scholarly journals THE MODEL OF THE DISTRIBUTION OF AIR FLOW IN THE MINE SHAFT

2020 ◽  
Vol 1 (3(72)) ◽  
pp. 48-53
Author(s):  
R.E. Leonov
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Three Dimensional ◽  
General Nature ◽  
Air Distribution ◽  
Air Concentration ◽  
Mine Shaft ◽  
Direct Flow ◽  
Distribution Equation

Under some assumptions of a general nature, a partial differential equation of the second strand is compiled with respect to the air concentration in the mine shaft for a direct-flow ventilation circuit. Boundary conditions are formulated and on their basis an analytical solution to the leakage air distribution equation is obtained. Received three-dimensional visualization of air distribution.

scholarly journals Analytical solution to bending of stiffened and continuous antisymmetric laminates

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Liecheng Sun ◽  
Issam E. Harik
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Displacement Function ◽  
The Other ◽  
Order Partial Differential Equation ◽  
Eighth Order ◽  
Coupling Problem ◽  
Simply Supported ◽  
General Response

AbstractAnalytical Strip Method is presented for the analysis of the bending-extension coupling problem of stiffened and continuous antisymmetric thin laminates. A system of three equations of equilibrium, governing the general response of antisymmetric laminates, is reduced to a single eighth-order partial differential equation (PDE) in terms of a displacement function. The PDE is then solved in a single series form to determine the displacement response of antisymmetric cross-ply and angle-ply laminates. The solution is applicable to rectangular laminates with two opposite edges simply supported and the other edges being free, clamped, simply supported, isotropic beam supports, or point supports.

SOLVING THE ASIAN OPTION PDE USING LIE SYMMETRY METHODS

2010 ◽  
Vol 13 (08) ◽  
pp. 1265-1277 ◽  
Author(s):  
NICOLETTE C. CAISTER ◽  
JOHN G. O'HARA ◽  
KESHLAN S. GOVINDER
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Stock Price ◽  
Ad Hoc ◽  
Lie Symmetry ◽  
Asian Option ◽  
Asian Options ◽  
Lie Point Symmetries ◽  
Reduced Order

Asian options incorporate the average stock price in the terminal payoff. Examination of the Asian option partial differential equation (PDE) has resulted in many equations of reduced order that in general can be mapped into each other, although this is not always shown. In the literature these reductions and mappings are typically acquired via inspection or ad hoc methods. In this paper, we evaluate the classical Lie point symmetries of the Asian option PDE. We subsequently use these symmetries with Lie's systematic and algorithmic methods to show that one can obtain the same aforementioned results. In fact we find a familiar analytical solution in terms of a Laplace transform. Thus, when coupled with their methodic virtues, the Lie techniques reduce the amount of intuition usually required when working with differential equations in finance.

scholarly journals Geometric shape features extraction using a steady state partial differential equation system

2019 ◽  
Vol 6 (4) ◽  
pp. 647-656 ◽  
Author(s):  
Takayuki Yamada
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Steady State ◽  
Analytical Solution ◽  
Geometric Shape ◽  
Geometrical Shape ◽  
Shape Features ◽  
Partial Differential ◽  
Topological Constraint ◽  
Numerical Examples

Abstract A unified method for extracting geometric shape features from binary image data using a steady-state partial differential equation (PDE) system as a boundary value problem is presented in this paper. The PDE and functions are formulated to extract the thickness, orientation, and skeleton simultaneously. The main advantage of the proposed method is that the orientation is defined without derivatives and thickness computation is not imposed a topological constraint on the target shape. A one-dimensional analytical solution is provided to validate the proposed method. In addition, two-dimensional numerical examples are presented to confirm the usefulness of the proposed method. Highlights A steady state partial differential equation for extraction of geometrical shape features is formulated. The functions for geometrical shape features are formulated by the solution of the proposed PDE. Analytical solution is provided in one-dimension. Numerical examples are provided in two-dimension.

