Properties of analytic solution and numerical solution of multi-pantograph equation

2004 ◽  
Vol 155 (3) ◽  
pp. 853-871 ◽  
Author(s):  
M.Z Liu ◽  
Dongsong Li
Keyword(s):  
Numerical Solution ◽  
Analytic Solution ◽  
Pantograph Equation

On the convective diffusion under partial slip at wall

1989 ◽  
Vol 54 (4) ◽  
pp. 967-980 ◽  
Author(s):  
Ondřej Wein ◽  
Petr Kučera
Keyword(s):  
Mass Transfer ◽  
Numerical Solution ◽  
Analytic Solution ◽  
Convective Diffusion ◽  
Linear Velocity ◽  
Partial Slip ◽  
Accurate Calculation ◽  
Transfer Coefficients ◽  
Empirical Formulas ◽  
Mass Transfer Coefficients

Extended Leveque problem is studied for linear velocity profiles, vx(z) = u + qz. The existing analytic solution is reconsidered and shown to be inapplicable for the accurate calculation of mean mass-transfer coefficients. A numerical solution is reported and its accuracy is checked in detail. Simple but fairly accurate empirical formulas are suggested for the calculating of local and mean mass-transfer coefficients.

Propagation of obliquely incident water waves over a trench

1983 ◽  
Vol 133 ◽  
pp. 47-63 ◽  
Author(s):  
James T. Kirby ◽  
Robert A. Dalrymple
Keyword(s):  
Numerical Solution ◽  
Water Waves ◽  
Analytic Solution ◽  
Small Scale ◽  
Obliquely Incident ◽  
Long Wave ◽  
Propagating Waves ◽  
Tank Experiment ◽  
Wave Region ◽  
Wave Limit

The diffraction of obliquely incident surface waves by an asymmetric trench is investigated using linearized potential theory. A numerical solution is constructed by matching particular solutions for each subregion of constant depth along vertical boundaries; the resulting matrix equation is solved numerically. Several cases where the trench-parallel wavenumber component in the incident-wave region exceeds the wavenumber for freely propagating waves in the trench are investigated and are found to result in large reductions in wave transmission; however, reflection is not total owing to the finiteness of the obstacle.Results for one case are compared with data obtained from a small-scale wave-tank experiment. An approximate solution based on plane-wave modes is derived and compared with the numerical solution and, in the long-wave limit, with a previous analytic solution.

Pendulum Motion Damped by Speed-Independent Friction

2021 ◽  
Vol 03 (03) ◽  
pp. 2150008
Author(s):  
Carl E. Mungan
Keyword(s):  
Numerical Solution ◽  
Turning Point ◽  
Analytic Solution ◽  
Angular Position ◽  
Rolling Friction ◽  
Pendulum Motion ◽  
Circular Trajectory ◽  
Air Drag ◽  
Small Block

A pendulum without a supporting string or rod is obtained if a small block or marble is released at the rim of a spherical bowl or cylindrical half-pipe. This setup also applies to the familiar loop-the-loop demonstration. However, the bob will then experience sliding or rolling friction, which is speed independent in contrast to the linear or quadratic air drag which is more commonly used to model damping of oscillators. An analytic solution can be found for the speed of the bob as a function of its angular position around the vertical circular trajectory. A numerical solution for the time that the object takes to move from one turning point to the next shows that it is smaller than it would be in the absence of friction.

scholarly journals Numerical Solution for Complex Systems of Fractional Order

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Rabha W. Ibrahim
Keyword(s):  
Complex Systems ◽  
Numerical Solution ◽  
Fractional Order ◽  
Analytic Solution ◽  
Approximate Solutions ◽  
Fractional Operators ◽  
Homotopy Perturbation

By using a complex transform, we impose a system of fractional order in the sense of Riemann-Liouville fractional operators. The analytic solution for this system is discussed. Here, we introduce a method of homotopy perturbation to obtain the approximate solutions. Moreover, applications are illustrated.

