scholarly journals MATHEMATICAL MODELLING OF AN ELONGATED MAGNETIC DROPLET IN A ROTATING MAGNETIC FIELD

2012 ◽  
Vol 17 (1) ◽  
pp. 47-57 ◽  
Author(s):  
Andrejs Cebers ◽  
Harijs Kalis
Keyword(s):  
Magnetic Field ◽  
Mathematical Modelling ◽  
Numerical Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Rotating Magnetic Field ◽  
Singularly Perturbed ◽  
Actual Shape ◽  
Initial Boundary Value ◽  
Magnetic Droplet

Dynamics of an elongated droplet under the action of a rotating magnetic field is considered by mathematical modelling. The actual shape of a droplet is obtained by solving the initial-boundary value problem of a nonlinear singularly perturbed partial differential equation (PDE). For the discretization in space the finite difference scheme (FDS) is applied. Time evolution of numerical solutions is obtained with MATLAB by solving a large system of ordinary differential equations (ODE).

scholarly journals NUMERICAL SIMULATION OF MAGNETIC DROPLET DYNAMICS IN A ROTATING FIELD

2013 ◽  
Vol 18 (1) ◽  
pp. 80-96
Author(s):  
Andrejs Cebers ◽  
Harijs Kalis
Keyword(s):  
Numerical Solutions ◽  
Nonlinear Partial Differential Equation ◽  
Method Of Lines ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Continuous Functions ◽  
Rotating Magnetic Field ◽  
Droplet Dynamics ◽  
Ill Posed ◽  
Initial Boundary Value

Dynamics and hysteresis of an elongated droplet under the action of a rotating magnetic field is considered for mathematical modelling. The shape of droplet is found by regularization of the ill-posed initial–boundary value problem for nonlinear partial differential equation (PDE). It is shown that two methods of the regularization – introduction of small viscous bending torques and construction of monotonous continuous functions are equivalent. Their connection with the regularization of the ill-posed reverse problems for the parabolic equation of heat conduction is remarked. Spatial discretization is carried out by the finite difference scheme (FDS). Time evolution of numerical solutions is obtained using method of lines for solving a large system of ordinary differential equations (ODE).

A numerical study of the generation of an azimuthal current in a plasma cylinder using a transverse rotating magnetic field

1981 ◽  
Vol 26 (3) ◽  
pp. 455-464 ◽  
Author(s):  
W. N. Hugrass ◽  
R. C. Grimm
Keyword(s):  
Magnetic Field ◽  
Numerical Study ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Threshold Value ◽  
Rotating Magnetic Field ◽  
Plasma Cylinder ◽  
Initial Boundary Value ◽  
Azimuthal Current ◽  
Cylindrical Plasma Column

The generation of a steady azimuthal current in a cylindrical plasma column using a rotating magnetic field is numerically investigated. The mixed initial-boundary-value problem is solved using a finite difference method. It is shown that substantial azimuthal current can be driven provided that the amplitude of the rotating magnetic field is greater than a certain threshold value which depends on the plasma resistivity.

Numerical Method for Singularly Perturbed Parabolic Equations in Unbounded Domains in the Case of Solutions Growing at Infinity

2009 ◽  
Vol 9 (1) ◽  
pp. 100-110
Author(s):  
G. I. Shishkin
Keyword(s):  
Boundary Layer ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Singularly Perturbed ◽  
Reaction Diffusion Equation ◽  
Maximum Norm ◽  
Lateral Part ◽  
Initial Boundary Value

AbstractAn initial-boundary value problem is considered in an unbounded do- main on the x-axis for a singularly perturbed parabolic reaction-diffusion equation. For small values of the parameter ε, a parabolic boundary layer arises in a neighbourhood of the lateral part of the boundary. In this problem, the error of a discrete solution in the maximum norm grows without bound even for fixed values of the parameter ε. In the present paper, the proximity of solutions of the initial-boundary value problem and of its numerical approximations is considered. Using the method of special grids condensing in a neighbourhood of the boundary layer, a special finite difference scheme converging ε-uniformly in the weight maximum norm has been constructed.

scholarly journals A Method for Generating Approximate Similarity Solutions of Nonlinear Partial Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Mazhar Iqbal ◽  
M. T. Mustafa ◽  
Azad A. Siddiqui
Keyword(s):  
Gas Flow ◽  
Numerical Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Similarity Solutions ◽  
Standard Application ◽  
Approximate Similarity ◽  
Initial Boundary Value ◽  
Unsteady Gas Flow

Standard application of similarity method to find solutions of PDEs mostly results in reduction to ODEs which are not easily integrable in terms of elementary or tabulated functions. Such situations usually demand solving reduced ODEs numerically. However, there are no systematic procedures available to utilize these numerical solutions of reduced ODE to obtain the solution of original PDE. A practical and tractable approach is proposed to deal with such situations and is applied to obtain approximate similarity solutions to different cases of an initial-boundary value problem of unsteady gas flow through a semi-infinite porous medium.

