The Two-Order Quasi-Linear Singular Perturbed Problems with Infinite Initial Conditions

2013 ◽  
Vol 353-356 ◽  
pp. 3248-3250
Author(s):  
Han Xu
Keyword(s):  
Asymptotic Analysis ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Uniqueness Of The Solution

In this paper a kind of quasi-linear singular perturbed problems with infinite initial conditions is investigated. The existence and uniqueness of the solution are proved.Therefore the asymptotic analysis of the solution can be obtained.

scholarly journals Existence and Uniqueness of Solution for a Class of Nonlinear Fractional Order Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Azizollah Babakhani ◽  
Dumitru Baleanu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Uniqueness Of Solution ◽  
Fractional Order Differential Equations ◽  
Banach Contraction ◽  
Uniqueness Of The Solution

We discuss the existence and uniqueness of solution to nonlinear fractional order ordinary differential equations(Dα-ρtDβ)x(t)=f(t,x(t),Dγx(t)),t∈(0,1)with boundary conditionsx(0)=x0,  x(1)=x1or satisfying the initial conditionsx(0)=0,  x′(0)=1, whereDαdenotes Caputo fractional derivative,ρis constant,1<α<2,and0<β+γ≤α. Schauder's fixed-point theorem was used to establish the existence of the solution. Banach contraction principle was used to show the uniqueness of the solution under certain conditions onf.

scholarly journals Discrete Mittag-Leffler Functions in Linear Fractional Difference Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
Jan Čermák ◽  
Tomáš Kisela ◽  
Luděk Nechvátal
Keyword(s):  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Linear Difference Equation ◽  
Discrete Fractional Calculus ◽  
Initial Value ◽  
Fractional Difference ◽  
Uniqueness Of The Solution ◽  
Discrete Analogues ◽  
Fractional Type ◽  
Special Time

This paper investigates some initial value problems in discrete fractional calculus. We introduce a linear difference equation of fractional order along with suitable initial conditions of fractional type and prove the existence and uniqueness of the solution. Then the structure of the solutions space is discussed, and, in a particular case, an explicit form of the general solution involving discrete analogues of Mittag-Leffler functions is presented. All our observations are performed on a special time scale which unifies and generalizes ordinary difference calculus andq-difference calculus. Some of our results are new also in these particular discrete settings.

scholarly journals STABILITY BY HOMOGENIZATION OF THERMOVISCOPLASTIC PROBLEMS

2010 ◽  
Vol 20 (09) ◽  
pp. 1591-1616 ◽  
Author(s):  
NICOLAS CHARALAMBAKIS ◽  
FRANCOIS MURAT
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Strain Rate ◽  
Existence And Uniqueness ◽  
Body Force ◽  
Initial Conditions ◽  
Rate Sensitivity ◽  
Effective Coefficients ◽  
Uniqueness Of The Solution ◽  
Theoretical Results

In this paper, we study the homogenization of the system of partial differential equations describing the quasistatic shearing of heterogeneous thermoviscoplastic materials. We first present the existence and uniqueness of the solution of the above system. We then define "stable by homogenization" models as the models where the equations in both the heterogeneous problems and the homogenized one are of the same form. Finally we show that the model with non-oscillating strain-rate sensitivity which is submitted to steady boundary shearing and body force, is stable by homogenization. In this model, the homogenized (effective) coefficients depend on the initial conditions and on the boundary shearing and body force. Those theoretical results are illustrated by one numerical example.

EXISTENCE AND UNIQUENESS OF THE SOLUTION TO THE EDGE PROBLEM IN A TWO-DIMENSIONAL REACTIVE BOUNDARY LAYER

1995 ◽  
Vol 05 (01) ◽  
pp. 1-27 ◽  
Author(s):  
ERIC BOILLAT
Keyword(s):  
Boundary Layer ◽  
Existence And Uniqueness ◽  
Initial Conditions ◽  
Tangential Velocity ◽  
Two Dimensional ◽  
Differential Problem ◽  
Boundary Layer Equations ◽  
Uniqueness Of The Solution ◽  
Thermodynamical Functions ◽  
Physical Quantities

We prove an existence and uniqueness theorem for the ordinary differential problem which characterizes the profiles of the different physical quantities at the edge of two-dimensional reactive boundary layer. The main difficulties to be circumvented are the nonlinearities due to the different thermodynamical functions involved in the reactive boundary layer equations and the degeneracy caused by the natural initial conditions, where the tangential velocity has to vanish. We conclude by making some mathematical considerations about the relations that exist between the reactive boundary layer equations and the corresponding equations which describe the boundary layer in chemical equilibrium.