Finite Analytic Numerical Solution Axisymmetric Navier-Stokes and Energy Equations

1983 ◽  
Vol 105 (3) ◽  
pp. 639-645 ◽  
Author(s):  
Ching-Jen Chen ◽  
Young Hwan Yoon
Keyword(s):  
Heat Transfer ◽  
Numerical Solution ◽  
Numerical Method ◽  
Stream Function ◽  
Analytic Solution ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Inlet Temperature ◽  
Local Analytic Solution

Connective heat transfer for steady-state laminar flow in axisymmetric coordinates is considered. Numerical solutions for flow pattern and temperature distribution are obtained by the finite analytic numerical method applied to the Navier-Stokes equations expressed in terms of vorticity and stream function, and the energy equation. The finite analytic numerical method differs from other numerical methods in that it utilizes a local analytic solution in an element of the problem to construct the total numerical solution. Finite analytic solutions of vorticity, stream function, temperature, and heat transfer coefficients for flow with Reynolds numbers of 5, 100, 1000, and 2000, and Prandtl numbers of 0.1, 1.0, and 10.0 with uniform grid sizes, are reported for an axisymmetric pipe with a sudden expansion and contraction. The wall temperature is considered to be isothermal and differs from the inlet temperature. It is shown that the finite analytic is stable, converges rapidly, and simulates the convection of fluid flow accurately, since the local analytic solution is capable of simulating automatically the influence of skewed convection through the element boundary on the interior nodal values, thereby minimizing the false numerical diffusion.

scholarly journals Four-Quadrant Riemann Problem for a 2×2 System Involving Delta Shock

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 138
Author(s):  
Jinah Hwang ◽  
Suyeon Shin ◽  
Myoungin Shin ◽  
Woonjae Hwang
Keyword(s):  
Numerical Solution ◽  
Riemann Problem ◽  
Hyperbolic Conservation Laws ◽  
Analytic Solution ◽  
Hyperbolic Conservation ◽  
Delta Shock ◽  
Initial Discontinuity ◽  
Planar Wave ◽  
Four Quadrant ◽  
Elementary Waves

In this paper, a four-quadrant Riemann problem for a 2×2 system of hyperbolic conservation laws is considered in the case of delta shock appearing at the initial discontinuity. We also remove the restriction in that there is only one planar wave at each initial discontinuity. We include the existence of two elementary waves at each initial discontinuity and classify 14 topologically distinct solutions. For each case, we construct an analytic solution and compute the numerical solution.

scholarly journals Fourier Operational Matrices of Differentiation and Transmission: Introduction and Applications

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
F. Toutounian ◽  
Emran Tohidi ◽  
A. Kilicman
Keyword(s):  
Exact Solution ◽  
Numerical Solution ◽  
Variable Coefficients ◽  
Operational Matrices ◽  
Residual Function ◽  
Numerical Examples ◽  
Pantograph Equation ◽  
Constant Coefficients ◽  
Collocation Scheme ◽  
Linear Differential

This paper introduces Fourier operational matrices of differentiation and transmission for solving high-order linear differential and difference equations with constant coefficients. Moreover, we extend our methods for generalized Pantograph equations with variable coefficients by using Legendre Gauss collocation nodes. In the case of numerical solution of Pantograph equation, an error problem is constructed by means of the residual function and this error problem is solved by using the mentioned collocation scheme. When the exact solution of the problem is not known, the absolute errors can be computed approximately by the numerical solution of the error problem. The reliability and efficiency of the presented approaches are demonstrated by several numerical examples, and also the results are compared with different methods.

The Research of the Seepage Regularity of Power-Law Fluid under Porosity Scale

2013 ◽  
Vol 423-426 ◽  
pp. 1722-1726
Author(s):  
Er Long Yang ◽  
Yu Xin Gao ◽  
Dong Liang Li
Keyword(s):  
Numerical Solution ◽  
Power Law ◽  
Analytic Solution ◽  
Capillary Flow ◽  
Resistance Coefficient ◽  
Numerical Results ◽  
Experimental Results ◽  
Quantitative Relation ◽  
Capillary Tubes ◽  
Power Law Index

There are various shapes of capillary tubes in the rock, and the flow in triangles and rectangles capillary tubes has been studied very little. In this paper, the approximate solution of the distribution of the velocity in triangles capillary tubes has been obtained using variational method. The quantitative relation between pressure difference and flow rate in the two kinds of capillary tubes has been obtained by integrating in the entire capillary tubes, which is similar to Poiseuille's law. this study also applies the Pdetool tool in Matlab to solve the laminar flow of different sections in capillary flow numerically and compares the variational results and numerical results with the literature analytic solution, numerical solution and the experimental results. On the foundation of comparing the variational results, the numerical results and the literature analytic solution, numerical solution and the experimental results, we analyze the influence of the power-law index and cross-section shape on the comprehensive resistance coefficient and flow pressure relationship.

Sign in / Sign up

Export Citation Format

Share Document