scholarly journals Heterogeneous Diffusion, Stability Analysis, and Solution Profiles for a MHD Darcy–Forchheimer Model

Mathematics ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 20
Author(s):  
José Luis Díaz ◽  
Saeed Rahman ◽  
Juan Miguel García-Haro
Keyword(s):  
Stability Analysis ◽  
Existence And Uniqueness ◽  
Stability Study ◽  
The Other ◽  
Spatial Structures ◽  
Spatial Derivative ◽  
Uniqueness Of Solutions ◽  
Forchheimer Model ◽  
Inner Region ◽  
Forchheimer Flow

In the presented analysis, a heterogeneous diffusion is introduced to a magnetohydrodynamics (MHD) Darcy–Forchheimer flow, leading to an extended Darcy–Forchheimer model. The introduction of a generalized diffusion was proposed by Cohen and Murray to study the energy gradients in spatial structures. In addition, Peletier and Troy, on one side, and Rottschäfer and Doelman, on the other side, have introduced a general diffusion (of a fourth-order spatial derivative) to study the oscillatory patterns close the critical points induced by the reaction term. In the presented study, analytical conceptions to a proposed problem with heterogeneous diffusions are introduced. First, the existence and uniqueness of solutions are provided. Afterwards, a stability study is presented aiming to characterize the asymptotic convergent condition for oscillatory patterns. Dedicated solution profiles are explored, making use of a Hamilton–Jacobi type of equation. The existence of oscillatory patterns may induce solutions to be negative, close to the null equilibrium; hence, a precise inner region of positive solutions is obtained.

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.

scholarly journals Existence and uniqueness of solutions to a first-order differential equation via fixed point theorem in orthogonal metric space

2019 ◽  
pp. 123
Author(s):  
Madjid Eshaghi Gordji ◽  
Hasti Habibi
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Metric Space ◽  
Existence And Uniqueness ◽  
Order Differential Equation ◽  
The Other ◽  
Uniqueness Theorems ◽  
Uniqueness Of Solutions ◽  
First Order

In this paper, among the other things, we show that the solution of the first-orderdifferential equation is a fixed point of an integral operator from an orthogonal metric space into itself. This approach provides a new proof of the classical existence and uniqueness theorems of solutions to a first-order differential equation.

scholarly journals Critical perturbations of elliptic operators by lower order terms

2021 ◽  
Author(s):  
◽  
Jose Luis Luna-Garcia
Keyword(s):  
Fundamental Solution ◽  
Elliptic Equations ◽  
Existence And Uniqueness ◽  
The Other ◽  
Layer Potentials ◽  
Lebesgue Spaces ◽  
Uniqueness Of Solutions ◽  
Linear Elliptic Operator ◽  
Lower Order Terms ◽  
Associated Solution

In this work we study issues of existence and uniqueness of solutions of certain boundary value problems for elliptic equations in the upper half-space. More specifically we treat the Dirichlet, Neumann, and Regularity problems for the general second order, linear, elliptic operator under a smallness assumption on the coefficients in certain critical Lebesgue spaces. Our results are perturbative in nature, asserting that if a certain operator L[subscript 0] has good properties (as far as boundedness and invertibility of certain associated solution operators), then the same is true for L[subscript 1], whenever the coefficients of these two operators are close in certain L[subscript p] spaces. Our approach is through the theory of layer potentials, though the lack of good estimates for solutions of L [equals] 0 force us to use a more abstract construction of these objects, as opposed to the more classical definition through the fundamental solution. On the other hand, these more general objects suggest a wider range of applications for these techniques. The results contained in this thesis were obtained in collaboration with Simon Bortz, Steve Hofmann, Svitlana Mayboroda, and Bruno Poggi. The resulting publications can be found in [BHL+a] and [BHL+b].

Integral Representations for the Solution of Dynamic Bending of a Plate with Displacement-Traction Boundary Data

2003 ◽  
Vol 10 (3) ◽  
pp. 467-480
Author(s):  
Igor Chudinovich ◽  
Christian Constanda
Keyword(s):  
Existence And Uniqueness ◽  
Boundary Data ◽  
Boundary Integral Equations ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Boundary Integral ◽  
Integral Representations ◽  
Transverse Shear Deformation ◽  
Uniqueness Of Solutions ◽  
Nonstationary Boundary

Abstract The existence of distributional solutions is investigated for the time-dependent bending of a plate with transverse shear deformation under mixed boundary conditions. The problem is then reduced to nonstationary boundary integral equations and the existence and uniqueness of solutions to the latter are studied in appropriate Sobolev spaces.

scholarly journals Existence and uniqueness of positive solutions for nonlinear fractional mixed problems

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Alberto Cabada ◽  
Om Kalthoum Wanassi
Keyword(s):  
Existence And Uniqueness ◽  
Fixed Point Theory ◽  
Mixed Boundary ◽  
Point Theory ◽  
Uniqueness Of Solutions ◽  
Nonlinear Part ◽  
Banach Contraction ◽  
Fractional Differential ◽  
Exhaustive Study ◽  
The Banach Contraction Principle

Abstract This paper is devoted to study the existence and uniqueness of solutions of a one parameter family of nonlinear Riemann–Liouville fractional differential equations with mixed boundary value conditions. An exhaustive study of the sign of the related Green’s function is carried out. Under suitable assumptions on the asymptotic behavior of the nonlinear part of the equation at zero and at infinity, and by application of the fixed point theory of compact operators defined in suitable cones, it is proved that there exists at least one solution of the considered problem. Moreover, the method of lower and upper solutions is developed and the existence of solutions is deduced by a combination of both techniques. In particular cases, the Banach contraction principle is used to ensure the uniqueness of solutions.

scholarly journals Stability Analysis and Existence of Solutions for a Modified SIRD Model of COVID-19 with Fractional Derivatives

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1431
Author(s):  
Bilal Basti ◽  
Nacereddine Hammami ◽  
Imadeddine Berrabah ◽  
Farid Nouioua ◽  
Rabah Djemiat ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Biological Models ◽  
Uniqueness Of Solutions ◽  
Caputo Fractional Derivatives ◽  
Single Form ◽  
Stability Results

This paper discusses and provides some analytical studies for a modified fractional-order SIRD mathematical model of the COVID-19 epidemic in the sense of the Caputo–Katugampola fractional derivative that allows treating of the biological models of infectious diseases and unifies the Hadamard and Caputo fractional derivatives into a single form. By considering the vaccine parameter of the suspected population, we compute and derive several stability results based on some symmetrical parameters that satisfy some conditions that prevent the pandemic. The paper also investigates the problem of the existence and uniqueness of solutions for the modified SIRD model. It does so by applying the properties of Schauder’s and Banach’s fixed point theorems.

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