scholarly journals Monotone Difference Schemes for Weakly Coupled Elliptic and Parabolic Systems

2017 ◽  
Vol 17 (2) ◽  
pp. 287-298 ◽  
Author(s):  
Piotr Matus ◽  
Francisco Gaspar ◽  
Le Minh Hieu ◽  
Vo Thi Kim Tuyen
Keyword(s):  
A Priori Estimate ◽  
Parabolic Systems ◽  
Uniform Norm ◽  
Difference Schemes ◽  
Scalar Case ◽  
Quasilinear Parabolic Equations ◽  
Positivity Condition ◽  
Weakly Coupled ◽  
Weakly Coupled System ◽  
Vector Difference

AbstractThe present paper is devoted to the development of the theory of monotone difference schemes, approximating the so-called weakly coupled system of linear elliptic and quasilinear parabolic equations. Similarly to the scalar case, the canonical form of the vector-difference schemes is introduced and the definition of its monotonicity is given. This definition is closely associated with the property of non-negativity of the solution. Under the fulfillment of the positivity condition of the coefficients, two-side estimates of the approximate solution of these vector-difference equations are established and the important a priori estimate in the uniform normCis given.

scholarly journals Growth of solutions of weakly coupled parabolic systems and Laplace's equation

1986 ◽  
Vol 41 (3) ◽  
pp. 391-403
Author(s):  
N. A. Watson
Keyword(s):  
Lebesgue Measure ◽  
Nonnegative Solution ◽  
Coupled System ◽  
Parabolic Systems ◽  
Parabolic Partial Differential Equations ◽  
Poisson Integrals ◽  
Weakly Coupled ◽  
Weakly Coupled System ◽  
Coupled Parabolic Systems ◽  
Growth Of Solutions

AbstractLet ui(x, t) be the ith component of a nonnegative solution of a weakly coupled system of second-order, linear, parabolic partial differential equations, for x ∈ Rn and 0 < t < T. We obtain lower estimates, near t = 0, for the Lebesgue measure of the set of x for which exceeds 1. Related results for Poisson integrals on a half-space are also described, some applications are given, and interesting comparisons emerge.

scholarly journals MONOTONE ECONOMICAL SCHEMES FOR QUASILINEAR PARABOLIC EQUATIONS

2002 ◽  
Vol 7 (2) ◽  
pp. 207-216
Author(s):  
N. V. Dzenisenko ◽  
A. P. Matus ◽  
P. P. Matus
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Quasilinear Parabolic Equation ◽  
Uniform Norm ◽  
Additive Scheme ◽  
Quasilinear Parabolic Equations ◽  
The Maximum Principle ◽  
Priori Estimates ◽  
The Difference ◽  
Quasilinear Parabolic

In order to approximate a multidimensional quasilinear parabolic equation with unlimited nonlinearity the economical vector‐additive scheme is constructed. It is shown that its solution satisfies the maximum principle and, hence, the scheme is monotone. The proof is based on the equivalence of the vector‐additive scheme and the scheme of summarized approximation (locally one‐dimensional scheme). The a priori estimates of the difference solution in the uniform norm are obtained.

Sign in / Sign up

Export Citation Format

Share Document