scholarly journals A method of fundamental solutions for the one-dimensional inverse Stefan problem

2011 ◽  
Vol 35 (9) ◽  
pp. 4367-4378 ◽  
Author(s):  
B.T. Johansson ◽  
D. Lesnic ◽  
T. Reeve
Keyword(s):  
Stefan Problem ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
One Dimensional ◽  
The One

A DISCREPANCY PRINCIPLE FOR THE SOURCE POINTS LOCATION IN USING THE MFS FOR SOLVING THE BHCP

2009 ◽  
Vol 06 (02) ◽  
pp. 181-197 ◽  
Author(s):  
Y. C. HON ◽  
M. LI
Keyword(s):  
Heat Conduction Problem ◽  
Linear Equations ◽  
Fundamental Solutions ◽  
Computational Method ◽  
Method Of Fundamental Solutions ◽  
Discrepancy Principle ◽  
Regularization Technique ◽  
One Dimensional ◽  
Curve Method ◽  
The One

Based on the discrepancy principle, we develop in this paper a new method of choosing the location of source points to solve the backward heat conduction problem (BHCP) by using the method of fundamental solutions (MFS). The standard Tikhonov regularization technique with the L curve method for an optimal regularized parameter is adopted for solving the resultant highly ill-conditioned system of linear equations. Numerical verifications of the proposed computational method are presented for both the one-dimensional and the two-dimensional BHCP.

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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