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Definitions of Real Order Integrals and Derivatives Using Operator Approach
Issue:
Volume 2, Issue 1, February 2013
Pages:
1-9
Received:
11 December 2012
Accepted:
Published:
20 February 2013
Abstract: The set E of functions f fulfilling some conditions is taken to be the definition domain of s-order integral operator J^s (iterative integral), first for any positive integer s and after for any positive s (fractional, transcendental π and e). The definition of k-order derivative operator D^k for any positive k (fractional, transcendental π and e) is derived from the definition of J^s. Some properties of J^sand D^k are given and demonstrated. The method is based on the properties of Euler’s gamma and beta functions.
Abstract: The set E of functions f fulfilling some conditions is taken to be the definition domain of s-order integral operator J^s (iterative integral), first for any positive integer s and after for any positive s (fractional, transcendental π and e). The definition of k-order derivative operator D^k for any positive k (fractional, transcendental π and e) ...
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Two Definitions of Fractional Derivative of Powers Functions
Raoelina Andriambololona,
Rakotoson Hanitriarivo,
Tokiniaina Ranaivoson,
Roland Raboanary
Issue:
Volume 2, Issue 1, February 2013
Pages:
10-19
Received:
11 December 2012
Accepted:
Published:
20 February 2013
Abstract: We consider the set of powers functions defined on R_+ and their linear combinations. After recalling some properties of the gamma function, we give two general definitions of derivatives of positive and negative integers, positive and negative fractional orders. Properties of linearity and commutativity are studied and the notions of semi-equality, semi-linearity and semi-commutativity are introduced. Our approach gives a unified definition of the common derivatives and integrals and their generalization.
Abstract: We consider the set of powers functions defined on R_+ and their linear combinations. After recalling some properties of the gamma function, we give two general definitions of derivatives of positive and negative integers, positive and negative fractional orders. Properties of linearity and commutativity are studied and the notions of semi-equality...
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Existence Results for A Nonlocal Problem Involving the p(x)-Laplacian
Issue:
Volume 2, Issue 1, February 2013
Pages:
20-27
Received:
14 January 2013
Accepted:
Published:
20 February 2013
Abstract: In the present paper, we deal with two different existence results of solutions for a nonlocal elliptic Dirichlet boundary value problem involving p(x)-Laplacian. The first one is based on the Brouwer fixed point theorem and the Galerkin method which gives a priori estimate of a nontrivial weak soltion. The second one is based on the variational methods. By using Mountain-Pass theorem, we obtain at least one nontrivial weak soltion.
Abstract: In the present paper, we deal with two different existence results of solutions for a nonlocal elliptic Dirichlet boundary value problem involving p(x)-Laplacian. The first one is based on the Brouwer fixed point theorem and the Galerkin method which gives a priori estimate of a nontrivial weak soltion. The second one is based on the variational me...
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Non Oscillatory Nonlinear Differential Systems with Slowly Varying Coefficients in Presence on Certain Damping Forces
Issue:
Volume 2, Issue 1, February 2013
Pages:
28-31
Received:
31 January 2013
Accepted:
Published:
20 February 2013
Abstract: Krylov-Bogoliubov-Mitropolskii method is modified and applied to certain damped nonlinear systems with slowly varying coefficients. The results obtained by this method show excellent coincidence with those obtained by numerical method. The method is illustrated by an example.
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About Solutions of Nonlinear Algebraic System with Two Variables
Issue:
Volume 2, Issue 1, February 2013
Pages:
32-37
Received:
25 January 2013
Accepted:
Published:
20 February 2013
Abstract: For nonlinear algebraic system with two variables sufficient conditions of existence of solutions are given. The proof of these statements is received as a corollary of more common reviewing considered in this paper. In particular, in this work the existence of multiple base on eigen and associated vectors of a two parameter system of operators in fi-nite-dimensional spaces is proved. Definitions of the associated vectors, multiple completeness of eigen and associated vectors of two-parameter not selfadjoint systems, nonlinearly depending on spectral parameters are introduced. At the proof of these results we essentially used the notion of the analog of an resultant of two polynomial bundles.
Abstract: For nonlinear algebraic system with two variables sufficient conditions of existence of solutions are given. The proof of these statements is received as a corollary of more common reviewing considered in this paper. In particular, in this work the existence of multiple base on eigen and associated vectors of a two parameter system of operators in ...
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Rhotrix Polynomials and Polynomial Rhotrices
Issue:
Volume 2, Issue 1, February 2013
Pages:
38-41
Received:
11 February 2013
Accepted:
Published:
20 February 2013
Abstract: In this piece of note, polynomials defined over the ring R of rhotrices of n-dimension and rhotrices defined over polynomials in were explored, the aim is to study their nature and present their properties. The hope is that these polynomials (or these rhotrices) will have wider applications than those polynomials defined over the non-commutative ring of n-square matrices (or those matrices defined over polynomials) since R is a commutative ring. The shortcomings of these polynomials and rhotrices were also confirmed as it was proved that the rings R[x] and R[f] are not integral domains.
Abstract: In this piece of note, polynomials defined over the ring R of rhotrices of n-dimension and rhotrices defined over polynomials in were explored, the aim is to study their nature and present their properties. The hope is that these polynomials (or these rhotrices) will have wider applications than those polynomials defined over the non-commutative ...
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Modeling the Epidemiology of Malaria and Control with Estimate of the Basic Repro-duction Number
Adamu Abdul Kareem,
Anande Richard Kimbir
Issue:
Volume 2, Issue 1, February 2013
Pages:
42-50
Received:
6 March 2013
Accepted:
Published:
20 February 2013
Abstract: Strategies for controlling the epidemiology of many infectious diseases such as malaria include a rapid reduc-tion in both the infected and susceptible population via treatment and vaccination. In this paper, we have modified the Tumwiine et al. (2007) mathematical model for the transmission of malaria by including a vaccination parameter. We have shown that the model has a unique disease-free equilibrium state which is locally and globally asymptotically stable, if 1, and that the endemic equilibrium exist provided > 1, where is a parameter which depends on the given model parameters. Numerical simulations of the modified model clearly show that, with a proper combination of treatment and vaccination, offered at about 65% each on the susceptible and infected population, malaria can be eradicated from the community.
Abstract: Strategies for controlling the epidemiology of many infectious diseases such as malaria include a rapid reduc-tion in both the infected and susceptible population via treatment and vaccination. In this paper, we have modified the Tumwiine et al. (2007) mathematical model for the transmission of malaria by including a vaccination parameter. We have ...
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