# STABILITY AND MONOTONICITY OF DIFFERENCE SCHEMES FOR NONLINEAR SCALAR CONSERVATION LAWS AND MULTIDIMENSIONAL QUASI-LINEAR PARABOLIC EQUATIONS

## Related Articles

- Study of Explicit Analytic Solutions for the Nonlinear Coupled Scalar Field Equations. Zhen-ya Yan; Hong-qing Zhang // Applied Mathematics & Mechanics;Jun2001, Vol. 22 Issue 6, p637
By using two different transformations, several types of exact analytic solutions for a class of nonlinear coupled scalar field equation are obtained, which contain soliton solutions, singular solitary wave solutions and triangle function solutions. These results can be applied to other...

- ON THE INSTABILITY OF THE SOLUTIONS OF SOME NONLINEAR VECTOR DIFFERENTIAL EQUATIONS OF FOURTH ORDER. Tunç, Cemil // Miskolc Mathematical Notes;2010, Vol. 11 Issue 2, p191
The purpose of this paper is to obtain some new sufficient conditions which guarantee the instability of the trivial solution of some certain nonlinear vector differential equations of fourth order. Our results improve and include some well known instability results, which were established on...

- A New Instability Result to Nonlinear Vector Differential Equations of Fifth Order. Tunç, Cemil; Karta, Melike // Discrete Dynamics in Nature & Society;2008, p1
By constructing a Lyapunov function, a new instability result is established, which guarantees that the trivial solution of a certain nonlinear vector differential equation of the fifth order is unstable. An example is also given to illustrate the importance of the result obtained. By this way,...

- CONVERGENCE OF THE LAX-FRIEDRICHS SCHEME AND STABILITY FOR CONSERVATION LAWS WITH A DISCONTINUOUS SPACE-TIME DEPENDENT FLUX. KARLSEN, K. H.; TOWERS, J. D. // Chinese Annals of Mathematics;Jul2004, Vol. 25 Issue 3, p287
The authors give the first convergence proof for the Lax-Friedrichs finite difference scheme for non-convex genuinely nonlinear scalar conservation laws of the form \[ u_t+f(k(x,t),u)_x=0\,, \] where the coefficient k(x,t) is allowed to be discontinuous along curves in the (x,t) plane. In...

- Monotonicity of Eventually Positive Solutions for a Second Order Nonlinear Difference Equation. Huiqin Chen; Zhen Jin; Shugui Kang // Discrete Dynamics in Nature & Society;2013, p1
We derive several sufficient conditions for monotonicity of eventually positive solutions on a class of second order perturbed nonlinear difference equation. Furthermore, we obtain a few nonexistence criteria for eventually positive monotone solutions of this equation. Examples are provided to...

- A class of systems of linear Fredholm integral equations of the third kind. Imanaliev, M. I.; Asanov, A.; Asanov, R. // Doklady Mathematics;Apr2011, Vol. 83 Issue 2, p227
The article focuses on the systems of linear Fredholm integral equations of the third kind. It examines various issues regarding the theory of integral equations, particularly the regularization of operators with the construction of Lavrent'ev for solving linear Fredholm integral equations of...

- MONOTONIC SOLUTIONS FOR QUADRATIC INTEGRAL EQUATIONS. Cichoń, Mieczysław; Metwali, Mohamed M.A. // Discussiones Mathematicae: Differential Inclusions, Control & Op;2011, Vol. 31 Issue 2, p157
Using the Darbo fixed point theorem associated with the measure of noncompactness, we establish the existence of monotonic integrable solution on a half-line R+ for a nonlinear quadratic functional integral equation.

- Higher-Order Method for the Solution of a Nonlinear Scalar Equation. Germani, A.; Manes, C.; Palumbo, P.; Sciandrone, M. // Journal of Optimization Theory & Applications;Dec2006, Vol. 131 Issue 3, p347
A new iterative method to find the root of a nonlinear scalar function f is proposed. The method is based on a suitable Taylor polynomial model of order n around the current point xk and involves at each iteration the solution of a linear system of dimension n. It is shown that the coefficient...

- SEMIDISCRETIZATION IN SPACE OF NONLINEAR DEGENERATE PARABOLIC EQUATIONS WITH BLOW-UP OF THE SOLUTIONS. Ishiwata, Tetsuya // Journal of Computational Mathematics;Nov2000, Vol. 18 Issue 6, p571
Semidiscretization in space of nonlinear degenerate parabolic equations of nondivergent form is presented, under zero Dirichlet boundary condition. It is shown that semidiscrete solutions blow up in finite time. In particular, the asymptotic behavior of blowing-up solutions, is discussed precisely.