scholarly journals Ultrasonic Waves in Bubbly Liquids: An Analytic Approach

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1309
Author(s):  
P. R. Gordoa ◽  
A. Pickering
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Acoustic Waves ◽  
Elliptic Functions ◽  
Coupled System ◽  
Ultrasonic Waves ◽  
Partial Differential ◽  
Special Cases ◽  
Spherical Bubbles

We consider the problem of the propagation of high-intensity acoustic waves in a bubble layer consisting of spherical bubbles of identical size with a uniform distribution. The mathematical model is a coupled system of partial differential equations for the acoustic pressure and the instantaneous radius of the bubbles consisting of the wave equation coupled with the Rayleigh–Plesset equation. We perform an analytic analysis based on the study of Lie symmetries for this system of equations, concentrating our attention on the traveling wave case. We then consider mappings of the resulting reductions onto equations defining elliptic functions, and special cases thereof, for example, solvable in terms of hyperbolic functions. In this way, we construct exact solutions of the system of partial differential equations under consideration. We believe this to be the first analytic study of this particular mathematical model.

scholarly journals Computational Study of the Coupled Mechanism of Thermophoretic Transportation and Mixed Convection Flow around the Surface of a Sphere

Molecules ◽  
2020 ◽  
Vol 25 (11) ◽  
pp. 2694
Author(s):  
Amir Abbas ◽  
Muhammad Ashraf ◽  
Yu-Ming Chu ◽  
Saqib Zia ◽  
Ilyas Khan ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Mixed Convection ◽  
Computational Study ◽  
Convection Flow ◽  
Mixed Convection Flow ◽  
Partial Differential ◽  
Primitive Form ◽  
The Mathematical Model

The main goal of the current work was to study the coupled mechanism of thermophoretic transportation and mixed convection flow around the surface of the sphere. To analyze the characteristics of heat and fluid flow in the presence of thermophoretic transportation, a mathematical model in terms of non-linear coupled partial differential equations obeying the laws of conservation was formulated. Moreover, the mathematical model of the proposed phenomena was approximated by implementing the finite difference scheme and boundary value problem of fourth order code BVP4C built-in scheme. The novelty point of this paper is that the primitive variable formulation is introduced to transform the system of partial differential equations into a primitive form to make the line of the algorithm smooth. Secondly, the term thermophoretic transportation in the mass equation is introduced in the mass equation and thus the effect of thermophoretic transportation can be calculated at different positions of the sphere. Basically, in this study, some favorite positions around the sphere were located, where the velocity field, temperature distribution, mass concentration, skin friction, and rate of heat transfer can be calculated simultaneously without any separation in flow around the surface of the sphere.

CONTINUOUS NEURAL NETWORKS APPLIED TO IDENTIFY A CLASS OF UNCERTAIN PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS

2010 ◽  
Vol 01 (04) ◽  
pp. 485-508
Author(s):  
R. FUENTES ◽  
A. POZNYAK ◽  
I. CHAIREZ ◽  
M. FRANCO ◽  
T. POZNYAK
Keyword(s):  
Neural Networks ◽  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Parametric Modeling ◽  
Parabolic Partial Differential Equations ◽  
Qualitative Behavior ◽  
Compact Set ◽  
Large Set ◽  
Partial Differential

There are a lot of examples in science and engineering that may be described using a set of partial differential equations (PDEs). Those PDEs are obtained applying a process of mathematical modeling using complex physical, chemical, etc. laws. Nevertheless, there are many sources of uncertainties around the aforementioned mathematical representation. It is well known that neural networks can approximate a large set of continuous functions defined on a compact set. If the continuous mathematical model is incomplete or partially known, the methodology based on Differential Neural Network (DNN) provides an effective tool to solve problems on control theory such as identification, state estimation, trajectories tracking, etc. In this paper, a strategy based on DNN for the no parametric identification of a mathematical model described by parabolic partial differential equations is proposed. The identification solution allows finding an exact expression for the weights' dynamics. The weights adaptive laws ensure the "practical stability" of DNN trajectories. To verify the qualitative behavior of the suggested methodology, a no parametric modeling problem for a couple of distributed parameter plants is analyzed: the plug-flow reactor model and the anaerobic digestion system. The results obtained in the numerical simulations confirm the identification capability of the suggested methodology.

Fluid and Structure Interaction in Cochlea’s Similar Geometry

2010 ◽  
Author(s):  
Mahmoud Hamadiche
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Acoustic Wave ◽  
Transverse Wave ◽  
Basilar Membrane ◽  
Striking Similarity ◽  
Partial Differential ◽  
Material Parameters ◽  
Non Linear

A non linear mathematical model addressing the passive mechanism of the cochlea is proposed in this work. In this respect, the interaction between the basilar membrane seen as an elastic solid and fluids in both scala vestibuli and tympani is developed. Via the fluid/solid interface, a full fluid/solid interaction is taking into account. Furthermore a significant improvement of the existing models has been made in both fluid flow modelling and solid modelling. In the present paper, the flow is three dimensional and the solid is non homogeneous two dimensional membrane where the material parameters depend only on the axial distance. The problem formulation leads to a system of non linear partial differential equations. Solution of the linearized system of partial differential equations of the proposed approach is presented. The numerical results obvious a lower and upper limits of the cochlea resonance frequency versus the material parameters of the basilar membrane. It is shown that a monochromatic acoustic wave energises only a portion of the basilar membrane and the location of the excited portion depends on the frequency of the incident acoustic wave. Those results explain the ability of the cochlea in deciphering the frequency of sound with high resolution in striking similarity with the known experimental results. The mathematical model shows that the excited strip of the basilar membrane by a monochromatic acoustic wave is very small when a transverse wave exists in the basilar membrane. Thus, a transverse wave improves highly the resolution of the cochlea in deciphering the high frequency of the incident acoustic wave.

