Results 1 
3 of
3
Asymptotic Laplace transforms and evolution equations
 Advances in Partial Differential Equations, Math. Topics
, 1998
"... Cauchy Problem This section is concerned with applications of asymptotic Laplace transform methods to abstract Cauchy problems u 0 (t) = Au(t); t 2 I ; u(0) = x; (ACP ) where A is a closed 14 linear operator with domain D(A) and range in a Banach space X , and either I = [0; T ] or I = [0; 1) ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
Cauchy Problem This section is concerned with applications of asymptotic Laplace transform methods to abstract Cauchy problems u 0 (t) = Au(t); t 2 I ; u(0) = x; (ACP ) where A is a closed 14 linear operator with domain D(A) and range in a Banach space X , and either I = [0; T ] or I = [0; 1). In this section, k will always be a regularizing function whose Laplace transform has no real zeros 15 ; i.e., k 2 L 1 loc ([0; 1); C ); lim sup !1 1 ln j k()j = 0; and k() 6= 0 for all ? !: Since A is closed and d dt (k u)(t) = (k u 0 )(t) + k(t)u(0), the convolution of k with (ACP ) yields (k u) 0 = k u 0 +k(t)x = A(k u)+k(t)x or k u = A(1 k u)+(1 k)(t)x. Thus, a function u 2 C [k] (I ; X) is called a kgeneralized (mild) solution of (ACP ) if v = k u 2 C 0 ([0; 1); X) 16 solves v(t) = A Z t 0 v(s) ds + (1 k)(t)(x); t 2 I : (3:1) Conversely, any v 2 C 0 (I ; X) satisfying (3:1) is called a kregularized solution of (ACP ). 17 One of the main to...
On the regularization and stabilization of approximation schemes for C0semigroups
 PARTIAL DIFFERENTIAL EQUATIONS AND SPECTRAL THEORY
, 2001
"... Many temporal discretization methods for linear evolution equations converge uniformly on compact time intervals at the rate 1/nα only for sufficiently smooth initial data. It is shown that these methods can be regularized such that the new schemes converge ‘in the average’ at the rate 1/nα for all ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
Many temporal discretization methods for linear evolution equations converge uniformly on compact time intervals at the rate 1/nα only for sufficiently smooth initial data. It is shown that these methods can be regularized such that the new schemes converge ‘in the average’ at the rate 1/nα for all initial data. Examples given include the CrankNicholson scheme and the alternating direction implicit method.