scholarly journals Borel summability of the heat equation with variable coefficients

2012 ◽  
Vol 252 (4) ◽  
pp. 3076-3092 ◽  
Author(s):  
O. Costin ◽  
H. Park ◽  
Y. Takei
Keyword(s):  
Heat Equation ◽  
Variable Coefficients ◽  
Borel Summability

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

scholarly journals On the Borel summability of divergent solutions of the heat equation

1999 ◽  
Vol 154 ◽  
pp. 1-29 ◽  
Author(s):  
D. A. Lutz ◽  
M. Miyake ◽  
R. Schäfke
Keyword(s):  
Heat Equation ◽  
Heat Kernel ◽  
Formal Solution ◽  
Cauchy Data ◽  
Borel Summability ◽  
Stripe Domain ◽  
Integral Expression ◽  
Irregular Singular Point ◽  
The Cauchy Problem ◽  
Continuation Property

AbstractIn recent years, the theory of Borel summability or multisummability of divergent power series of one variable has been established and it has been proved that every formal solution of an ordinary differential equation with irregular singular point is multisummable. For partial differential equations the summability problem for divergent solutions has not been studied so well, and in this paper we shall try to develop the Borel summability of divergent solutions of the Cauchy problem of the complex heat equation, since the heat equation is a typical and an important equation where we meet diveregent solutions. In conclusion, the Borel summability of a formal solution is characterized by an analytic continuation property together with its growth condition of Cauchy data to infinity along a stripe domain, and the Borel sum is nothing but the solution given by the integral expression by the heat kernel. We also give new ways to get the heat kernel from the Borel sum by taking a special Cauchy data.

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