Hypothesis on the Solvability of Parabolic Equations with nonlocal Condition

2002 ◽  
Vol 7 (1) ◽  
pp. 93-104 ◽  
Author(s):  
Mifodijus Sapagovas
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Nonlocal Condition ◽  
Nonlocal Conditions ◽  
Numerical Algorithms ◽  
Uniqueness Of The Solution

Numerous and different nonlocal conditions for the solvability of parabolic equations were researched in many articles and reports. The article presented analyzes such conditions imposed, and observes that the existence and uniqueness of the solution of parabolic equation is related mainly to ”smallness” of functions, involved in nonlocal conditions. As a consequence the hypothesis has been made, stating the assumptions on functions in nonlocal conditions are related to numerical algorithms of solving parabolic equations, and not to the parabolic equation itself.

scholarly journals Correctness conditions for high-order differential equations with unbounded coefficients

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kordan N. Ospanov
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Order Linear Differential Equation ◽  
Unbounded Coefficients ◽  
Weighted Norms ◽  
Uniqueness Of The Solution ◽  
Linear Differential

AbstractWe give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

scholarly journals Solutions for impulsive fractional pantograph differential equation via generalized anti-periodic boundary condition

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Idris Ahmed ◽  
Poom Kumam ◽  
Jamilu Abubakar ◽  
Piyachat Borisut ◽  
Kanokwan Sitthithakerngkiet
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Fixed Point ◽  
Fractional Differential Equation ◽  
Existence And Uniqueness ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Fixed Point Theorems ◽  
Uniqueness Of The Solution ◽  
Fractional Differential

Abstract This study investigates the solutions of an impulsive fractional differential equation incorporated with a pantograph. This work extends and improves some results of the impulsive fractional differential equation. A differential equation of an impulsive fractional pantograph with a more general anti-periodic boundary condition is proposed. By employing the well-known fixed point theorems of Banach and Krasnoselskii, the existence and uniqueness of the solution of the proposed problem are established. Furthermore, two examples are presented to support our theoretical analysis.

scholarly journals Lidstone–Euler Second-Type Boundary Value Problems: Theoretical and Computational Tools

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Francesco Aldo Costabile ◽  
Maria Italia Gualtieri ◽  
Anna Napoli
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Interpolation Problem ◽  
Boundary Value ◽  
Computational Tools ◽  
Numerical Examples ◽  
Odd Order ◽  
Uniqueness Of The Solution ◽  
Type Boundary

AbstractGeneral nonlinear high odd-order differential equations with Lidstone–Euler boundary conditions of second type are treated both theoretically and computationally. First, the associated interpolation problem is considered. Then, a theorem of existence and uniqueness of the solution to the Lidstone–Euler second-type boundary value problem is given. Finally, for a numerical solution, two different approaches are illustrated and some numerical examples are included to demonstrate the validity and applicability of the proposed algorithms.

scholarly journals On Existence Theorems to Symmetric Functional Set-Valued Differential Equations

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1219
Author(s):  
Marek T. Malinowski
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Existence Theorems ◽  
Type Theorem ◽  
Integral Representations ◽  
Uniqueness Of The Solution ◽  
Nonempty Compact ◽  
Picard Type Theorem ◽  
Convex Subsets

In this paper, we consider functional set-valued differential equations in their integral representations that possess integrals symmetrically on both sides of the equations. The solutions have values that are the nonempty compact and convex subsets. The main results contain a Peano type theorem on the existence of the solution and a Picard type theorem on the existence and uniqueness of the solution to such equations. The proofs are based on sequences of approximations that are constructed with appropriate Hukuhara differences of sets. An estimate of the magnitude of the solution’s values is provided as well. We show the closeness of the unique solutions when the equations differ slightly.

NEW MODELS IN MICROPOLAR FLUID AND THEIR APPLICATION TO LUBRICATION

2005 ◽  
Vol 15 (03) ◽  
pp. 343-374 ◽  
Author(s):  
GUY BAYADA ◽  
NADIA BENHABOUCHA ◽  
MICHÈLE CHAMBAT
Keyword(s):  
Boundary Condition ◽  
Friction Coefficient ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Micropolar Fluid ◽  
Reynolds Equation ◽  
Slip Boundary Condition ◽  
Slip Boundary ◽  
New Models ◽  
Uniqueness Of The Solution

A thin micropolar fluid with new boundary conditions at the fluid-solid interface, linking the velocity and the microrotation by introducing a so-called "boundary viscosity" is presented. The existence and uniqueness of the solution is proved and, by way of asymptotic analysis, a generalized micropolar Reynolds equation is derived. Numerical results show the influence of the new boundary conditions for the load and the friction coefficient. Comparisons are made with other works retaining a no slip boundary condition.

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