scholarly journals AN INTEGRO-DIFFERENTIAL CONSERVATION LAW ARISING IN A MODEL OF GRANULAR FLOW

2012 ◽  
Vol 09 (01) ◽  
pp. 105-131 ◽  
Author(s):  
DEBORA AMADORI ◽  
WEN SHEN
Keyword(s):  
Granular Flow ◽  
Conservation Law ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Standard Theory ◽  
Step Method ◽  
Fractional Step Method ◽  
Time Step ◽  
Global Existence Of Solutions

We study a scalar integro-differential conservation law which was recently derived by the authors as the slow erosion limit of a granular flow. Considering a set of more general erosion functions, we study the initial boundary value problem for which one cannot adapt the standard theory of conservation laws. We construct approximate solutions with a fractional step method, by recomputing the integral term at each time step. A prioriL∞bound and total variation estimates yield the convergence and global existence of solutions with bounded variation. Furthermore, we present a well-posedness analysis which establishes that these solutions are stable in the L1norm with respect to the initial data.

THE INITIAL BOUNDARY VALUE PROBLEM FOR THE FOUR-VELOCITY PLANE BROADWELL MODEL

1991 ◽  
Vol 01 (03) ◽  
pp. 293-310 ◽  
Author(s):  
GIUSEPPE TOSCANI ◽  
WŁODZIMIERZ WALUŚ
Keyword(s):  
Boundary Value Problem ◽  
Specular Reflection ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Time Interval ◽  
Step Method ◽  
Fractional Step Method ◽  
Uniqueness And Existence ◽  
Initial Boundary Value

The initial boundary value problem for the four-velocity Broadwell equations on a square with specular reflection on the boundary is considered. Uniqueness and existence of solutions on finite time interval is established. The length of the time interval depends on the initial data. The proof of existence uses the procedure known in numerical analysis as the fractional step method. It shows that the method is convergent justifying its usage in computations.

Numerical Simulation of Solitary Wave Using the Fully Lagrangian Method of Moving Particle Semi Implicit

2014 ◽  
Author(s):  
Mohammad Amin Nabian ◽  
Leila Farhadi
Keyword(s):  
Numerical Simulation ◽  
Free Surface ◽  
Solitary Wave ◽  
Stokes Equations ◽  
Rock Avalanche ◽  
Water Tank ◽  
Step Method ◽  
Fractional Step Method ◽  
Time Step ◽  
Moving Particle

A mesh-less numerical approach, called the moving particle semi implicit method (MPS), is presented to solve inviscid Navier-Stokes equations in a fully Lagrangian form using a fractional step method. This method consists of splitting each time step in two steps. The fluid is represented with particles and the motion of each particle is calculated through interactions with neighboring particles by means of a kernel function. In this paper, the MPS method is used to simulate a dynamic system consisting of a heavy box sinking vertically into a water tank, known as Scott Russell’s wave generator problem. This problem is an example of a falling rock avalanche into natural or artificial reservoirs. The box sinks into water tank and as a result the water is heaved up to form a solitary wave and a reverse plunging wave which forms a vortex. This vortex follows the solitary wave down the water tank. The good agreement between the numerical simulation and the analytical solution confirms the accuracy of the model. This proves the applicability of the present model in simulating complex free surface problems. The number of particles on free surface is presented as an indicator of stability of the model.

Landau–Ginzburg-type equations on the half-line in the critical case

2005 ◽  
Vol 135 (6) ◽  
pp. 1241-1262 ◽  
Author(s):  
Elena I. Kaikina ◽  
Hector F. Ruiz-Paredes
Keyword(s):  
Critical Case ◽  
Linear Part ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Inverse Laplace Transform ◽  
Half Line ◽  
Global Existence Of Solutions ◽  
Representation Of Solutions ◽  
Image Position ◽  
Initial Boundary Value

We study nonlinear Landau–Ginzburg-type equations on the half-line in the critical case where β ∈ C, ρ > 2. The linear operator K is a pseudodifferential operator defined by the inverse Laplace transform with dissipative symbol K(p) = αpρ, M = [1/2ρ]. The aim of this paper is to prove the global existence of solutions to the initial–boundary-value problem and to find the main term of the asymptotic representation of solutions in the critical case, when the time decay of the nonlinearity has the same rate as that of the linear part of the equation.

