scholarly journals Analytical Solution and Numerical Method

1971 ◽  
Vol 74 (629) ◽  
pp. 631-638
Author(s):  
Yoshiaki YAMADA
Keyword(s):  
Analytical Solution ◽  
Numerical Method

Convergence and approximate calculation of average degree under different network sizes for decreasing random birth-and-death networks

2018 ◽  
Vol 32 (15) ◽  
pp. 1850159
Author(s):  
Yin Long ◽  
Xiao-Jun Zhang ◽  
Kui Wang
Keyword(s):  
Analytical Solution ◽  
Numerical Method ◽  
Power Law ◽  
Approximate Calculation ◽  
Approximate Expression ◽  
Network Size ◽  
Average Degree ◽  
Large Network ◽  
Network Link ◽  
Theoretical Results

In this paper, convergence and approximate calculation of average degree under different network sizes for decreasing random birth-and-death networks (RBDNs) are studied. First, we find and demonstrate that the average degree is convergent in the form of power law. Meanwhile, we discover that the ratios of the back items to front items of convergent reminder are independent of network link number for large network size, and we theoretically prove that the limit of the ratio is a constant. Moreover, since it is difficult to calculate the analytical solution of the average degree for large network sizes, we adopt numerical method to obtain approximate expression of the average degree to approximate its analytical solution. Finally, simulations are presented to verify our theoretical results.

scholarly journals ON ONE NUMERICAL METHOD FOR THE COMPUTING OF SEDIMENTATION OF THE SURFACE OF THE SOIL MASSIVE CAUSED BY THE CONSTRUCTION OF THE SHELL LINING OF THE TUNNEL

2018 ◽  
Vol 14 (1) ◽  
pp. 78-91
Author(s):  
Sergey B. Kosytsyn ◽  
Vladimir Y. Akulich
Keyword(s):  
Analytical Solution ◽  
Numerical Method ◽  
Strain State ◽  
Soil Mass ◽  
Stress Strain ◽  
Stress Strain State ◽  
The Earth ◽  
Surrounding Soil

The distinctive work is aimed at the geotechnical forecast of the influence of the construction of the tunnel on the change in the stress-strain state of the surrounding soil mass, namely, the precipitations that arise on the surface of the earth. The work assumes both a numerical and an analytical solution with subsequent com-parative analysis

scholarly journals The Positivity of Numerical Method for Susceptible-Infected-Recovered-Susceptible Epidemic Model

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Feng Feng ◽  
Zong Wang
Keyword(s):  
Analytical Solution ◽  
Numerical Solution ◽  
Numerical Method ◽  
Epidemic Model ◽  
Numerical Technique ◽  
Environmental Perturbations ◽  
Sirs Epidemic Model ◽  
Sirs Model ◽  
Balanced Implicit Method ◽  
Stochastic Sirs Epidemic Model

Sudden environmental perturbations may affect the positivity of the solution of the susceptible-infected-recovered-susceptible (SIRS) model. Most of the SIRS epidemic models have no analytical solution. Thus, in order to find the appropriate solution, the numerical technique becomes more essential for us to solve the dynamic behavior of epidemics. In this paper, we are concerned with the positivity of the numerical solution of a stochastic SIRS epidemic model. A new numerical method that is the balanced implicit method (BIM) is set, which preserves the positivity under given conditions. The BIM method can maintain positive numerical solution. An illustrative numerical instance is presented for the numerical BIM of the stochastic SIRS model.

scholarly journals Time-Continuous Real-Time Trajectory Generation for Safe Autonomous Flight of a Quadrotor in Unknown Environment

2021 ◽  
Vol 11 (7) ◽  
pp. 3238
Author(s):  
Yonghee Park ◽  
Woosung Kim ◽  
Hyungpil Moon
Keyword(s):  
Analytical Solution ◽  
Numerical Method ◽  
Trajectory Generation ◽  
Local Minima ◽  
Computationally Efficient ◽  
Cluttered Environment ◽  
Order Polynomial ◽  
Path Planner ◽  
Global And Local ◽  
Global Trajectory

