The single-phase Stefan problem for a general one-dimensional equation of parabolic type. The generalized solution of the Stefan problem

2016 ◽  
pp. 295-323
Keyword(s):  
Stefan Problem ◽  
Single Phase ◽  
Parabolic Type ◽  
Generalized Solution ◽  
One Dimensional ◽  
Dimensional Equation

Asymptotic derivation of a refined equation for an elastic beam resting on a Winkler foundation

2021 ◽  
pp. 108128652110238
Author(s):  
Barış Erbaş ◽  
Julius Kaplunov ◽  
Isaac Elishakoff
Keyword(s):  
Ad Hoc ◽  
Moving Load ◽  
Low Frequency ◽  
Elastic Beam ◽  
Winkler Foundation ◽  
One Dimensional ◽  
Beam Equations ◽  
Dimensional Equation ◽  
Phase And Group Velocities ◽  
Group Velocities

A two-dimensional mixed problem for a thin elastic strip resting on a Winkler foundation is considered within the framework of plane stress setup. The relative stiffness of the foundation is supposed to be small to ensure low-frequency vibrations. Asymptotic analysis at a higher order results in a one-dimensional equation of bending motion refining numerous ad hoc developments starting from Timoshenko-type beam equations. Two-term expansions through the foundation stiffness are presented for phase and group velocities, as well as for the critical velocity of a moving load. In addition, the formula for the longitudinal displacements of the beam due to its transverse compression is derived.

scholarly journals NEW L1-GRADIENT TYPE ESTIMATES OF SOLUTIONS TO ONE-DIMENSIONAL QUASILINEAR PARABOLIC SYSTEMS

2010 ◽  
Vol 12 (01) ◽  
pp. 85-106 ◽  
Author(s):  
S. N. ANTONTSEV ◽  
J. I. DÍAZ
Keyword(s):  
Type Equation ◽  
Parabolic Type ◽  
Parabolic Systems ◽  
Diffusion Term ◽  
One Dimensional ◽  
First Order ◽  
Estimates Of Solutions ◽  
Gradient Type ◽  
Degenerate Type ◽  
Parabolic Type Equation

We consider a general class of one-dimensional parabolic systems, mainly coupled in the diffusion term, which, in fact, can be of the degenerate type. We derive some new L1-gradient type estimates for its solutions which are uniform in the sense that they do not depend on the coefficients nor on the size of the spatial domain. We also give some applications of such estimates to gas dynamics, filtration problems, a p-Laplacian parabolic type equation and some first order systems of Hamilton–Jacobi or conservation laws type.

scholarly journals An integral relationship for a fractional one-phase Stefan problem

2018 ◽  
Vol 21 (4) ◽  
pp. 901-918 ◽  
Author(s):  
Sabrina Roscani ◽  
Domingo Tarzia
Keyword(s):  
Boundary Condition ◽  
Stefan Problem ◽  
One Dimensional ◽  
Similarity Type ◽  
Integral Relationship ◽  
Temperature Boundary ◽  
Liouville Derivative ◽  
The One ◽  
Stefan Condition ◽  
Temperature Boundary Condition

Abstract A one-dimensional fractional one-phase Stefan problem with a temperature boundary condition at the fixed face is considered by using the Riemann–Liouville derivative. This formulation is more convenient than the one given in Roscani and Santillan (Fract. Calc. Appl. Anal., 16, No 4 (2013), 802–815) and Tarzia and Ceretani (Fract. Calc. Appl. Anal., 20, No 2 (2017), 399–421), because it allows us to work with Green’s identities (which does not apply when Caputo derivatives are considered). As a main result, an integral relationship between the temperature and the free boundary is obtained which is equivalent to the fractional Stefan condition. Moreover, an exact solution of similarity type expressed in terms of Wright functions is also given.

Oscillatory instabilities and dynamics of multi-spike patterns for the one-dimensional Gray-Scott model

2009 ◽  
Vol 20 (2) ◽  
pp. 187-214 ◽  
Author(s):  
WAN CHEN ◽  
MICHAEL J. WARD
Keyword(s):  
Hopf Bifurcation ◽  
Stefan Problem ◽  
Differential Algebraic Equations ◽  
Finite Domain ◽  
Quasi Equilibrium ◽  
Algebraic Equations ◽  
Source Terms ◽  
One Dimensional ◽  
The One ◽  
Oscillatory Instabilities

The dynamics and oscillatory instabilities of multi-spike solutions to the one-dimensional Gray-Scott reaction–diffusion system on a finite domain are studied in a particular parameter regime. In this parameter regime, a formal singular perturbation method is used to derive a novel ODE–PDE Stefan problem, which determines the dynamics of a collection of spikes for a multi-spike pattern. This Stefan problem has moving Dirac source terms concentrated at the spike locations. For a certain subrange of the parameters, this Stefan problem is quasi-steady and an explicit set of differential-algebraic equations characterizing the spike dynamics is derived. By analysing a nonlocal eigenvalue problem, it is found that this multi-spike quasi-equilibrium solution can undergo a Hopf bifurcation leading to oscillations in the spike amplitudes on an O(1) time scale. In another subrange of the parameters, the spike motion is not quasi-steady and the full Stefan problem is solved numerically by using an appropriate discretization of the Dirac source terms. These numerical computations, together with a linearization of the Stefan problem, show that the spike layers can undergo a drift instability arising from a Hopf bifurcation. This instability leads to a time-dependent oscillatory behaviour in the spike locations.

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