Some existence results for conservation laws with source-term

2002 ◽  
Vol 25 (13) ◽  
pp. 1149-1160 ◽  
Author(s):  
Jo�o-Paulo Dias ◽  
Philippe G. LeFloch
Keyword(s):  
Conservation Laws ◽  
Source Term ◽  
Existence Results

scholarly journals OPTIMAL DECAY RATES TO CONSERVATION LAWS WITH DIFFUSION-TYPE TERMS OF REGULARITY-GAIN AND REGULARITY-LOSS

2012 ◽  
Vol 22 (07) ◽  
pp. 1250012 ◽  
Author(s):  
RENJUN DUAN ◽  
LIZHI RUAN ◽  
CHANGJIANG ZHU
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Source Term ◽  
Decay Rates ◽  
Heat Diffusion ◽  
Time Behavior ◽  
Time Decay ◽  
Diffusion Type ◽  
Linear Solution ◽  
Optimal Decay Rates

We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term related to an index s ∈ ℝ over the whole space ℝn for any spatial dimension n ≥ 1. Here, the diffusion-type source term behaves as the usual diffusion term over the low frequency domain while it admits on the high frequency part a feature of regularity-gain and regularity-loss for s < 1 and s > 1, respectively. For all s ∈ ℝ, we not only obtain the Lp–Lq time-decay estimates on the linear solution semigroup but also establish the global existence and optimal time-decay rates of small-amplitude classical solutions to the nonlinear Cauchy problem. In the case of regularity-loss, the time-weighted energy method is introduced to overcome the weakly dissipative property of the equation. Moreover, the large-time behavior of solutions asymptotically tending to the heat diffusion waves is also studied. The current results have general applications to several concrete models arising from physics.

scholarly journals ON DUALITY, SYMMETRY AND SYMMETRY LOST IN SOLID MECHANICS

2014 ◽  
Vol 11 (03) ◽  
pp. 1343004
Author(s):  
HUY DUONG BUI
Keyword(s):  
Fracture Mechanics ◽  
Conservation Laws ◽  
Boundary Value Problems ◽  
Solid Mechanics ◽  
Source Term ◽  
Boundary Value ◽  
Duality Symmetry ◽  
Classical Formulation ◽  
True Path ◽  
Adjoint Variables

The paper recalls the concept of duality in mathematics and extends it to solid mechanics. One important application of duality is to restore some symmetry between current fields and their adjoint ones. This leads to many alternative schemes for numerical analyses, different from the classical one as used in classical formulation of boundary value problems (finite element method). Usually, conservation laws in fracture mechanics make use of the current fields, displacement and stress. Many conservation laws of this type are not free of the source term. Consequently, one cannot derive path-independent integrals for use in fracture mechanics. The introduction of variables and dual or adjoint variables leads to true path-independent integrals. Duality also introduces some anti-symmetry in current fields and adjoint ones for some boundary value problems. The symmetry is lost between fields and adjoint fields. The last notion enables us to derive new variational formulation on dual subspaces and to exactly solve inverse problems for detecting cracks and volume defects.

scholarly journals Picard Type Iterative Scheme with Initial Iterates in Reverse Order for a Class of Nonlinear Three Point BVPs

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Mandeep Singh ◽  
Amit K. Verma
Keyword(s):  
Source Term ◽  
Iterative Scheme ◽  
Upper And Lower Solutions ◽  
Sufficient Conditions ◽  
Existence Results ◽  
Monotone Iterative Technique ◽  
Reverse Order ◽  
Monotone Iterative Method ◽  
Iterative Technique ◽  
Monotone Iterative

We consider the following class of three point boundary value problemy′′(t)+f(t,y)=0,0<t<1,y′(0)=0,y(1)=δy(η), whereδ>0,0<η<1, the source termf(t,y)is Lipschitz and continuous. We use monotone iterative technique in the presence of upper and lower solutions for both well-order and reverse order cases. Under some sufficient conditions, we prove some new existence results. We use examples and figures to demonstrate that monotone iterative method can efficiently be used for computation of solutions of nonlinear BVPs.

Symmetries and conservation laws of a damped Boussinesq equation

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640012 ◽  
Author(s):  
María Luz Gandarias ◽  
María Rosa
Keyword(s):  
Differential Equations ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Source Term ◽  
Boussinesq Equation ◽  
Lie Symmetries ◽  
Multiplier Method ◽  
Independent Source ◽  
Damped Equation

In this work, we consider a damped equation with a time-independent source term. We derive the classical Lie symmetries admitted by the equation as well as the reduced ordinary differential equations. We also present some exact solutions. Conservation laws for this equation are constructed by using the multiplier method.

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