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Détails du magazine
Format
Magazine
eISSN
1338-9750
Première publication
12 Nov 2012
Période de publication
3 fois par an
Langues
Anglais

Chercher

Volume 79 (2021): Edition 2 (December 2021)

Détails du magazine
Format
Magazine
eISSN
1338-9750
Première publication
12 Nov 2012
Période de publication
3 fois par an
Langues
Anglais

Chercher

12 Articles
Accès libre

Application of the Extended Fan Sub-Equation Method to Time Fractional Burgers-Fisher Equation

Publié en ligne: 01 Jan 2022
Pages: 1 - 12

Résumé

Abstract

In this paper, the extended Fan sub-equation method to obtain the exact solutions of the generalized time fractional Burgers-Fisher equation is applied. By applying this method, we obtain different solutions that are benefit to further comprise the concepts of complex nonlinear physical phenomena. This method is simple and can be applied to several nonlinear equations. Fractional derivatives are taken in the sense of Jumarie’s modified Riemann-Liouville derivative. A comparative study with the other methods approves the validity and effectiveness of the technique, and on the other hand, for suitable parameter values, we plot 2D and 3D graphics of the exact solutions by using the extended Fan sub-equation method. In this work, we use Mathematica for computations and programming.

Mots clés

  • extended Fan sub-equation method
  • time fractional Burgers-Fisher equation
  • solitary wave solution
Accès libre

Some Results Involving the Airy Functions and Airy Transforms

Publié en ligne: 01 Jan 2022
Pages: 13 - 32

Résumé

Abstract

In the present work, the author studied some properties of the modified Bessel’s functions and Airy functions. It is worth mentioning that the Airy functions are used in many fields of physics. They are applied in many branches of classical and quantum physics. The author also studied certain properties of the Airy transform and derived some new integral relations involving the Airy functions. Non-trivial illustrative examples are provided as well. All the results are presented in lucid and comprehensible language.

Mots clés

  • Airy function
  • Airy transform
  • modified Bessel functions
  • Riccati differential equation
  • Parseval-Plancherel identity
Accès libre

Two Non Algebraic Limit Cycles of a Class of Polynomial Differential Systems with Non-Elementary Equilibrium Point

Publié en ligne: 01 Jan 2022
Pages: 33 - 46

Résumé

Abstract

The problems of existence of limit cycles and their numbers are the most difficult problems in the dynamical planar systems. In this paper, we study the limit cycles for a family of polynomial differential systems of degree 6k + 1, k ∈ ℕ*, with the non-elementary singular point. Under some suitable conditions, we show our system exhibiting two non algebraic or two algebraic limit cycles explicitly given. To illustrate our results we present some examples.

Mots clés

  • Algebraic and non–algebraic limit cycle
  • planar polynomial differential system
  • first integral
Accès libre

Explicit Non Algebraic Limit Cycle for a Discontinuous Piecewise Differential Systems Separated by One Straight Line and Formed by Linear Center and Linear System Without Equilibria

Publié en ligne: 01 Jan 2022
Pages: 47 - 58

Résumé

Abstract

In this paper, we deal with the discontinuous piecewise differential linear systems formed by two differential systems separated by a straight line when one of these two differential systems is a linear without equilibria and the other is a linear center. We are going to show that the maximum number of crossing limit cycles is one, and if exists, it is non algebraic and analytically given.

Mots clés

  • limit cycle
  • first integral
  • discontinuous piecewise linear differential system
Accès libre

Controllability of Nonlocal Impulsive Functional Differential Equations with Measure of Noncompactness in Banach Spaces

Publié en ligne: 01 Jan 2022
Pages: 59 - 80

Résumé

Abstract

This paper is concerned with the controllability of impulsive differential equations with nonlocal conditions. First, we establish a property of measure of noncompactness in the space of piecewise continuous functions. Then, by using this property and Darbo-Sadovskii’s Fixed Point Theorem, we get the controllability of nonlocal impulsive differential equations under compactness conditions, Lipschitz conditions and mixed-type conditions, respectively.

