scholarly journals Existence and Uniqueness of BVPs Defined on Semi-Infinite Intervals: Insight from the Iterative Transformation Method

2021 ◽  
Vol 26 (1) ◽  
pp. 18
Author(s):  
Riccardo Fazio
Keyword(s):  
Existence And Uniqueness ◽  
Numerical Test ◽  
Transformation Method ◽  
Uniqueness Of Solutions ◽  
Boundary Problems ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Uniqueness Question ◽  
Infinite Intervals ◽  
Definition Of

This work is concerned with the existence and uniqueness of boundary value problems defined on semi-infinite intervals. These kinds of problems seldom admit exactly known solutions and, therefore, the theoretical information on their well-posedness is essential before attempting to derive an approximate solution by analytical or numerical means. Our utmost contribution in this context is the definition of a numerical test for investigating the existence and uniqueness of solutions of boundary problems defined on semi-infinite intervals. The main result is given by a theorem relating the existence and uniqueness question to the number of real zeros of a function implicitly defined within the formulation of the iterative transformation method. As a consequence, we can investigate the existence and uniqueness of solutions by studying the behaviour of that function. Within such a context, the numerical test is illustrated by two examples where we find meaningful numerical results.

scholarly journals Existence and Uniqueness of BVPs Defined on Semi-Infinite Intervals: Insight from the Iterative Transformation Method

2021 ◽  
Author(s):  
Riccardo Fazio
Keyword(s):  
Existence And Uniqueness ◽  
Numerical Test ◽  
Transformation Method ◽  
Uniqueness Of Solutions ◽  
Boundary Problems ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Uniqueness Question ◽  
Infinite Intervals ◽  
Definition Of

This work is concerned with the existence and uniqueness of boundary value problems defined on semi-infinite intervals. These kinds of problems seldom admit exactly known solutions and, therefore, the theoretical information on their well-posedness is essential before attempting to derive an approximate solution by analytical or numerical means. Our utmost contribution in this context is the definition of a numerical test for investigating the existence and uniqueness of solutions of boundary problems defined on semi-infinite intervals. The main result is given by a theorem relating the existence and uniqueness question to the number of real zeros of a function implicitly defined within the formulation of the iterative transformation method. As a consequence, we can investigate the existence and uniqueness of solutions by studying the behaviour of that function. Within such a context the numerical test is illustrated by two examples where we find meaningful numerical results.

scholarly journals Theory of existence and uniqueness for the nonlinear Maxwell-Boltzmann equation I

1977 ◽  
Vol 16 (3) ◽  
pp. 379-414 ◽  
Author(s):  
Aleksander Glikson
Keyword(s):  
Boltzmann Equation ◽  
Existence And Uniqueness ◽  
Broad Class ◽  
Physical Space ◽  
Uniqueness Result ◽  
Cauchy Problems ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Boltzmann Gas ◽  
Definition Of

A review of the development of the theory of existence and uniqueness of solutions to initial-value problems for mostly reduced versions of the nonlinear Maxwell-Boltzmann equation with a cut-off of intermolecular interaction, precedes the formulation and discussion of a somewhat generalized initial-value problem for the full nonlinear Maxwell-Boltzmann equation, with or without a cut-off. This is followed by a derivation of a new existence-uniqueness result for a particular Cauchy problem for the full nonlinear Maxwell-Boltzmann equation with a cut-off, under the assumption that the monatomic Boltzmann gas in the unbounded physical space X is acted upon by a member of a broad class of external conservative forces with sufficiently well-behaved potentials, defined on X and bounded from below. The result represents a significant improvement of an earlier theorem by this author which was until now the strongest obtained for Cauchy problems for the full Maxwell-Boltzmann equation. The improvement is basically due to the introduction of equivalent norms in a Banach space, the definition of which is connected with an exponential function of the total energy of a free-streaming molecule.

scholarly journals On Different Classes of Algebraic Polynomials with Random Coefficients

