scholarly journals On the generalized heat-equation

1984 ◽  
Vol 27 (3) ◽  
pp. 261-273
Author(s):  
C. Nasim ◽  
B. D. Aggarwala
Keyword(s):  
Heat Equation ◽  
Fixed Number ◽  
Fixed Positive Number ◽  
Generalized Heat Equation ◽  
Image Position

The general heat equation is defined aswhere v is a fixed positive number and α is a fixed number. If v = α = 0, then (1.1) reduces to the ordinary heat equationwhere u(x,t) is regarded as the temperature at a point x at time t, in an infinite insulated rod extended along the x-axis in the xt-plane. If we set , then (1.1) becomes

Series Expansions of Generalized Temperature Functions in N Dimensions

1966 ◽  
Vol 18 ◽  
pp. 794-802 ◽  
Author(s):  
Deborah Tepper Haimo
Keyword(s):  
Heat Equation ◽  
Polynomial Solution ◽  
Higher Dimensions ◽  
Series Expansions ◽  
Fixed Positive Number ◽  
Generalized Heat Equation ◽  
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The generalized heat equation is given by1.1where Δxf(x) = f″(x) + (2v/x)f′(x), v a fixed positive number. In a recent paper (5), the author established criteria for representing solutions of (1.1) in either the form1.2or1.3where Pn,v(x, t) is t he polynomial solution of (1.1) given explicitly by1.4and Wn,v(x, t) is its Appell transform; cf. (1). Our object is to generalize these results by extending them to higher dimensions. D. V. Widder (8) studied the problem for the ordinary heat equation.

scholarly journals On quantum theory in terms of white noise

1977 ◽  
Vol 68 ◽  
pp. 21-34 ◽  
Author(s):  
T. Hida ◽  
L. Streit
Keyword(s):  
Quantum Theory ◽  
Probability Measure ◽  
Heat Equation ◽  
Correlation Functions ◽  
Time Parameter ◽  
Imaginary Time ◽  
Dynamical Models ◽  
Quantum Dynamical ◽  
Reverse Transition ◽  
Generalized Heat Equation

It has often been pointed out that a much more manageable structure is obtained from quantum theory if the time parameter t is chosen imaginary instead of real. Under a replacement of t by i·t the Schrödinger equation turns into a generalized heat equation, time ordered correlation functions transform into the moments of a probability measure, etc. More recently this observation has become extremely important for the construction of quantum dynamical models, where criteria were developed by E. Nelson, by K. Osterwalder and R. Schrader and others [8] which would permit the reverse transition to real time after one has constructed an imaginary time (“Euclidean”) model.

scholarly journals Some Remarks on a Recent Paper by Borch

1961 ◽  
Vol 1 (5) ◽  
pp. 265-272 ◽  
Author(s):  
Paul Markham Kahn
Keyword(s):  
Distribution Function ◽  
Random Variable ◽  
Fixed Number ◽  
Point Of View ◽  
Optimum Amount ◽  
Measurable Transformation ◽  
The Difference ◽  
Image Position ◽  
Stop Loss ◽  
Condition B

In his recent paper, “An Attempt to Determine the Optimum Amount of Stop Loss Reinsurance”, presented to the XVIth International Congress of Actuaries, Dr. Karl Borch considers the problem of minimizing the variance of the total claims borne by the ceding insurer. Adopting this variance as a measure of risk, he considers as the most efficient reinsurance scheme that one which serves to minimize this variance. If x represents the amount of total claims with distribution function F (x), he considers a reinsurance scheme as a transformation of F (x). Attacking his problem from a different point of view, we restate and prove it for a set of transformations apparently wider than that which he allows.The process of reinsurance substitutes for the amount of total claims x a transformed value Tx as the liability of the ceding insurer, and hence a reinsurance scheme may be described by the associated transformation T of the random variable x representing the amount of total claims, rather than by a transformation of its distribution as discussed by Borch. Let us define an admissible transformation as a Lebesgue-measurable transformation T such thatwhere c is a fixed number between o and m = E (x). Condition (a) implies that the insurer will never bear an amount greater than the actual total claims. In condition (b), c represents the reinsurance premium, assumed fixed, and is equal to the expected value of the difference between the total amount of claims x and the total retained amount of claims Tx borne by the insurer.

