The Resolvent Operator

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Laplace Transform ◽  
Heat Kernel ◽  
Resolvent Operator ◽  
Contour Integration ◽  
Resolvent Kernel ◽  
The Laplace Transform ◽  
Elliptic Estimates ◽  
The Resolvent Operator

This chapter describes the construction of a resolvent operator using the Laplace transform of a parametrix for the heat kernel and a perturbative argument. In the equation (μ‎-L) R(μ‎) f = f, R(μ‎) is a right inverse for (μ‎-L). In Hölder spaces, these are the natural elliptic estimates for generalized Kimura diffusions. The chapter first constructs the resolvent kernel using an induction over the maximal codimension of bP, and proves various estimates on it, along with corresponding estimates for the solution operator for the homogeneous Cauchy problem. It then considers holomorphic semi-groups and uses contour integration to construct the solution to the heat equation, concluding with a discussion of Kimura diffusions where all coefficients have the same leading homogeneity.

Holder Estimates for Higher Dimensional Corner Models

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Cauchy Problem ◽  
Laplace Transform ◽  
Heat Kernel ◽  
Resolvent Operator ◽  
Higher Dimensions ◽  
Euclidean Case ◽  
Higher Dimensional ◽  
The Laplace Transform ◽  
The Resolvent Operator ◽  
Hölder Estimates

This chapter establishes Hölder space estimates for higher dimensional corner model problems. It first explains the homogeneous Cauchy problem before estimating the solution of the inhomogeneous problem in a n-dimensional corner. It then reduces the proof of an estimate in higher dimensions to the estimation of a product of 1-dimensional integrals. Using the “1-variable-at-a-time” method, the chapter proves the higher dimensional estimates in several stages by considering the “pure corner” case where m = 0, and then turns to the Euclidean case, where n = 0. It also discusses the resolvent operator as the Laplace transform of the heat kernel.

Holder Estimates for Euclidean Models

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Cauchy Problem ◽  
Heat Kernel ◽  
Resolvent Operator ◽  
Second Derivative ◽  
Kernel Estimates ◽  
First Derivative ◽  
Essential Components ◽  
The Laplace Transform ◽  
The Resolvent Operator ◽  
Hölder Estimates

This chapter presents the Hölder space estimates for Euclidean model problems. It first considers the homogeneous Cauchy problem and the inhomogeneous problem before defining the resolvent operator as the Laplace transform of the heat kernel. It then describes the 1-dimensional kernel estimates that form essential components of the proofs of the Hölder estimates for the general model problems; these include basic kernel estimates, first derivative estimates, and second derivative estimates. The proofs of these estimates are elementary. The chapter concludes by proving estimates on the resolvent and investigating the off-diagonal behavior of the heat kernel in many variables.

scholarly journals The inversion of a transform related to the Laplace transform and to heat conduction

1964 ◽  
Vol 4 (1) ◽  
pp. 1-14 ◽  
Author(s):  
David V. Widder
Keyword(s):  
Heat Conduction ◽  
Fundamental Solution ◽  
Heat Equation ◽  
Laplace Transform ◽  
Physical Interpretation ◽  
Integral Transform ◽  
The Laplace Transform ◽  
Image Position

In a recent paper [7] the author considered, among other things, the integral transform where is the fundamental solution of the heat equation There we gave a physical interpretation of the transform (1.1). Here we shall choose a slightly different interpretation, more convenient for our present purposes. If then u(O, t) = f(t). That is, the function f(t) defined by equation (1.1) is the temperature at the origin (x = 0) of an infinite bar along the x-axis t seconds after it was at a temperature defined by the equation .