GLOBAL WEAK SOLUTIONS OF AN INITIAL BOUNDARY VALUE PROBLEM FOR SCREW PINCHES IN PLASMA PHYSICS

2009 ◽  
Vol 19 (06) ◽  
pp. 833-875 ◽  
Author(s):  
JIANWEN ZHANG ◽  
SONG JIANG ◽  
FENG XIE
Keyword(s):  
Magnetic Field ◽  
Boundary Value Problem ◽  
Plasma Physics ◽  
Weak Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Field Equations ◽  
Initial Boundary Value

This paper is concerned with an initial-boundary value problem for screw pinches arisen from plasma physics. We prove the global existence of weak solutions to this physically very important problem. The main difficulties in the proof lie in the presence of 1/x-singularity in the equations at the origin and the additional nonlinear terms induced by the magnetic field. Solutions will be obtained as the limit of the approximate solutions in annular regions between two cylinders. Under certain growth assumption on the heat conductivity, we first derive a number of regularities of the approximate physical quantities in the fluid region, as well as a lot of uniform integrability in the entire spacetime domain. By virtue of these estimates we then argue in a similar manner as that in Ref. 20 to take the limit and show that the limiting functions are indeed a weak solution which satisfies the mass, momentum and magnetic field equations in the entire spacetime domain in the sense of distributions, but satisfies the energy equation only in the compact subsets of the fluid region. The analysis in this paper allows the possibility that energy is absorbed into the origin, i.e. the total energy be possibly lost in the limit as the inner radius goes to zero.

Global Existence of Solutions for some Nonlinear Wave Equation

2014 ◽  
Vol 638-640 ◽  
pp. 1700-1704
Author(s):  
Yue Hu
Keyword(s):  
Wave Equation ◽  
Global Existence ◽  
Boundary Value Problem ◽  
Nonlinear Wave Equation ◽  
Existence Of Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Dissipative Term ◽  
Global Existence Of Solutions ◽  
Initial Boundary Value

In this paper, we consider the existence of global solution to the initial-boundary value problem for some hyperbolic equation with P-Laplace operator and a nonlinear dissipative term using the compactness criteria and the monotone mapping’s method.

FRACTIONAL LANDAU–GINZBURG EQUATIONS ON A SEGMENT

2008 ◽  
Vol 10 (06) ◽  
pp. 1151-1181
Author(s):  
ELENA I. KAIKINA
Keyword(s):  
Global Existence ◽  
Boundary Value Problem ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Existence Of Solutions ◽  
Asymptotic Representation Of Solutions ◽  
Representation Of Solutions ◽  
Initial Boundary Value

We study the initial-boundary value problem for the fractional Landau–Ginzburg equations on a segment. The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem and to find the main term of the asymptotic representation of solutions.

scholarly journals Approximate Solutions of Delay Parabolic Equations with the Dirichlet Condition

2012 ◽  
Vol 2012 ◽  
pp. 1-31 ◽  
Author(s):  
Deniz Agirseven
Keyword(s):  
Finite Difference ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Dirichlet Condition ◽  
Difference Schemes ◽  
Parabolic Partial Differential Equation ◽  
Analysis Methods ◽  
Homotopy Analysis ◽  
Initial Boundary Value

Finite difference and homotopy analysis methods are used for the approximate solution of the initial-boundary value problem for the delay parabolic partial differential equation with the Dirichlet condition. The convergence estimates for the solution of first and second orders of difference schemes in Hölder norms are obtained. A procedure of modified Gauss elimination method is used for the solution of these difference schemes. Homotopy analysis method is applied. Comparison of finite difference and homotopy analysis methods is given on the problem.

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