scholarly journals On the Existence of Solutions of a Two-Layer Green Roof Mathematical Model

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1608
Author(s):  
J. Ignacio Tello ◽  
Lourdes Tello ◽  
María Luisa Vilar
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Existence Of Solutions ◽  
Green Roof ◽  
Maximal Monotone ◽  
Equation Modeling ◽  
Partial Differential ◽  
Maximal Monotone Graph ◽  
Monotone Graph

The aim of this article is to fill part of the existing gap between the mathematical modeling of a green roof and its computational treatment, focusing on the mathematical analysis. We first introduce a two-dimensional mathematical model of the thermal behavior of an extensive green roof based on previous models and secondly we analyze such a system of partial differential equations. The model is based on an energy balance for buildings with vegetation cover and it is presented for general shapes of roofs. The model considers a vegetable layer and the substratum and the energy exchange between them. The unknowns of the problem are the temperature of each layer described by a coupled system of two partial differential equations of parabolic type. The equation modeling the evolution of the temperature of the substratum also considers the change of phase of water described by a maximal monotone graph. The main result of the article is the proof of the existence of solutions of the system which is given in detail by using a regularization of the maximal monotone graph. Appropriate estimates are obtained to pass to the limit in a weak formulation of the problem. The result goes one step further from modeling to validate future numerical results.

scholarly journals On the solution of reaction-diffusion equations with double diffusivity

1987 ◽  
Vol 10 (1) ◽  
pp. 163-172
Author(s):  
B. D. Aggarwala ◽  
C. Nasim
Keyword(s):  
Heat Conduction ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Partial Differential ◽  
Coupled Partial Differential Equations ◽  
Special Cases ◽  
Two Temperatures

In this paper, solution of a pair of Coupled Partial Differential equations is derived. These equations arise in the solution of problems of flow of homogeneous liquids in fissured rocks and heat conduction involving two temperatures. These equations have been considered by Hill and Aifantis, but the technique we use appears to be simpler and more direct, and some new results are derived. Also, discussion about the propagation of initial discontinuities is given and illustrated with graphs of some special cases.

scholarly journals Sensitivity analysis and practical identifiability of the mathematical model for partial differential equations

2021 ◽  
Vol 2092 (1) ◽  
pp. 012012
Author(s):  
O Krivorotko ◽  
D Andornaya
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Sensitivity Matrix ◽  
Partial Differential ◽  
Identifiability Analysis ◽  
Practical Identifiability ◽  
Eigenvalue Method ◽  
Simulation Results ◽  
The Mathematical Model

Abstract A sensitivity-based identifiability analysis of mathematical model for partial differential equations is carried out using an orthogonal method and an eigenvalue method. These methods are used to study the properties of the sensitivity matrix and the effects of changes in the model coefficients on the simulation results. Practical identifiability is investigated to determine whether the coefficients can be reconstructed with noisy experimental data. The analysis is performed using correlation matrix method with allowance for Gaussian noise in the measurements. The results of numerical calculations to obtain identifiable sets of parameters for the mathematical model arising in social networks are presented and discussed.

scholarly journals STOCHASTIC SOLUTIONS OF A CLASS OF HIGHER ORDER CAUCHY PROBLEMS IN ℝd

2010 ◽  
Vol 10 (03) ◽  
pp. 341-366 ◽  
Author(s):  
ERKAN NANE
Keyword(s):  
Partial Differential Equations ◽  
Markov Process ◽  
Differential Equations ◽  
Stable Process ◽  
Higher Order ◽  
Cauchy Problems ◽  
Partial Differential ◽  
Iterated Brownian Motion ◽  
Special Cases ◽  
First Time

We study solutions of a class of higher order partial differential equations in bounded domains. These partial differential equations appeared first time in the papers of Allouba and Zheng [4], Baeumer, Meerschaert and Nane [10], Meerschaert, Nane and Vellaisamy [37], and Nane [42]. We express the solutions by subordinating a killed Markov process by a hitting time of a stable subordinator of index 0 < β < 1, or by the absolute value of a symmetric α-stable process with 0 < α ≤ 2, independent of the Markov process. In some special cases we represent the solutions by running composition of k independent Brownian motions, called k-iterated Brownian motion for an integer k ≥ 2. We make use of a connection between fractional-time diffusions and higher order partial differential equations established first by Allouba and Zheng [4] and later extended in several directions by Baeumer, Meerschaert and Nane [10].

scholarly journals On the influence of the initial data in a combustion problem

1980 ◽  
Vol 22 (2) ◽  
pp. 193-209 ◽  
Author(s):  
K. K. Tam
Keyword(s):  
Partial Differential Equations ◽  
Initial Data ◽  
Differential Equations ◽  
Temporal Evolution ◽  
Initial Conditions ◽  
Coupled System ◽  
Parabolic Partial Differential Equations ◽  
Partial Differential ◽  
Approximate Expressions ◽  
Combustion Problem

AbstractThe combustion of a material can be modelled by two coupled parabolic partial differential equations for the temperature and concentration of the material. This paper deals with properties of the solution of these equations inside a cylinder or a sphere and under given initial conditions. Bounds for the variation of the temperature with the initial conditions are first established by considering a decoupled form of the equations. Then the coupled system is used to obtain approximate expressions for the temporal evolution of temperature and concentration.

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