Holder Estimates for General Models

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Cauchy Problem ◽  
General Model ◽  
Resolvent Operator ◽  
Heat Equations ◽  
Second Derivative ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Long Time ◽  
The Resolvent Operator ◽  
Hölder Estimates

This chapter presents the Hölder estimates for general model problems. It first estimates solutions to heat equations for both the homogeneous Cauchy problem and the inhomogeneous problem, obtaining first and second derivative estimates in the latter case, before discussing a general result describing the off-diagonal and long-time behavior of the solution kernel for the general model. It also states a proposition summarizing the properties of the resolvent operator as an operator on the Hölder spaces. In contrast to the case of the heat equation, there is no need to assume that the data has compact support in the x-variables to prove estimates when k > 0.

scholarly journals On the Lauwerier formulation of the temperature field problem in oil strata

2005 ◽  
Vol 2005 (10) ◽  
pp. 1577-1588 ◽  
Author(s):  
Lyubomir Boyadjiev ◽  
Ognian Kamenov ◽  
Shyam Kalla
Keyword(s):  
Porous Medium ◽  
Temperature Field ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Laplace Transform ◽  
Special Functions ◽  
Integral Form ◽  
Field Problem ◽  
Fractional Heat Equation ◽  
The Laplace Transform

The paper is concerned with the fractional extension of the Lauwerier formulation of the problem related to the temperature field description in a porous medium (sandstone) saturated with oil (strata). The boundary value problem for the fractional heat equation is solved by means of the Caputo differintegration operatorD∗(α)of order0<α≤1and the Laplace transform. The solution is obtained in an integral form, where the integrand is expressed in terms of a convolution of two special functions of Wright type.

Influence of Moving Heat Source on Skin Tissue in the Context of Two-Temperature Caputo–Fabrizio Heat Transport Law

2019 ◽  
Vol 11 (02) ◽  
pp. 2050002 ◽  
Author(s):  
Abhik Sur ◽  
Sudip Mondal ◽  
M. Kanoria
Keyword(s):  
Heat Source ◽  
Heat Equation ◽  
Laplace Transform ◽  
Heat Transport ◽  
Biological Tissues ◽  
Moving Heat Source ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
Temperature Parameter ◽  
Two Temperature

Modeling and understanding heat transport and temperature variations within biological tissues and body organs are key issues in medical thermal therapeutic applications, such as hyperthermia cancer treatment. In this analysis, the bio-heat equation has been studied in the context of a new formulation using Caputo–Fabrizio (CF) heat transport law. The heat equation for this problem involving the Three-Phase (3P)-lag model under two-temperature theory. The Laplace transform technique is implemented to solve the governing equations. The influences of the CF order, the two-temperature parameter and moving heat source velocity on the temperature of skin tissues have been precisely investigated. The numerical inversion of the Laplace transform is carried out using the Zakian method. The numerical outcomes of conductive temperature and thermodynamic temperatures have been represented graphically. Excellent predictive capability is demonstrated for identification of appropriate procedure to select different CF order to reach effective heating in hyperthermia treatment. Significant effect of thermal therapy is reported due to the effect of CF order, the two-temperature parameter and the velocity of moving heat source as well.

Holder Estimates for the 1-dimensional Model Problems

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Heat Kernel ◽  
Resolvent Operator ◽  
Heat Kernels ◽  
Dimensional Model ◽  
Kernel Estimates ◽  
Schauder Estimates ◽  
Diffusion Operator ◽  
Higher Dimensional ◽  
The Resolvent Operator ◽  
Hölder Estimates

This chapter establishes Hölder space estimates for the 1-dimensional model problems. It gives a detailed treatment of the 1-dimensional case, in part because all of the higher dimensional estimates are reduced to estimates on heat kernels for the 1-dimensional model problems. It also presents the proof of parabolic Schauder estimates for the generalized Kimura diffusion operator using the explicit formula for the heat kernel, along with standard tools of analysis. Finally, it considers kernel estimates for degenerate model problems, explains how Hölder estimates are obtained for the 1-dimensional model problems, and describes the properties of the resolvent operator.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj&gt; 0 for eachj&gt; 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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