Mots clés

  • controllability
  • impulsive differential equations
  • nonlocal conditions
  • measure of non compactness
  • fixed point theorem
Accès libre

Oscillation Tests for Linear Difference Equations with Non-Monotone Arguments

Publié en ligne: 01 Jan 2022
Pages: 81 - 100

Résumé

Abstract

This paper presents sufficient conditions involving limsup for the oscillation of all solutions of linear difference equations with general deviating argument of the form Δx(n)+p(n)x(τ(n))=0,n0[x(n)q(n)x(σ(n))=0,n],\[\Delta x(n) + p(n)x(\tau (n)) = 0,\,n \in {_0}\quad [\nabla x(n) - q(n)x(\sigma (n)) = 0,\,n \in ],\ , where (p(n))n0 and (q(n))n1 are sequences of nonnegative real numbers and (τ(n))n0,(σ(n))n1\[{(\tau (n))_{n \ge 0}},\quad {(\sigma (n))_{n \ge 1}}\] are (not necessarily monotone) sequences of integers. The results obtained improve all well-known results existing in the literature and an example, numerically solved in MATLAB, illustrating the significance of these results is provided.

Mots clés

  • non-monotone argument
  • retarded argument
  • advanced argument
  • oscillation
  • Grönwall inequality
Accès libre

Oscillation Behaviour of Solutions for a Class of a Discrete Nonlinear Fractional-Order Derivatives

Publié en ligne: 01 Jan 2022
Pages: 101 - 118

Résumé

Abstract

Based on the generalized Riccati transformation technique and some inequality, we study some oscillation behaviour of solutions for a class of a discrete nonlinear fractional-order derivative equation Δ[γ()[α()+β()Δμu()]η]+ϕ()f[G()]=0,N0+1μ, \[\Delta [\gamma (\ell ){[\alpha (\ell ) + \beta (\ell ){\Delta ^\mu }u(\ell )]^\eta }] + \phi (\ell )f[G(\ell )] = 0,\ell \in {N_{{\ell _0} + 1 - \mu }},\] where 0>0,G()=j=01+μ(j1)(μ)u(j)\[{\ell _0} > 0,\quad G(\ell ) = \sum\limits_{j = {\ell _0}}^{\ell - 1 + \mu } {{{(\ell - j - 1)}^{( - \mu )}}u(j)} \] and Δμ is the Riemann-Liouville (R-L) difference operator of the derivative of order μ, 0 < μ ≤ 1 and η is a quotient of odd positive integers. Illustrative examples are given to show the validity of the theoretical results.

Mots clés

  • oscillation
  • Riemann-Liouville fractional derivatives
  • difference equations
Accès libre

Oscillatory Behaviour of Second-Order Nonlinear Differential Equations with Mixed Neutral Terms

Publié en ligne: 01 Jan 2022
Pages: 119 - 134

Résumé

Abstract

The authors examine the oscillation of second-order nonlinear differential equations with mixed nonlinear neutral terms. They present new oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are illustrated by some examples.

Mots clés

  • oscillation
  • second order
  • neutral differential equations
  • mixed neutral term
Accès libre

Properties of the Katugampola Fractional Operators

Publié en ligne: 01 Jan 2022
Pages: 135 - 148

Résumé

Abstract

In this work, there are considered higher order fractional operators defined in the sense of Katugampola. There are proved some fundamental properties of the Katugampola fractional operators of any arbitrary real order. Moreover, there are given conditions ensuring existence of the higher order Katugampola fractional derivative in space of the absolutely continuous functions.

Mots clés

  • fractional calculus
  • Katugampola fractional operators
  • higher order
  • existence
Accès libre

Existence of The Asymptotically Periodic Solution to the System of Nonlinear Neutral Difference Equations

Publié en ligne: 01 Jan 2022
Pages: 149 - 162

Résumé

Abstract

The system of nonlinear neutral difference equations with delays in the form { Δ(yi(n)+pi(n)yi(nτi))=ai(n)fi(yi+1(n))+gi(n),Δ(ym(n)+pm(n)ym(nτm))=am(n)fm(y1(n))+gm(n),\[\left\{ \begin{array}{l} \Delta ({y_i}(n) + {p_i}(n){y_i}(n - {\tau _i})) = {a_i}(n){f_i}({y_{i + 1}}(n)) + {g_i}(n),\\ \Delta ({y_m}(n) + {p_m}(n){y_m}(n - {\tau _m})) = {a_m}(n){f_m}({y_1}(n)) + {g_m}(n), \end{array} \right.\] for i = 1, . . . , m − 1, m ≥ 2, is studied. The sufficient conditions for the existence of an asymptotically periodic solution of the above system, are established. Here sequences (pi(n)), i = 1,..., m, are bounded away from -1. The presented results are illustrated by theoretical and numerical examples.

