scholarly journals Sharp boundary trace inequalities

2014 ◽  
Vol 144 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Giles Auchmuty
Keyword(s):  
Weight Function ◽  
Sharp Boundary ◽  
Bounded Region ◽  
Boundary Trace ◽  
Special Cases ◽  
Sobolev Functions ◽  
Trace Inequalities

This paper describes sharp inequalities for the trace of Sobolev functions on the boundary of a bounded region Ω ⊂ ℝN. The inequalities bound (semi-)norms of the boundary trace by certain norms of the function, its gradient on the region and by two specific constants κρ and κΩ associated with the domain and a weight function, respectively. These inequalities are sharp in that there exist functions for which equality holds. Explicit inequalities in some special cases when the region is a ball, or the region between two balls, are evaluated.

scholarly journals On dual integral equations with Hankel kernel and an arbitrary weight function

1986 ◽  
Vol 9 (2) ◽  
pp. 293-300 ◽  
Author(s):  
C. Nasim
Keyword(s):  
Weight Function ◽  
Integral Equations ◽  
Single Equation ◽  
Dual Integral Equations ◽  
Fredholm Type ◽  
Operational Procedure ◽  
Mellin Transforms ◽  
Special Cases ◽  
General Order ◽  
Arbitrary Weight

In this paper we deal with dual integral equations with an arbitrary weight function and Hankel kernels of distinct and general order. We propose an operational procedure, which depends on exploiting the properties of the Mellin transforms, and readily reduces the dual equations to a single equation. This then can be inverted by the Hankel inversion to give us an equation of Fredholm type, involving the unknown function. Most of the known results are then derived as special cases of our general result.

Numerical solutions and analysis of diffusion for new generalized fractional Burgers equation

2013 ◽  
Vol 16 (3) ◽  
Author(s):  
Yufeng Xu ◽  
Om Agrawal
Keyword(s):  
Finite Difference ◽  
Diffusion Process ◽  
Weight Function ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
Weight Functions ◽  
Linear Recurrence ◽  
Scale Function ◽  
Special Cases ◽  
Wide Range

AbstractIn this paper, numerical solutions of Burgers equation defined by using a new Generalized Time-Fractional Derivative (GTFD) are discussed. The numerical scheme uses a finite difference method. The new GTFD is defined using a scale function and a weight function. Many existing fractional derivatives are the special cases of it. A linear recurrence relationship for the numerical solutions of the resulting system of linear equations is found via finite difference approach. Burgers equations with different fractional orders and coefficients are computed which show that this numerical method is simple and effective, and is capable of solving the Burgers equation accurately for a wide range of viscosity values. Furthermore, we study the influence of the scale and the weight functions on the diffusion process of Burgers equation. Numerical simulations illustrate that a scale function can stretch or contract the diffusion on the time domain, while a weight function can change the decay velocity of the diffusion process.

Orthogonal Polynomials With Weight Function (1 - x)α( l + x)β + Mδ(x + 1) + Nδ(x - 1)

1984 ◽  
Vol 27 (2) ◽  
pp. 205-214 ◽  
Author(s):  
Tom H. Koornwinder
Keyword(s):  
Differential Equation ◽  
Weight Function ◽  
Orthogonal Polynomials ◽  
Fourth Order ◽  
Hypergeometric Functions ◽  
Order Differential Equation ◽  
Second Order Differential Equations ◽  
Special Cases ◽  
Delta Functions ◽  
Fourth Order Differential Equation

AbstractWe study orthogonal polynomials for which the weight function is a linear combination of the Jacobi weight function and two delta functions at 1 and — 1. These polynomials can be expressed as 4F3 hypergeometric functions and they satisfy second order differential equations. They include Krall’s Jacobi type polynomials as special cases. The fourth order differential equation for the latter polynomials is derived in a more simple way.

scholarly journals INTEGRAL TRANSFORMATION IN A PARTIALLY BOUNDED REGION WITH A RADIAL THERMAL FLOW

2018 ◽  
Vol 13 (6) ◽  
pp. 89-96
Author(s):  
E. M. Kartashov
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Green's Functions ◽  
Green’S Functions ◽  
Integral Transformation ◽  
Bounded Region ◽  
Integral Transformations ◽  
Functional Relations ◽  
Special Cases ◽  
Unsteady Heat Conduction

A mathematical theory is developed for constructing integral transformations in a partially bounded region with a radial heat flow - a massive body bounded from the inside by a cylindrical cavity. Constructed: an integral transformation, the image of the operator on the right side of the equation of unsteady heat conduction, the inversion formula for the image of the desired function. The proposed approach favorably differs from the classical theory of differential equations of mathematical physics for the construction of generalized integral transformations based on the eigenfunctions of the corresponding singular Sturm-Liouville problems. The developed method is based on the operational solution of the initial boundary problems of unsteady heat conduction with an initial function of a general form L2(r0,∞) belonging to the r > r0 region and homogeneous boundary conditions and is associated with the calculation of the Riemann-Mellin contour integrals from images containing various combinations of modified Bessel functions. At the same time, for the above-mentioned region, the method of Green's functions was developed by constructing integral representations of analytical solutions of the first, second and third boundary value problems through inhomogeneities in the initial formulation of the problem (boundary conditions, source function in the initial equation). Mathematical models for finding the corresponding Green's functions are formulated, and functional relations of all three Green functions included in the presented integral formula are written out with the help of the developed theory of integral transformations. The functional relations constructed in the article can be used when considering numerous special cases of practical thermal physics. The specific possible applications of the presented results in many areas of science and technology are given.

