Tangential derivatives and higher-order regularizing properties of the double layer heat potential

Analysis ◽  
2019 ◽  
Vol 38 (4) ◽  
pp. 167-193 ◽  
Author(s):  
Massimo Lanza de Cristoforis ◽  
Paolo Luzzini
Keyword(s):  
Integral Operator ◽  
Explicit Formula ◽  
Double Layer ◽  
Higher Order ◽  
Time Variable ◽  
High Order ◽  
Hölder Spaces ◽  
Heat Potential ◽  
Holder Spaces ◽  
Derivatives Of

Abstract We prove an explicit formula for the tangential derivatives of the double layer heat potential. By exploiting such a formula, we prove the validity of a regularizing property for the integral operator associated to the double layer heat potential in spaces of functions with high-order derivatives in parabolic Hölder spaces defined on the boundary of parabolic cylinders which are unbounded in the time variable.

scholarly journals Ball Convergence of an Efficient Eighth Order Iterative Method Under Weak Conditions

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 260 ◽  
Author(s):  
Janak Sharma ◽  
Ioannis Argyros ◽  
Sunil Kumar
Keyword(s):  
Iterative Method ◽  
Iterative Methods ◽  
Higher Order ◽  
High Order ◽  
Convergence Order ◽  
Eighth Order ◽  
First Derivative ◽  
Ball Convergence ◽  
Derivatives Of

The convergence order of numerous iterative methods is obtained using derivatives of a higher order, although these derivatives are not involved in the methods. Therefore, these methods cannot be used to solve equations with functions that do not have such high-order derivatives, since their convergence is not guaranteed. The convergence in this paper is shown, relying only on the first derivative. That is how we expand the applicability of some popular methods.

scholarly journals Moduli of continuity, functional spaces,\break and elliptic boundary value problems. The full regularity spaces Cα0,λ(Ω̅)

2018 ◽  
Vol 7 (1) ◽  
pp. 15-34 ◽  
Author(s):  
Hugo Beirão da Veiga
Keyword(s):  
Modulus Of Continuity ◽  
Elliptic Boundary ◽  
Continuous Functions ◽  
Second Order ◽  
Hölder Spaces ◽  
Elliptic Boundary Value ◽  
New Class ◽  
Holder Spaces ◽  
Derivatives Of ◽  
Full Regularity

AbstractLet {\boldsymbol{L}} be a second order uniformly elliptic operator, and consider the equation {\boldsymbol{L}u=f} under the boundary condition {u=0}. We assume data f in generical subspaces of continuous functions {D_{\overline{\omega}}} characterized by a given modulus of continuity{\overline{\omega}(r)}, and show that the second order derivatives of the solution u belong to functional spaces {D_{\widehat{\omega}}}, characterized by a modulus of continuity{\widehat{\omega}(r)} expressed in terms of {\overline{\omega}(r)}. Results are optimal. In some cases, as for Hölder spaces, {D_{\widehat{\omega}}=D_{\overline{\omega}}}. In this case we say that full regularity occurs. In particular, full regularity occurs for the new class of functional spaces {C^{0,\lambda}_{\alpha}(\overline{\Omega})} which includes, as a particular case, the classical Hölder spaces {C^{0,\lambda}(\overline{\Omega})=C^{0,\lambda}_{0}(\overline{\Omega})}. Few words, concerning the possibility of generalizations and applications to non-linear problems, are expended at the end of the introduction and also in the last section.

Differentiation of Mass and Stiffness Matrices for High Order Sensitivity Calculations in Finite Element-Based Equilibrium Problems

1993 ◽  
Vol 115 (4) ◽  
pp. 829-832 ◽  
Author(s):  
J. E. Bernard ◽  
S. K. Kwon ◽  
J. A. Wilson
Keyword(s):  
Finite Element ◽  
Design Variable ◽  
Equilibrium Problems ◽  
Higher Order ◽  
High Order ◽  
Stiffness Matrices ◽  
Higher Order Terms ◽  
Fixed Boundaries ◽  
Cubic Polynomials ◽  
Derivatives Of

Extension of sensitivity methods to include higher order terms depends on the ability to compute higher order derivatives of the mass and stiffness matrices. This paper presents a method based on the use of cubic polynomials to fit mass and stiffness matrices across a range of interest of the design variable. The method is illustrated through an example which uses Pade´ approximants to expand the solution to a statics problem. The design variable is the thickness of one part of a plate with fixed boundaries. The solution gives a very good approximation over fivefold change in the value of the design variable.

