On the behavior of the fundamental solution of the heat equation with variable coefficients

2010 ◽  
Vol 20 (2) ◽  
pp. 431-455 ◽  
Author(s):  
S. R. S. Varadhan
Keyword(s):  
Fundamental Solution ◽  
Heat Equation ◽  
Variable Coefficients

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 Kesten's bound for subexponential densities on the real line and its multi-dimensional analogues

2018 ◽  
Vol 50 (2) ◽  
pp. 373-395 ◽  
Author(s):  
Dmitri Finkelshtein ◽  
Pasha Tkachov
Keyword(s):  
Fundamental Solution ◽  
Heat Equation ◽  
Probability Density ◽  
Real Line ◽  
The Real ◽  
Probability Densities ◽  
Tail Asymptotic

Abstract We study the tail asymptotic of subexponential probability densities on the real line. Namely, we show that the n-fold convolution of a subexponential probability density on the real line is asymptotically equivalent to this density multiplied by n. We prove Kesten's bound, which gives a uniform in n estimate of the n-fold convolution by the tail of the density. We also introduce a class of regular subexponential functions and use it to find an analogue of Kesten's bound for functions on ℝd. The results are applied to the study of the fundamental solution to a nonlocal heat equation.

Asymptotic behaviour of the fundamental solution to ∂ u /∂ t = — ( — ∆) m u

1993 ◽  
Vol 441 (1912) ◽  
pp. 423-432 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Positive Integer ◽  
Fundamental Solution ◽  
Heat Equation ◽  
Asymptotic Behaviour ◽  
Infinite Number ◽  
Fourier Integral ◽  
Number Of Zeros

An asymptotic expansion is derived for the Fourier integral f ^ ( x ) = 1 ( 2 π ) n / 2 ∫ R n exp ( − | q | 2 m + i x ⋅ q ) d q , x ε R n as | x | →∞, where m is a positive integer. From this, it is deduced that the fundamental solution to the ‘heat’ equation ∂ u / ∂ t = − ( − Δ ) m u has an infinite number of zeros tending to infinity.

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
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