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.

Numerical analysis and simulation of delay parabolic partial differential equation involving a small parameter

2019 ◽  
Vol 37 (1) ◽  
pp. 289-312 ◽  
Author(s):  
Lolugu Govindarao ◽  
Jugal Mohapatra
Keyword(s):  
Boundary Value Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Shishkin Mesh ◽  
Second Order ◽  
Singularly Perturbed ◽  
Hybrid Scheme ◽  
Content Type ◽  
Convection Diffusion ◽  
Initial Boundary Value

Purpose The purpose of this paper is to provide an efficient and robust second-order monotone hybrid scheme for singularly perturbed delay parabolic convection-diffusion initial boundary value problem. Design/methodology/approach The delay parabolic problem is solved numerically by a finite difference scheme consists of implicit Euler scheme for the time derivative and a monotone hybrid scheme with variable weights for the spatial derivative. The domain is discretized in the temporal direction using uniform mesh while the spatial direction is discretized using three types of non-uniform meshes mainly the standard Shishkin mesh, the Bakhvalov–Shishkin mesh and the Gartland Shishkin mesh. Findings The proposed scheme is shown to be a parameter-uniform convergent scheme, which is second-order convergent and optimal for the case. Also, the authors used the Thomas algorithm approach for the computational purposes, which took less time for the computation, and hence, more efficient than the other methods used in literature. Originality/value A singularly perturbed delay parabolic convection-diffusion initial boundary value problem is considered. The solution of the problem possesses a regular boundary layer. The authors solve this problem numerically using a monotone hybrid scheme. The error analysis is carried out. It is shown to be parameter-uniform convergent and is of second-order accurate. Numerical results are shown to verify the theoretical estimates.

scholarly journals Singularly perturbed hyperbolic problems on metric graphs: asymptotics of solutions

2017 ◽  
Vol 15 (1) ◽  
pp. 404-419 ◽  
Author(s):  
Yuriy Golovaty ◽  
Volodymyr Flyud
Keyword(s):  
Differential Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Singularly Perturbed ◽  
Hyperbolic Problems ◽  
Metric Graphs ◽  
Asymptotics Of Solutions ◽  
Layer Method ◽  
Boundary Layer Method ◽  
Initial Boundary Value

Abstract We are interested in the evolution phenomena on star-like networks composed of several branches which vary considerably in physical properties. The initial boundary value problem for singularly perturbed hyperbolic differential equation on a metric graph is studied. The hyperbolic equation becomes degenerate on a part of the graph as a small parameter goes to zero. In addition, the rates of degeneration may differ in different edges of the graph. Using the boundary layer method the complete asymptotic expansions of solutions are constructed and justified.

Unsteady, Combined Radiation and Conduction in an Absorbing, Scattering, and Emitting Medium

1973 ◽  
Vol 95 (3) ◽  
pp. 357-364 ◽  
Author(s):  
K. C. Weston ◽  
J. L. Hauth
Keyword(s):  
Boundary Value Problem ◽  
Energy Equation ◽  
Numerical Solutions ◽  
Gaussian Quadrature ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Isotropic Scattering ◽  
Radiation Parameter ◽  
Parallel Plates ◽  
Initial Boundary Value

The transient cooldown of a gray, absorbing, isotropic scattering, emitting, and conducting medium bounded by gray, diffusely emitting and reflecting parallel plates is considered. Numerical solutions are obtained for the initial boundary-value problem with a discontinuous decrease in temperature at one boundary. The quasi-steady equation of radiative transfer is solved using Gaussian quadrature and a matrix eigenvector technique together with explicit numerical solution of the unsteady energy equation. Temperature and energy flux distributions are presented for variations of optical thickness, boundary emissivity, albedo, and conduction–radiation parameter.

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