scholarly journals PROPERTIES OF INTEGRALS WHICH HAVE THE TYPE OF DERIVATIVES OF VOLUME POTENTIALS FOR DEGENERATED $\overrightarrow{2\lowercase{b}}$ - PARABOLIC EQUATION OF KOLMOGOROV TYPE

2021 ◽  
Vol 9 (2) ◽  
pp. 7-21
Author(s):  
V. Dron' ◽  
I. Medyns'kyi
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equation ◽  
Homogeneous Equation ◽  
Hölder Spaces ◽  
Kolmogorov Type ◽  
Estimates Of Solutions ◽  
The Cauchy Problem ◽  
The Given ◽  
Holder Spaces ◽  
Derivatives Of

In weight Holder spaces it is studied the smoothness of integrals, which have the structure and properties of derivatives of volume potentials which generated by fundamental solution of the Cauchy problem for degenerated $\overrightarrow{2b}$-parabolic equation of Kolmogorov type. The coefficients in this equation depend only on the time variable. Special distances and norms are used for constructing of the weight Holder spaces. The results of the paper can be used for establishing of the correct solvability of the Cauchy problem and estimates of solutions of the given non-homogeneous equation in corresponding weight Holder spaces.

scholarly journals Properties of integrals which have the type of derivatives of volume potentials for one ultraparabolic arbitrary order equation

2019 ◽  
Vol 11 (2) ◽  
pp. 268-280
Author(s):  
V.S. Dron' ◽  
S.D. Ivasyshen ◽  
I.P. Medyns'kyi
Keyword(s):  
Cauchy Problem ◽  
Arbitrary Order ◽  
Homogeneous Equation ◽  
Fundamental Solutions ◽  
Order Equation ◽  
Hölder Spaces ◽  
The Cauchy Problem ◽  
Weighted Hölder Spaces ◽  
Holder Spaces ◽  
Derivatives Of

In weighted Hölder spaces it is studied the smoothness of integrals, which have the structure and properties of derivatives of volume potentials which generated by fundamental solutions of the Cauchy problem for one ultraparabolic arbitrary order equation of the Kolmogorov type. The coefficients in this equation depend only on the time variable. Special distances and norms are used for constructing of the weighted Hölder spaces. The results of the paper can be used for establishing of the correct solvability of the Cauchy problem and estimates of solutions of the given non-homogeneous equation in corresponding weighted Hölder spaces.

Some properties of the Hermite polynomials

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Feng Qi ◽  
Bai-Ni Guo
Keyword(s):  
Explicit Formula ◽  
Generating Function ◽  
Generating Functions ◽  
Recurrence Relations ◽  
Hermite Polynomials ◽  
Higher Order ◽  
Bell Polynomials ◽  
Explicit Formulas ◽  
Faà Di Bruno Formula ◽  
Derivatives Of

Abstract In this paper, by the Faà di Bruno formula and properties of Bell polynomials of the second kind, the authors reconsider the generating functions of Hermite polynomials and their squares, find an explicit formula for higher-order derivatives of the generating function of Hermite polynomials, and derive explicit formulas and recurrence relations for Hermite polynomials and their squares.

Differentiation of Mass and Stiffness Matrices for High Order Sensitivity Calculations in Finite Element-Based Equilibrium Problems

1990 ◽  
Author(s):  
J. E. Bernard ◽  
S. K. Kwon ◽  
J. A. Wilson
Keyword(s):  
Finite Element ◽  
Design Variable ◽  
Equilibrium Problems ◽  
Higher Order ◽  
High Order ◽  
Stiffness Matrices ◽  
Higher Order Terms ◽  
Fixed Boundaries ◽  
Cubic Polynomials ◽  
Derivatives Of

Abstract Extension of sensitivity methods to include higher order terms depends on the ability to compute higher order derivatives of the mass and stiffness matrices. This paper presents a method based on the use of cubic polynomials to fit mass and stiffness matrices across a range of interest of the design variable. The method is illustrated through an example which uses Padé approximants to expand the solution to a statics problem. The design variable is the thickness of one part of a plate with fixed boundaries. The solution gives a very good approximation over five fold change in the value of the design variable.

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