scholarly journals Comparison of Analytical Solution of DGLAP Equations forF2NS(x,t)at Smallxby Two Methods

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Neelakshi N. K. Borah ◽  
D. K. Choudhury ◽  
P. K. Sahariah
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Structure Function ◽  
Method Of Characteristics ◽  
Dglap Equation ◽  
Partial Differential ◽  
Chi Square ◽  
Data Set ◽  
The Method Of Characteristics

The DGLAP equation for the nonsinglet structure functionF2NS(x,t)at LO is solved analytically at lowxby converting it into a partial differential equation in two variables: Bjorkenxandt  (t=ln(Q2/Λ2)and then solved by two methods: Lagrange’s auxiliary method and the method of characteristics. The two solutions are then compared with the available data on the structure function. The relative merits of the two solutions are discussed calculating the chi-square with the used data set.

Three-Dimensional Elastic Solution of a Transversely Isotropic Functionally Graded Rectangular Plate

2012 ◽  
Vol 591-593 ◽  
pp. 2655-2660 ◽  
Author(s):  
Guo Jun Nie ◽  
Zhao Yang Feng ◽  
Jun Tao Shi ◽  
Ying Ya Lu ◽  
Zheng Zhong
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Rectangular Plate ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Functionally Graded ◽  
Elastic Solution ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Present Solution

Three-dimensional elastic solution of a simply supported, transversely isotropic functionally graded rectangular plate is presented in this paper. Suppose that all elastic coefficients of the material have the same power-law dependence on the thickness coordinate. By introducing two new displacement functions, three equations of equilibrium in terms of displacements are reduced to two uncoupled partial differential equations. Exact solution for a second-order partial differential equation expressed by one of displacement functions is obtained and analytical solution for another fourth-order partial differential equation expressed by another displacement function is found by employing the Frobenius method. The validity of the present solution is first investigated. And the effect of the gradation of material properties on the mechanical behavior of the plate is studied through numerical examples.

An evaluation of a dynamic model of the secondary clarifier

1996 ◽  
Vol 34 (5-6) ◽  
pp. 19-26 ◽  
Author(s):  
Ulf Jeppsson ◽  
Stefan Diehl
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Mathematical Modelling ◽  
Dynamic Behaviour ◽  
Linear Partial Differential Equation ◽  
Layer Model ◽  
Important Conclusion ◽  
Secondary Clarifier ◽  
Theoretical Results

The main objective of the paper is to support and illustrate recent theoretical results on the mathematical modelling of the secondary clarifier. A new settler model is compared with a traditional layer model by means of numerical simulations. Emphasis is put on the numerical solution's ability to approximate the analytical solution of the conservation law written as a non-linear partial differential equation. The new settler model is consistent in this respect. Another important conclusion is that a layer model dividing the settler into only ten layers (normally used in settler models) is too crude an approximation to capture the detailed dynamic behaviour of the settler. All simulations presented are performed with the settler models coupled to the IAWQ Activated Sludge Model No. 1.

scholarly journals Modified Kudrayshov Method to Solve Generalized Kuramoto–Sivashinsky Equation

2018 ◽  
Author(s):  
Adem Kilicman ◽  
Rathinavel Silambarasan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Exact Solutions ◽  
Exact Analytical Solution ◽  
Nonlinear Partial Differential Equation ◽  
Complex Structures ◽  
Algebraic Equations ◽  
Uniqueness Of Solutions ◽  
Sivashinsky Equation

The generalized Kuramoto–Sivashinsky equation is investigated using the modified Kudrayshov method for the exact analytical solution. The modified Kudrayshov method converts the nonlinear partial differential equation to algebraic equations, as a result of various steps, which on solving the so obtained equation systems yields the analytical solution. By this way various exact solutions including complex structures are found and drawn their behaviour in complex plane by Maple to compare the uniqueness of solutions.

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