Optimization and α-Disfocality for Ordinary Differential Equations

1985 ◽  
Vol 37 (2) ◽  
pp. 310-323 ◽  
Author(s):  
M. Essén
Keyword(s):  
Distribution Function ◽  
Continuous Function ◽  
Differential Equations ◽  
Convex Hull ◽  
Ordinary Differential Equations ◽  
Lebesgue Measure ◽  
Fixed Positive Number ◽  
Image Position

For f ∊ L−1(0, T), we define the distribution functionwhere T is a fixed positive number and |·| denotes Lebesgue measure. Let Φ:[0, T] → [0, m] be a nonincreasing, right continuous function. In an earlier paper [3], we discussed the equation(0.1)when the coefficient q was allowed to vary in the classWe were in particular interested in finding the supremum and infimum of y(T) when q was in or in the convex hull Ω(Φ) of (see below).

On backward self-similar blow-up solutions to a supercritical semilinear heat equation

2010 ◽  
Vol 140 (4) ◽  
pp. 821-831 ◽  
Author(s):  
Noriko Mizoguchi
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Blow Up ◽  
Radial Solution ◽  
Similar Solution ◽  
Semilinear Heat Equation ◽  
Blowing Up ◽  
Self Similar Solution ◽  
Self Similar ◽  
Image Position

We are concerned with a Cauchy problem for the semilinear heat equationthen u is called a backward self-similar solution blowing up at t = T. Let pS and pL be the Sobolev and the Lepin exponents, respectively. It was shown by Mizoguchi (J. Funct. Analysis257 (2009), 2911–2937) that k ≡ (p − 1)−1/(p−1) is a unique regular radial solution of (P) if p > pL. We prove that it remains valid for p = pL. We also show the uniqueness of singular radial solution of (P) for p > pS. These imply that the structure of radial backward self-similar blow-up solutions is quite simple for p ≥ pL.

XL.—The Linear Difference-differential Equation with Constant Coefficients

1949 ◽  
Vol 62 (4) ◽  
pp. 387-393 ◽  
Author(s):  
E. M. Wright
Keyword(s):  
Differential Equation ◽  
Fixed Number ◽  
Initial Interval ◽  
Constant Coefficients ◽  
Image Position

SummaryUnder the condition that one at least of the leading coefficients amn, a0n differs from zero, the equationhas as solution a series convergent for all x greater (or all x less) than a fixed number. The coefficients of the various terms in the series are expressed in terms of the arbitrary values of the solution and its first n derivatives in an initial interval of appropriate length.This paper was assisted in publication by a grant from the Carnegie Trust for the Universities of Scotland.

A Hausdorff measure classification of polar sets for the heat equation

1985 ◽  
Vol 97 (2) ◽  
pp. 325-344 ◽  
Author(s):  
S. J. Taylor ◽  
N. A. Watson
Keyword(s):  
Heat Equation ◽  
Hausdorff Measure ◽  
Image Position

Our main purpose is to give criteria for determining which subsets of Rn+1 are polar relative to the heat equation

scholarly journals On the strong maximum principle for parabolic differential equations

1986 ◽  
Vol 29 (1) ◽  
pp. 93-96 ◽  
Author(s):  
Wolfgang Walter
Keyword(s):  
Maximum Principle ◽  
Differential Equations ◽  
Heat Equation ◽  
Nonlinear Equations ◽  
Strong Maximum Principle ◽  
Parabolic Differential Equations ◽  
Carry Over ◽  
Image Position

In a recent paper [2], D. Colton has given a new proof for the strong maximum principle with regard to the heat equation ut = Δu. His proof depends on the analyticity (in x) of solutions. For this reason it does not carry over to the equationor to more general equations. But in order to tread mildly nonlinear equations such asut = Δu + f(u) which are important in many applications, it is essential to have the strong maximum principle at least for equation (*). It should also be said that this proof uses nontrivial facts about the heat equation.

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