loaded, large-deflection cantilever beams. The path coordinates are parameterized in a single parameter called the pseudo-rigid-body angle. The approximations are accurate to within 0.5 percent of the closed-form elliptic integral solutions. A physical model is associated with the method, and may be used to simplify complex problems. The method proves to be particularly useful in the analysis and design of compliant mechanisms.

scholarly journals New Explicit Solutions to the Fractional-Order Burgers’ Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
M. Hafiz Uddin ◽  
Mohammad Asif Arefin ◽  
M. Ali Akbar ◽  
Mustafa Inc
Keyword(s):  
Differential Equations ◽  
Rational Function ◽  
Acoustic Waves ◽  
Burgers Equation ◽  
Three Dimensional ◽  
Wall Friction ◽  
Weakly Nonlinear ◽  
Bubbly Liquids ◽  
Fractional Differential ◽  
Accuracy Of Results

The closed-form wave solutions to the time-fractional Burgers’ equation have been investigated by the use of the two variables G ′ / G , 1 / G -expansion, the extended tanh function, and the exp-function methods translating the nonlinear fractional differential equations (NLFDEs) into ordinary differential equations. In this article, we ascertain the solutions in terms of tanh , sech , sinh , rational function, hyperbolic rational function, exponential function, and their integration with parameters. Advanced and standard solutions can be found by setting definite values of the parameters in the general solutions. Mathematical analysis of the solutions confirms the existence of different soliton forms, namely, kink, single soliton, periodic soliton, singular kink soliton, and some other types of solitons which are shown in three-dimensional plots. The attained solutions may be functional to examine unidirectional propagation of weakly nonlinear acoustic waves, the memory effect of the wall friction through the boundary layer, bubbly liquids, etc. The methods suggested are direct, compatible, and speedy to simulate using algebraic computation schemes, such as Maple, and can be used to verify the accuracy of results.

scholarly journals On the evolution of solutions of Burgers equation on the positive quarter-plane

2019 ◽  
Vol 52 (1) ◽  
pp. 237-248
Author(s):  
Esen Hanaç
Keyword(s):  
Boundary Condition ◽  
Analytic Solution ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Condition ◽  
Quarter Plane ◽  
Initial Boundary Value ◽  
Good Agreement

AbstractIn this paper we investigate an initial-boundary value problem for the Burgers equation on the positive quarter-plane; $\matrix{ {{v_t} + v{v_x} - {v_{xx}} = 0,\,\,\,x > 0,\,\,\,t > 0,} \cr {v\left( {x,0} \right) = {u_ + },\,\,\,x > 0,} \cr {v\left( {0,t} \right) = {u_b},\,\,t > 0,} \cr }$ where x and t represent distance and time, respectively, and u+ is an initial condition, ub is a boundary condition which are constants (u+ ≠ ub). Analytic solution of above problem is solved depending on parameters (u+ and ub) then compared with numerical solutions to show there is a good agreement with each solutions.

An investment model with entry and exit decisions

2000 ◽  
Vol 37 (2) ◽  
pp. 547-559 ◽  
Author(s):  
J. Kate Duckworth ◽  
Mihail Zervos
Keyword(s):  
Dynamic Programming ◽  
Closed Form ◽  
Production Scheduling ◽  
Analytic Solution ◽  
Investment Model ◽  
Programming Approach ◽  
Entry And Exit ◽  
Dynamic Programming Approach ◽  
Exit Decisions ◽  
Parameter Values

We consider an investment model which generalizes a number of models that have been studied in the literature. The model involves entry and exit decisions as well as decisions relating to production scheduling. We then address the problem of its valuation from the standpoint of the dynamic programming approach. Our analysis results in a closed form analytic solution that can take qualitatively different forms depending on parameter values.

scholarly journals Quantum Calcium-Ion Interactions with EEG

Sci ◽  
2018 ◽  
Vol 1 (1) ◽  
pp. 7 ◽  
Author(s):  
Lester Ingber
Keyword(s):  
Closed Form ◽  
Path Integral ◽  
Stochastic Systems ◽  
Analytic Solution ◽  
Short Term Memory ◽  
Mathematical Physics ◽  
Modest Improvement ◽  
Classical Physics ◽  
Arbitrary Time ◽  
Eeg Data

Background: Previous papers have developed a statistical mechanics of neocortical interactions (SMNI) fit to short-term memory and EEG data. Adaptive Simulated Annealing (ASA) has been developed to perform fits to such nonlinear stochastic systems. An N-dimensional path-integral algorithm for quantum systems, qPATHINT, has been developed from classical PATHINT. Both fold short-time propagators (distributions or wave functions) over long times. Previous papers applied qPATHINT to two systems, in neocortical interactions and financial options. Objective: In this paper the quantum path-integral for Calcium ions is used to derive a closed-form analytic solution at arbitrary time that is used to calculate interactions with classical-physics SMNI interactions among scales. Using fits of this SMNI model to EEG data, including these effects, will help determine if this is a reasonable approach. Method: Methods of mathematical-physics for optimization and for path integrals in classical and quantum spaces are used for this project. Studies using supercomputer resources tested various dimensions for their scaling limits. In this paper the quantum path-integral is used to derive a closed-form analytic solution at arbitrary time that is used to calculate interactions with classical-physics SMNI interactions among scales. Results: The mathematical-physics and computer parts of the study are successful, in that there is modest improvement of cost/objective functions used to fit EEG data using these models. Conclusions: This project points to directions for more detailed calculations using more EEG data and qPATHINT at each time slice to propagate quantum calcium waves, synchronized with PATHINT propagation of classical SMNI.

Some Properties of R. G. D. Allen's Treatment of Kalecki's 1935 Model of Business Cycles

2002 ◽  
Vol 24 (4) ◽  
pp. 463-474
Author(s):  
John V. Baxley ◽  
John C. Moorhouse
Keyword(s):  
Business Cycles ◽  
Business Cycle ◽  
Mathematical Models ◽  
Closed Form ◽  
Analytic Solution ◽  
John Maynard Keynes

More than sixty-five years have passed since Michal Kalecki (1935) published one of the first formal mathematical models of business cycles. His paper presents a closed-form analytic solution. This characteristic, among others, sets Kalecki's work apart from that of contemporary literary business cycle theorists such as Friedrich A. Hayek (1935) and John Maynard Keynes (1936).

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