On the diffusion equation and diffusion wavelets on flat cylinders and the n-torus

Bernstein, Swanhild; Ebert, Svend; Kraußhar, Rolf Sören

doi:10.1002/mma.1369

Cited by 4 publications

(8 citation statements)

References 18 publications

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“…(see, eg, Pinsky, Chapter 4.2 for the case

n = 1

and Bernstein et al for

n \geq 2

). Then, we are in the position to introduce the

q

‐periodic in space layer heat potentials.…”

Section: Introductionmentioning

confidence: 99%

Regularizing properties of space‐periodic layer heat potentials and applications to boundary value problems in periodic domains

Luzzini

2020

Math Methods in App Sciences

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We introduce space-periodic layer heat potentials and we prove some regularizing properties in parabolic Schauder spaces defined on the boundary of infinite parabolic cylinders. Then, we show how to exploit these mapping properties for the space-periodic layer potentials in order to solve two initial-boundary value problems for the heat equation in an unbounded periodic domain. KEYWORDS heat equation, integral equations method, integral operators in parabolic Schauder spaces, periodic domains, space-periodic layer heat potentials MSC CLASSIFICATION 35K05; 31B10; 45A05; 35K20

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“…(see, eg, Pinsky, Chapter 4.2 for the case

n = 1

and Bernstein et al for

n \geq 2

). Then, we are in the position to introduce the

q

‐periodic in space layer heat potentials.…”

Section: Introductionmentioning

confidence: 99%

Regularizing properties of space‐periodic layer heat potentials and applications to boundary value problems in periodic domains

Luzzini

2020

Math Methods in App Sciences

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“…Let us consider the notation

{normalΩ}_{t} = ({}, x : (x, t) \in \underset{normalΩ}{false} \times I) \subset {double-struckR}^{n}

. Following the ideas of , we introduce the following definition: Definition For a function

u \in \overset{\circ}{W_{p}^{1}} (normalΩ)

, we define the forward/backward parabolic Dirac operator as

D_{\pm} u = ((), D + i \partial_{t} \pm α i^{†}) u .

…”

Section: Preliminariesmentioning

confidence: 99%

“…Applying the multiplication rules , we obtain the factorization of a family of time‐dependent operators, that is,

{(D_{\pm})}^{2} u = (- normalΔ \pm α \partial_{t}) u .

From , we have that the fundamental solution of D + reads

E_{+} (x, t) = e_{+} (x, t) D_{+} = \frac{H (t)}{{(4 παt)}^{\frac{n}{2}}} \exp ((), - α \frac{| x |^{2}}{4 t}) ((), - \frac{x}{2 t} + i ((), \frac{| x |^{2}}{4 t^{2}} + \frac{n}{2 t}) + α i^{†}),

where H ( t ) denotes the Heaviside function and

e_{+} (x, t) = αe (x, t) = \frac{α H (t)}{{(4 παt)}^{\frac{n}{2}}} \exp ((), - α \frac{| x |^{2}}{4 t})

is the fundamental solution of −Δ+ α ∂ t . The kernels and can be used to solve special boundary value problems.…”

Section: Preliminariesmentioning

confidence: 99%

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Time‐dependent operators on some non‐orientable projective orbifolds

Kraußhar

Rodrigues

Vieira

2015

Math Methods in App Sciences

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In this paper, we present an explicit construction for the fundamental solution of the heat operator, the Schrödinger operator, and related first-order parabolic Dirac operators on a class of some conformally flat non-orientable orbifolds. More concretely, we treat a class of projective cylinders and tori where we can study parabolic monogenic sections with values in different pin bundles. We present integral representation formulas together with some elementary tools of harmonic analysis that enable us to solve boundary value problems on these orbifolds. 5305-5319 5305 R. S. KRAUßHAR, M. M. RODRIGUES AND N. VIEIRA example, 'upper' and 'lower' parts with each other. This is translated into an additional symmetry structure, which actually leads to a singularity-in general, we are now dealing with orbifolds in this context. We explain how we can construct associated Cauchy kernels for the parabolic Dirac operator, as well as heat and Schrödinger kernels for this class of orbifolds. Similarly to the oriented case, we can consider distinct pin bundles in the non-oriented case. We show how arbitrary pin sections can be represented by these kernels. We also set up integral representation formulas for the sections in the kernel of the parabolic Dirac (resp. heat and Schrödinger) operator on these orbifolds.This paper also provides a continuation of [11] where explicit formulas for the Cauchy and Green's kernels for the elliptic Dirac and Laplace operators have been developed.The structure of the paper is as follows. In Section 2, we recall some basic facts about Clifford algebras and about factorization and regularization of time-dependent operators. In Section 3, we define the spin and the pin structures that are necessary for the development of our work. In the following section, we study the non-orientable projective counterparts of cylinders and tori. We start by presenting the geometric context. Then we construct the Cauchy and heat kernels, as well as the corresponding integral formulas. Section 5 is dedicated to the study of the Schrödinger operator. Here, we start by studying the inhomogeneous Schrödinger equation in orbifolds in terms of a proper Teodorescu and Cauchy operators. In this last section, we finally present a Hodge-type decomposition for the regularized case, and we will study the behavior of our result when the regularization parameter tends to zero.

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“…Denoting the Gegenbauer coefficients of the kernel by Kp (m) the coefficients of the corresponding wavelet of order zero are given by ^'p(fe) = -(a(p))^^^^p(fe)(see [5], [8]). This fact allows us to prove the following theorem.…”

Section: Ipf= [ Vkp{){y)f{y)hl{y)da{y)mentioning

confidence: 99%