scholarly journals Matrix Elements of the Boltzmann Collision Operator in a Basis Determined by an Anisotropic Maxwellian Weight Function including Drift

1980 ◽  
Vol 33 (2) ◽  
pp. 449 ◽  
Author(s):  
Kailash Kumar
Keyword(s):  
Weight Function ◽  
Cross Sections ◽  
Collision Operator ◽  
Weight Functions ◽  
Anisotropic Pressure ◽  
Matrix Elements ◽  
Methods Of Calculation ◽  
Special Cases ◽  
The Matrix ◽  
Boltzmann Collision Operator

The matrix elements of the linear Boltzmann collision operator are calculated in a Burnett-function basis determined by a weight function which itself describes a velocity distribution with a net drift and an anisotropic pressure (or temperature) tensor. Three different methods of calculation are described, leading to three different types of formulae. Two of these involve infinite summations, while the third involves only finite sums, but at the cost of greater complications in the summands and the integrals over cross sections. Both elastic and inelastic collisions are treated. Special cases arising from particular choices of the parameters in the weight functions are pointed out. The structure of the formulae is illustrated by means of diagrams. The work is a contribution towards establishing efficient methods of calculation based upon a better understanding of the matrix elements in such bases.

Eigenvalue asymptotics for an elliptic boundary problem

2007 ◽  
Vol 137 (2) ◽  
pp. 281-302 ◽  
Author(s):  
M. Faierman ◽  
M. Möller
Keyword(s):  
Boundary Conditions ◽  
Asymptotic Behaviour ◽  
Weight Function ◽  
Boundary Problem ◽  
Spectral Parameter ◽  
Elliptic Boundary ◽  
Bounded Region ◽  
Eigenvalue Asymptotics ◽  
Elliptic Boundary Problem ◽  
Parameter Condition

We consider an elliptic boundary problem in a bounded region Ω ⊂ ℝn wherein the spectral parameter is multiplied by a real-valued weight function with the property that it, together with its reciprocal, is essentially bounded in Ω. The problem is considered under limited smoothness assumptions and under an ellipticity with parameter condition. Then, fixing our attention upon the operator induced on L2(Ω) by the boundary problem under null boundary conditions, we establish results pertaining to the asymptotic behaviour of the eigenvalues of this operator under weaker smoothness assumptions than have hitherto been supposed.

scholarly journals Boundary trace inequalities and rearrangements

2008 ◽  
Vol 105 (1) ◽  
pp. 241-265 ◽  
Author(s):  
Andrea Cianchi ◽  
Ron Kerman ◽  
Luboš Pick
Keyword(s):  
Boundary Trace ◽  
Trace Inequalities

scholarly journals Midpoint Inequalities in Fractional Calculus Defined Using Positive Weighted Symmetry Function Kernels

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 550
Author(s):  
Pshtiwan Othman Mohammed ◽  
Hassen Aydi ◽  
Artion Kashuri ◽  
Y. S. Hamed ◽  
Khadijah M. Abualnaja
Keyword(s):  
Fractional Calculus ◽  
Weight Function ◽  
Convex Functions ◽  
Fractional Integral ◽  
Integral Kernel ◽  
Type Inequality ◽  
Special Cases ◽  
Symmetry Function ◽  
Increasing Function

The aim of our study is to establish, for convex functions on an interval, a midpoint version of the fractional HHF type inequality. The corresponding fractional integral has a symmetric weight function composed with an increasing function as integral kernel. We also consider a midpoint identity and establish some related inequalities based on this identity. Some special cases can be considered from our main results. These results confirm the generality of our attempt.

scholarly journals On Riemann problem in weighted Smirnov classes with general weight

2021 ◽  
Vol 25 (1) ◽  
pp. 33-56
Author(s):  
Bilal Bilalov ◽  
Aysel Guliyeva ◽  
Sabina Sadigova
Keyword(s):  
Weight Function ◽  
Unbounded Domains ◽  
Orthogonality Condition ◽  
Measurable Coefficient ◽  
Type Condition ◽  
Power Type ◽  
Special Cases ◽  
General Weight ◽  
Smirnov Classes ◽  
Piecewise Continuous

Weighted Smirnov classes in bounded and unbounded domains are defined in this work. Nonhomogeneous Riemann problems with a measurable coefficient whose argument is a piecewise continuous function are considered in these classes. A Muckenhoupt type condition is imposed on the weight function and the orthogonality condition is found for the solvability of nonhomogeneous problem in weighted Smirnov classes, and the formula for the index of the problem is derived. Some special cases with power type weight function are also considered,and conditions on degeneration order are found.

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