On the Uniqueness of Solution of the Initial Value Problem in Softening Materials

1985 ◽  
Vol 52 (3) ◽  
pp. 649-653 ◽  
Author(s):  
K. C. Valanis
Keyword(s):  
Constitutive Equation ◽  
Unique Solution ◽  
Initial Value Problem ◽  
Constitutive Equations ◽  
Uniqueness Of Solution ◽  
Initial Value ◽  
Material Models ◽  
Softening Behavior ◽  
Softening Material

The initial value problem in finite “softening” material domains is discussed. An inequality that is true of all materials irrespective of their constitution is first established. It is then shown that the solution to this problem is unique when the attending constitutive equation satisfies another inequality, which is characteristic of material models that we call “positive.” A number of constitutive equations that give rise to realistic softening behavior are shown to lead to a unique solution of the initial value problem.

scholarly journals On the n-parameter abstract Cauchy problem

2004 ◽  
Vol 69 (3) ◽  
pp. 383-394
Author(s):  
M. Janfada ◽  
A. Niknam
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Unique Solution ◽  
Initial Value Problems ◽  
Abstract Cauchy Problem ◽  
Linear Operators ◽  
Uniqueness Of Solution ◽  
Initial Value ◽  
Bounded Linear ◽  
Closed Operators

Let Hi(i = 1, 2, …, n), be closed operators in a Banach space X. The generalised initialvalue problem of the abstract Cauchy problem is studied. We show that the uniqueness of solution u: [0, T1] × [0, T2] × … × [0, Tn] → X of this n-abstract Cauchy problem is closely related to C0-n-parameter semigroups of bounded linear operators on X. Also as another application of C0-n-parameter semigroups, we prove that many n-parameter initial value problems cannot have a unique solution for some initial values.

scholarly journals Well-Posedness and Time Regularity for a System of Modified Korteweg-de Vries-Type Equations in Analytic Gevrey Spaces

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 809
Author(s):  
Aissa Boukarou ◽  
Kaddour Guerbati ◽  
Khaled Zennir ◽  
Sultan Alodhaibi ◽  
Salem Alkhalaf
Keyword(s):  
Unique Solution ◽  
Initial Value Problem ◽  
Coupled System ◽  
Well Posedness ◽  
Initial Value ◽  
Gevrey Spaces ◽  
De Vries ◽  
Gevrey Space

Studies of modified Korteweg-de Vries-type equations are of considerable mathematical interest due to the importance of their applications in various branches of mechanics and physics. In this article, using trilinear estimate in Bourgain spaces, we show the local well-posedness of the initial value problem associated with a coupled system consisting of modified Korteweg-de Vries equations for given data. Furthermore, we prove that the unique solution belongs to Gevrey space G σ × G σ in x and G 3 σ × G 3 σ in t. This article is a continuation of recent studies reflected.

UNSTEADY DIFFUSION OF FLUIDS THROUGH SOLIDS UNDERGOING LARGE DEFORMATIONS

1991 ◽  
Vol 01 (03) ◽  
pp. 311-346 ◽  
Author(s):  
L. TAO ◽  
K.R. RAJAGOPAL ◽  
A.S. WINEMAN
Keyword(s):  
Initial Value Problem ◽  
Constitutive Equations ◽  
Large Deformations ◽  
Singular Surface ◽  
Jump Conditions ◽  
Singular Surfaces ◽  
Initial Value ◽  
Diffusion Problems

In this work, we develop a theory for studying the unsteady diffusion problems which involve the motion of singular surfaces within the context of the theory of interacting continua. The theory allows for the possibility of mixtures with different properties on either side of the surface. Constitutive equations have to be postulated for the mixtures and the singular surface. The jump conditions across the singular surface are obtained by extending techniques developed in the case of a single continuum. To evaluate the validity of the theory we solve a typical boundary-initial value problem employing the theory.

scholarly journals Fixed-Point Theorems in Complete Gauge Spaces and Applications to Second-Order Nonlinear Initial-Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Meryam Cherichi ◽  
Bessem Samet ◽  
Calogero Vetro
Keyword(s):  
Fixed Point ◽  
Initial Value Problem ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Fixed Point Theorems ◽  
Second Order ◽  
Uniqueness Of Solution ◽  
Contractive Condition ◽  
Initial Value ◽  
Gauge Space

We establish fixed-point results for mappings and cyclic mappings satisfying a generalized contractive condition in a complete gauge space. Our theorems generalize and extend some fixed-point results in the literature. We apply our obtained results to the study of existence and uniqueness of solution to a second-order nonlinear initial-value problem.

scholarly journals Numerical solutions of nonstandard first order initial value problems

1992 ◽  
Vol 5 (1) ◽  
pp. 69-82 ◽  
Author(s):  
M. Venkatesulu ◽  
P. D. N. Srinivasu
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Unique Solution ◽  
Initial Value Problem ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Initial Value ◽  
First Order ◽  
Physical Phenomena

Differential equations of the form y′=f(t,y,y′), where f is not necessarily linear in its arguments, represent certain physical phenomena and are known for quite some time. The well known Clairut's and Chrystal's equations fall into this category. Earlier, we established the existence of a (unique) solution of the nonstandard initial value problem (NSTD IV P) y′=f(t,y,y′), y(t0)=y0 under certain natural hypotheses on f. In this paper we present some first order convergent numerical methods for finding the approximate solutions of the NST D I V Ps.

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