In this paper, we present an efficient global and local replanning method for a quadrotor to complete a flight mission in a cluttered and unmapped environment. A minimum-snap global path planner generates a global trajectory that comprises some waypoints in a cluttered environment. When facing unexpected obstacles, our method modifies the global trajectory using geometrical planning and closed-form formulation for an analytical solution with 9th-order polynomial. The proposed method provides an analytical solution, not a numerical one, and it is computationally efficient without falling into a local minima problem. In a simulation, we show that the proposed method can fly a quadrotor faster than the numerical method in a cluttered environment. Furthermore, we show in experiments that the proposed method can provide safer and faster trajectory generation than the numerical method in a real environment.

scholarly journals TORSIONAL SHEAR STRESS IN PRISMATIC BEAMS WITH ARBITRARY CROSS-SECTIONS USING FINITE ELEMENT METHOD

2021 ◽  
Vol 30 (2) ◽  
Author(s):  
Dang-Bao Tran
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Shear Stress ◽  
Numerical Methods ◽  
Analytical Solution ◽  
Numerical Method ◽  
Cross Sections ◽  
Structural Element ◽  
Quadrilateral Elements ◽  
Torsional Shear

Determining the shear stress of a structural element caused by torsion is a vital problem. The analytical solution of the Saint-Venant torsion is only suitable for simple cross-sections. The numerical method to evaluate the shear stress of complicated cross-sections is indispensable. Many numerical methods have been studied by scientists. Among these studies, Gruttmann proposed an excellent numerical method, which inherited the Saint-Venant theory. However, the use of isoparametric four-noded quadrilateral elements made the method not to reach the best optimization. The objective of this paper is to improve Gruttmann‘s method by using isoparametric eight-noded quadrilateral elements. MATLAB is the language for programming the numerical method. The validated examples have demonstrated that the author’s numerical method is more effective than Gruttmann‘s method.

scholarly journals NUMERICAL CALCULATION OF NONSTATIONARY FRACTIONAL DIFFERENTIAL EQUATION IN PROBLEMS OF MODELING TOXIC SUBSTANCES DISTRIBUTION IN GROUND WATERS

2019 ◽  
pp. 70-80
Author(s):  
Alexandra Alekseyevna Afanasyeva ◽  
Tatyana Nikolayevna Shvetsova-Shilovskaya ◽  
Dmitriy Evgenevich Ivanov ◽  
Denis Igorevich Nazarenko ◽  
Elena Victorovna Kazarezova
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Differential Equations ◽  
Difference Scheme ◽  
Numerical Method ◽  
Fractional Differential Equation ◽  
Numerical Solutions ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Fractional Differential

At present, the theory of fractional calculus is widely used in many fields of science for modeling various processes. Differential equations with fractional derivatives are used to model the migration of pollutants in porous inhomogeneous media and allow a more correct description of the behavior of pollutants at large distances from the source. The analytical solution of differential equations with fractional order derivatives is often very complicated or even impossible. There has been proposed a numerical method for solving fractional differential equations in partial derivatives with respect to time to describe the migration of pollutants in groundwater. An implicit difference scheme is developed for the numerical solution of a non-stationary fractional differential equation, which is an analogue of the well-known implicit Crank-Nicholson difference scheme. The system of difference equations is presented in matrix form. The solution of the problem is reduced to the multiple solution of a tridiagonal system of linear algebraic equations by the tridiagonal matrix algorithm. The results of evaluating the spread of pollutant in groundwater based on the numerical method for model examples are presented. The concentrations of the substance obtained on the basis of the analytical and numerical solutions of the unsteady one-dimensional fractional differential equation are compared. The results obtained using the proposed method and on the basis of the well-known analytical solution of the fractional differential equation are in fairly good agreement with each other. The relative error is on average 9%. In contrast to the well-known analytical solution, the developed numerical method can be used to model the spread of pollutants in groundwater, taking into account their biodegradation.

