Problem with Integral Conditions in the Time Variable for a Sobolev-Type System of Equations with Constant Coefficients

2017 ◽  
Vol 69 (4) ◽  
pp. 621-645 ◽  
Author(s):  
A. M. Kuz’ ◽  
B. I. Ptashnyk
Keyword(s):  
Type System ◽  
Time Variable ◽  
Sobolev Type ◽  
System Of Equations ◽  
Integral Conditions ◽  
Constant Coefficients

scholarly journals Problem for hyperbolic system of equations having constant coefficients with integral conditions with respect to the time variable

2014 ◽  
Vol 6 (2) ◽  
pp. 282-299 ◽  
Author(s):  
A.M. Kuz ◽  
B.Yo. Ptashnyk
Keyword(s):  
Hyperbolic System ◽  
Cartesian Product ◽  
Sufficient Conditions ◽  
Time Variable ◽  
System Of Equations ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Integral Conditions ◽  
Small Denominators ◽  
Constant Coefficients

In a domain specified in the form of a Cartesian product of a segment $\left[0,T\right]$ and the space ${\mathbb R}^{p}$, we study a problem with integral conditions with respect to the time variable for  hyperbolic system with constant coefficients in a class of almost periodic functions in the space variables. A criterion for the unique solvability of this problem and sufficient conditions for the existence of its solution are established. To solve the problem of small denominators arising in the construction of solutions of the posed problem, we use the metric approach.

scholarly journals A multi-species chemotaxis system: Lyapunov functionals, duality, critical mass

2017 ◽  
Vol 29 (3) ◽  
pp. 515-542 ◽  
Author(s):  
N. I. KAVALLARIS ◽  
T. RICCIARDI ◽  
G. ZECCA
Keyword(s):  
Total Population ◽  
Critical Mass ◽  
Type System ◽  
Type Inequality ◽  
Chemical Substance ◽  
Sobolev Type ◽  
Lyapunov Functionals ◽  
Single Chemical ◽  
Chemotaxis System ◽  
Sobolev Type Inequality

We introduce a multi-species chemotaxis type system admitting an arbitrarily large number of population species, all of which are attracted versus repelled by a single chemical substance. The production versus destruction rates of the chemotactic substance by the species is described by a probability measure. For such a model, we investigate the variational structures, in particular, we prove the existence of Lyapunov functionals, we establish duality properties as well as a logarithmic Hardy–Littlewood–Sobolev type inequality for the associated free energy. The latter inequality provides the optimal critical value for the conserved total population mass.

A boundary integral investigation for unsteady modified Helmholtz problems of some other classes of anisotropic functionally graded materials

2021 ◽  
Vol 10 (01) ◽  
pp. 2150005
Author(s):  
Moh. Ivan Azis
Keyword(s):  
Functionally Graded Materials ◽  
Numerical Solutions ◽  
Free Variable ◽  
Variable Coefficients ◽  
Boundary Integral ◽  
Functionally Graded ◽  
Time Variable ◽  
Graded Materials ◽  
The Laplace Transform ◽  
Constant Coefficients

Numerical solutions for a class of unsteady modified Helmholtz problems of anisotropic functionally graded materials are sought. The governing equation which is a variable coefficients equation is transformed to a constant coefficients equation. The time variable is transformed using the Laplace transform. The resulted partial differential equation of constant coefficients and time free variable is then converted to a boundary integral equation, from which boundary element solutions can be obtained. Some examples are considered to verify the accuracy, convergence and consistency of the numerical solutions. The results show that the numerical solutions are accurate, convergent and consistent.

scholarly journals Rothe time-discretization method for a nonlocal problem arising in thermoelasticity

2005 ◽  
Vol 2005 (1) ◽  
pp. 13-28 ◽  
Author(s):  
Nabil Merazga ◽  
Abdelfatah Bouziani
Keyword(s):  
Continuous Dependence ◽  
Nonlocal Problem ◽  
Time Discretization ◽  
Boundary Integral ◽  
Time Variable ◽  
Discretization Method ◽  
Integral Conditions ◽  
Rothe Method ◽  
Semidiscrete Approximation ◽  
The Given

We investigate a model parabolic mixed problem with purely boundary integral conditions arising in the context of thermoelasticity. Using the Rothe method which is based on a semidiscretization of the given problem with respect to the time variable, the questions of existence, uniqueness, and continuous dependence upon data of a weak solution are proved. Moreover, we establish convergence and derive an error estimate for a semidiscrete approximation.

