Existence and uniqueness of solutions to a concentrated capacity problem with change of phase

1990 ◽  
Vol 1 (4) ◽  
pp. 339-351 ◽  
Author(s):  
Daniele Andreucci
Keyword(s):  
Heat Equation ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Weak Sense ◽  
Uniqueness Of Solutions ◽  
Limiting Case ◽  
Multidimensional Domain ◽  
Diffusion Problems

A concentrated capacity problem is posed for the heat equation in a multidimensional domain. In the concentrated capacity (i.e. in a portion of the boundary of the domain) a change of phase takes place, and a Stefan-like problem is posed. This scheme has been introduced in the literature as the formal limiting case of a certain class of diffusion problems.Our main result is a theorem of continuous dependence of the solution on the data. It is also used to prove the existence of the solution (in a weak sense), assuming only integrability of the data. The solution is found as the limit of the solutions of the approximating problems.

EXISTENCE OF SOLUTIONS FOR A CONTINUOUS MULTIGRAIN MODEL FOR POLYMERIZATION

2000 ◽  
Vol 10 (08) ◽  
pp. 1263-1276
Author(s):  
DANIELE ANDREUCCI ◽  
ANTONIO FASANO ◽  
RICCARDO RICCI
Keyword(s):  
Existence And Uniqueness ◽  
Existence Of Solutions ◽  
The Other ◽  
Uniqueness Of Solutions ◽  
Small Times ◽  
Catalyst Pellets ◽  
Multigrain Model ◽  
Diffusion Problems

We prove the existence and uniqueness of solutions, for small times, for a mathematical scheme modeling the Ziegler–Natta process of polymerization. The model consists, essentially, of two diffusion problems at two different space scales, one relative to the microscopical catalyst pellets, the other to the macroscopical aggregate of those pellets. The coupling between the two scales is of nonstandard nature.

QUANTUM INTEGRAL EQUATIONS OF VOLTERRA TYPE WITH GENERALIZED OPERATOR-VALUED KERNELS

2000 ◽  
Vol 03 (04) ◽  
pp. 505-517 ◽  
Author(s):  
ZHIYUAN HUANG ◽  
CAISHI WANG ◽  
XIANGJUN WANG
Keyword(s):  
Integral Equation ◽  
Explicit Expression ◽  
Integral Equations ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Uniqueness Of Solutions ◽  
Volterra Type ◽  
Quantum Integral ◽  
Generalized Operator

Quantum integral equation of Volterra type with generalized operator-valued kernel is introduced. Existence and uniqueness of solutions are established, explicit expression of the solution is given, the continuity, continuous dependence on free terms and other properties of the solution are proved.

scholarly journals On the evolution equation with a dynamic Hardy-type potential

2021 ◽  
Author(s):  
Jann-Long Chern ◽  
Gyeongha Hwang ◽  
Jin Takahashi ◽  
Eiji Yanagida
Keyword(s):  
Integral Equation ◽  
Singular Point ◽  
Heat Equation ◽  
Existence And Uniqueness ◽  
Singular Potential ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Hardy Type ◽  
Radial Heat ◽  
Type Potential

Abstract Motivated by the celebrated paper of Baras and Goldstein (1984), we study the heat equation with a dynamic Hardy-type singular potential. In particular, we are interested in the case where the singular point moves in time. Under appropriate conditions on the potential and initial value, we show the existence, non-existence and uniqueness of solutions, and obtain a sharp lower and upper bound near the singular point. Proofs are given by using solutions of the radial heat equation, some precise estimates for an equivalent integral equation and the comparison principle.

scholarly journals Prevalence of backward stochastic differential equations with unique solution

2004 ◽  
Vol 2004 (2) ◽  
pp. 123-136
Author(s):  
K. Bahlali ◽  
B. Mezerdi ◽  
Y. Ouknine
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Baire Category ◽  
Backward Stochastic Differential Equations ◽  
Successive Approximations ◽  
Uniqueness Of Solutions ◽  
Almost All ◽  
Continuous Dependence Of Solutions

We prove that in the sense of Baire category, almost all backward stochastic differential equations (BSDEs) with bounded and continuous coefficient have the properties of existence and uniqueness of solutions as well as the continuous dependence of solutions on the coefficient and the L2-convergence of their associated successive approximations.

scholarly journals To solving the fractionally loaded heat equation

2021 ◽  
Vol 101 (1) ◽  
pp. 65-77
Author(s):  
M.T. Kosmakova ◽  
◽  
S.A. Iskakov ◽  
L.Zh. Kasymova ◽  
◽  
...  
Keyword(s):  
Integral Equation ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Fractional Derivative ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Continuous Functions ◽  
Uniqueness Of Solutions ◽  
Loaded Term ◽  
Differential Part

In this paper we consider a boundary value problem for a fractionally loaded heat equation in the class of continuous functions. Research methods are based on an approach to the study of boundary value problems, based on their reduction to integral equations. The problem is reduced to a Volterra integral equation of the second kind by inverting the differential part. We also carried out a study the limit cases for the fractional derivative order of the term with a load in the heat equation of the boundary value problem. It is shown that the existence and uniqueness of solutions to the integral equation depends on the order of the fractional derivative in the loaded term.

Quasi-Linearisation Methods for a Non-Linear Heat Equation with Functional Dependence

2000 ◽  
Vol 7 (1) ◽  
pp. 97-116 ◽  
Author(s):  
Henryk Leszczyński
Keyword(s):  
Heat Equation ◽  
Unknown Function ◽  
Existence And Uniqueness ◽  
Functional Dependence ◽  
Sufficient Conditions ◽  
Convergent Sequence ◽  
Successive Approximations ◽  
Uniqueness Of Solutions ◽  
Volterra Type ◽  
Non Linear

Abstract We consider a heat equation with the non-linear right-hand side which depends on certain Volterra-type functionals acting on the unknown function and on its gradient. We give some natural sufficient conditions for the existence and uniqueness of solutions to this equation. The solution is obtained as a limit of a fast convergent sequence of successive approximations obtained by the quasi-linearisation method.

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