2008 ◽  
Vol 2008 ◽  
pp. 1-8 ◽  
Author(s):  
K. Farahmand ◽  
A. Grigorash ◽  
B. McGuinness
Keyword(s):  
Asymptotic Relation ◽  
Random Variables ◽  
Random Coefficients ◽  
Algebraic Polynomials ◽  
Expected Number ◽  
Level Crossings ◽  
Gaussian Random Variables ◽  
Real Zeros ◽  
Number Of Real Zeros ◽  
Definition Of

The expected number of real zeros of the polynomial of the form , where is a sequence of standard Gaussian random variables, is known. For large it is shown that this expected number in is asymptotic to . In this paper, we show that this asymptotic value increases significantly to when we consider a polynomial in the form instead. We give the motivation for our choice of polynomial and also obtain some other characteristics for the polynomial, such as the expected number of level crossings or maxima. We note, and present, a small modification to the definition of our polynomial which improves our result from the above asymptotic relation to the equality.

scholarly journals Fractional Cauchy Problems for Infinite Interval Case-II

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1165
Author(s):  
Mohammed Al Horani ◽  
Mauro Fabrizio ◽  
Angelo Favini ◽  
Hiroki Tanabe
Keyword(s):  
Existence And Uniqueness ◽  
Direct Problem ◽  
Abstract Cauchy Problem ◽  
Continuous Functions ◽  
Cauchy Problems ◽  
Fractional Powers ◽  
Uniqueness Of Solutions ◽  
Infinite Intervals ◽  
Abstract Cauchy Problems ◽  
Densely Defined

We consider fractional abstract Cauchy problems on infinite intervals. A fractional abstract Cauchy problem for possibly degenerate equations in Banach spaces is considered. This form of degeneration may be strong and some convenient assumptions about the involved operators are required to handle the direct problem. Required conditions on spaces are also given, guaranteeing the existence and uniqueness of solutions. The fractional powers of the involved operator B X have been investigated in the space which consists of continuous functions u on [ 0 , ∞ ) without assuming u ( 0 ) = 0 . This enables us to refine some previous results and obtain the required abstract results when the operator B X is not necessarily densely defined.

scholarly journals α-Coupled Fixed Points and Their Application in Dynamic Programming

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
J. Harjani ◽  
J. Rocha ◽  
K. Sadarangani
Keyword(s):  
Dynamic Programming ◽  
Fixed Point ◽  
Fixed Points ◽  
Existence And Uniqueness ◽  
Functional Equations ◽  
Coupled Fixed Point ◽  
Uniqueness Of Solutions ◽  
Bounded Functions ◽  
Definition Of ◽  
Coupled Fixed Points

We introduce the definition ofα-coupled fixed point in the space of the bounded functions on a setSand we present a result about the existence and uniqueness of such points. Moreover, as an application of our result, we study the problem of existence and uniqueness of solutions for a class of systems of functional equations arising in dynamic programming.

scholarly journals Existence and uniqueness of solutions for multi-term fractional q-integro-differential equations via quantum calculus

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Sotiris K. Ntouyas ◽  
Mohammad Esmael Samei
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Contraction Mapping Principle ◽  
Uniqueness Of Solutions ◽  
Quantum Calculus ◽  
Numerical Result ◽  
Banach Contraction ◽  
Definition Of ◽  
Fixed Point Techniques

AbstractIn this investigation, by applying the definition of the fractional q-derivative of the Caputo type and the fractional q-integral of the Riemann–Liouville type, we study the existence and uniqueness of solutions for a multi-term nonlinear fractional q-integro-differential equations under some boundary conditions ${}^{c}D_{q}^{\alpha} x(t) = w ( t, x(t), (\varphi_{1} x)(t), (\varphi_{2} x)(t), {}^{c}D_{q} ^{ \beta_{1}} x(t), {}^{c}D_{q}^{\beta_{2}} x(t), \ldots, {}^{c}D _{q}^{ \beta_{n}}x(t) )$Dqαcx(t)=w(t,x(t),(φ1x)(t),(φ2x)(t),cDqβ1x(t),cDqβ2x(t),…,cDqβnx(t)). Our results are based on some classical fixed point techniques, as Schauder’s fixed point theorem and Banach contraction mapping principle. Besides, some instances are exhibited to illustrate our results and we report all algorithms required along with the numerical result obtained.

Sign in / Sign up

Export Citation Format

Share Document