The behavior at infinity of a solution to a Sobolev type system

2007 ◽  
Vol 48 (5) ◽  
pp. 784-797
Author(s):  
L. N. Bondar
Keyword(s):  
Type System ◽  
Sobolev Type

Symmetry and Regularity of Solutions to the Weighted Hardy–Sobolev Type System

2016 ◽  
Vol 16 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Lu Chen ◽  
Zhao Liu ◽  
Guozhen Lu
Keyword(s):  
Lipschitz Continuity ◽  
Type System ◽  
Regularity Of Solutions ◽  
Sobolev Type ◽  
Sobolev Inequalities ◽  
Plane Method ◽  
Integral Forms ◽  
Special Cases ◽  
Formula Group ◽  
Moving Plane

AbstractHardy–Littlewood–Sobolev inequalities and the Hardy–Sobolev type system play an important role in analysis and PDEs. In this paper, we consider the very general weighted Hardy–Sobolev type system$u(x)=\int_{\mathbb{R}^{n}}\frac{1}{|x|^{\tau}|x-y|^{n-\alpha}|y|^{t}}f_{1}(u(y% ),v(y))dy,\quad v(x)=\int_{\mathbb{R}^{n}}\frac{1}{|x|^{t}|x-y|^{n-\alpha}|y|^% {\tau}}f_{2}(u(y),v(y))dy,$where$\displaystyle f_{1}(u(y),v(y))=\lambda_{1}u^{p_{1}}(y)+\mu_{1}v^{q_{1}}(y)+% \gamma_{1}u^{\alpha_{1}}(y)v^{\beta_{1}}(y),$$\displaystyle f_{2}(u(y),v(y))=\lambda_{2}u^{p_{2}}(y)+\mu_{2}v^{q_{2}}(y)+% \gamma_{2}u^{\alpha_{2}}(y)v^{\beta_{2}}(y).$Only the special cases when ${\gamma_{1}=\gamma_{2}=0}$ and one of ${\lambda_{i}}$ and ${\mu_{i}}$ is zero (for both ${i=1}$ and ${i=2}$) have been considered in the literature. We establish the integrability of the solutions to the above Hardy–Sobolev type system and the ${C^{\infty}}$ regularity of solutions to this system away from the origin, which improves significantly the Lipschitz continuity in most works in the literature. Moreover, we also use the moving plane method of [8] in integral forms developed in [6] to prove that each pair ${(u,v)}$ of positive solutions of the above integral system is radially symmetric and strictly decreasing about the origin.

Parallel functional programming on recursively defined data via data-parallel recursion

1999 ◽  
Vol 9 (4) ◽  
pp. 427-462 ◽  
Author(s):  
SUSUMU NISHIMURA ◽  
ATSUSHI OHORI
Keyword(s):  
Parallel Processing ◽  
Functional Programming ◽  
Formal Semantics ◽  
Operational Semantics ◽  
Type System ◽  
Parallel Execution ◽  
System Of Equations ◽  
Functional Language ◽  
Data Parallel ◽  
Evaluation Mechanism

This article proposes a new language mechanism for data-parallel processing of dynamically allocated recursively defined data. Different from the conventional array-based data- parallelism, it allows parallel processing of general recursively defined data such as lists or trees in a functional way. This is achieved by representing a recursively defined datum as a system of equations, and defining new language constructs for parallel transformation of a system of equations. By integrating them with a higher-order functional language, we obtain a functional programming language suitable for describing data-parallel algorithms on recursively defined data in a declarative way. The language has an ML style polymorphic type system and a type sound operational semantics that uniformly integrates the parallel evaluation mechanism with the semantics of a typed functional language. We also show the intended parallel execution model behind the formal semantics, assuming an idealized distributed memory multicomputer.

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