Some Applications of CDL-BIEM

1994 ◽  
Author(s):  
Nhan Phan-Thien ◽  
Sangtae Kim
Keyword(s):  
Analytical Solution ◽  
Numerical Method ◽  
Analytical Solutions ◽  
Spherical Inclusion ◽  
Three Dimensional ◽  
Simple Problem ◽  
Benchmark Problem ◽  
Bounding Surface ◽  
Rigid Displacement ◽  
Particulate Solids

This chapter presents some selected three-dimensional applications of the CDL-BIEM in elasticity and Stokes flows, especially to particulate solids for which the method is devised. It is paramount that any numerical method should be validated against known analytical solutions. The method will therefore be benchmarked against known simple solutions of the type reported in chapters 2 and 5. Some selected nontrivial examples, where no analytical solutions are available, will also be presented. The translating sphere is a simple problem with known analytical solution and smooth bounding surface; it is a popular benchmark problem for boundary element codes. Here a rigid spherical inclusion of radius a, centered at x = 0, is displaced by either (1) a constant vector U or (2) acted on by a force F, and we seek the force in the case of problem (1), or the rigid displacement in problem (2).

scholarly journals Multi scale numerical approach for micro-cracked viscoelastic masonry

2018 ◽  
Vol 251 ◽  
pp. 04031
Author(s):  
T.T.Nga Nguyen ◽  
S.Tuan Nguyen ◽  
N.Quang Vu ◽  
T.Ta. Nguyen ◽  
N.Hung Tran ◽  
...  
Keyword(s):  
Analytical Solution ◽  
Numerical Method ◽  
Viscoelastic Properties ◽  
Numerical Approach ◽  
Two Dimensional ◽  
Periodic Homogenization ◽  
Periodic Cell ◽  
Multi Scale ◽  
Homogenization Approach ◽  
Incremental Formulation

A multi-scale numerical method for viscoelastic micro-cracked masonry is proposed. Firstly, the effective viscoelastic properties of the masonry are modelled by a periodic homogenization approach. The Modified Maxwell (MM) model is chosen for the creep. Secondly, an incremental procedure is proposed. Thirdly, an incremental formulation is used to get the overall viscoelastic behaviour of the two dimensional periodic cell. Finally, the result of the method is validated against analytical solution.

scholarly journals Solving Richards’ nonlinear PDE modeling dynamic from water in unsaturated zone by a new numerical method called SBA

2017 ◽  
Vol 6 (1) ◽  
pp. 20
Author(s):  
Ben-Sthal Sakoma Yelingue ◽  
Wenddabo Olivier Sawadogo ◽  
Blaise Some
Keyword(s):  
Analytical Solution ◽  
Numerical Method ◽  
Partial Derivative ◽  
Unsaturated Zone ◽  
Nonlinear Pde ◽  
Saturated Zone ◽  
Strongly Nonlinear

In this paper the partial derivative equation strongly nonlinear of Richards which models the dynamics of water in the Un-saturated Zone (UZ) was linearized and solved by a new numerical method called SBA. The analytical solution has been simulated in order to be applied later to the following in unsaturated zone with the aquifers of Bangui and its boundaries.

An Approximate Analytical Model of Thermohydrodynamic Estimation of Finite Length Journal Bearings

2005 ◽  
Author(s):  
Mihail G. Ionescu ◽  
Vladimir-Codrin M. Ionescu
Keyword(s):  
Analytical Solution ◽  
Numerical Method ◽  
Analytical Model ◽  
Finite Length ◽  
Journal Bearings ◽  
Bearing Estimation

The present paper proposes an original analytical solution of the thermohydrodynamic lubrication, and the validation of the model in the case of journal bearings. The simplicity and accuracy of the method and the computing times of the order of seconds necessary for bearing estimation constitute important advantages of this model over the numerical method.

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