Mots clés

  • Difference equation
  • delays
  • neutral type
  • periodicity
  • asymptotic behaviour
Accès libre

Certain Singular Distributions and Fractals

Publié en ligne: 01 Jan 2022
Pages: 163 - 198

Résumé

Abstract

In the presented paper, the main attention is given to fractal sets whose elements have certain restrictions on using digits or combinations of digits in their own nega-P-representation. Topological, metric, and fractal properties of images of certain self-similar fractals under the action of some singular distributions, are investigated.

Mots clés

  • Fractal
  • Moran structure
  • Hausdorff dimension
  • range of values of a function
  • continuous function
Accès libre

Existence and Multiplicity of Positive Solutions for a Third-Order Two-Point Boundary Value Problem

Publié en ligne: 01 Jan 2022
Pages: 199 - 212

Résumé

Abstract

We study the existence and multiplicity of positive solutions for a third-order two-point boundary value problem by applying Krasnosel’skii’s fixed point theorem. To illustrate the applicability of the obtained results, we consider some examples.

Mots clés

  • Nonlinear boundary value problems
  • existence and multiplicity of positive solutions
  • Green’s function
  • Krasnosel’skii’s fixed point theorem
12 Articles
Accès libre

Application of the Extended Fan Sub-Equation Method to Time Fractional Burgers-Fisher Equation

Publié en ligne: 01 Jan 2022
Pages: 1 - 12

Résumé

Abstract

In this paper, the extended Fan sub-equation method to obtain the exact solutions of the generalized time fractional Burgers-Fisher equation is applied. By applying this method, we obtain different solutions that are benefit to further comprise the concepts of complex nonlinear physical phenomena. This method is simple and can be applied to several nonlinear equations. Fractional derivatives are taken in the sense of Jumarie’s modified Riemann-Liouville derivative. A comparative study with the other methods approves the validity and effectiveness of the technique, and on the other hand, for suitable parameter values, we plot 2D and 3D graphics of the exact solutions by using the extended Fan sub-equation method. In this work, we use Mathematica for computations and programming.

Mots clés

  • extended Fan sub-equation method
  • time fractional Burgers-Fisher equation
  • solitary wave solution
Accès libre

Some Results Involving the Airy Functions and Airy Transforms

Publié en ligne: 01 Jan 2022
Pages: 13 - 32

Résumé

Abstract

In the present work, the author studied some properties of the modified Bessel’s functions and Airy functions. It is worth mentioning that the Airy functions are used in many fields of physics. They are applied in many branches of classical and quantum physics. The author also studied certain properties of the Airy transform and derived some new integral relations involving the Airy functions. Non-trivial illustrative examples are provided as well. All the results are presented in lucid and comprehensible language.

Mots clés

  • Airy function
  • Airy transform
  • modified Bessel functions
  • Riccati differential equation
  • Parseval-Plancherel identity
Accès libre

Two Non Algebraic Limit Cycles of a Class of Polynomial Differential Systems with Non-Elementary Equilibrium Point

Publié en ligne: 01 Jan 2022
Pages: 33 - 46

Résumé

Abstract

The problems of existence of limit cycles and their numbers are the most difficult problems in the dynamical planar systems. In this paper, we study the limit cycles for a family of polynomial differential systems of degree 6k + 1, k ∈ ℕ*, with the non-elementary singular point. Under some suitable conditions, we show our system exhibiting two non algebraic or two algebraic limit cycles explicitly given. To illustrate our results we present some examples.

Mots clés

  • Algebraic and non–algebraic limit cycle
  • planar polynomial differential system
  • first integral
Accès libre

Explicit Non Algebraic Limit Cycle for a Discontinuous Piecewise Differential Systems Separated by One Straight Line and Formed by Linear Center and Linear System Without Equilibria

Publié en ligne: 01 Jan 2022
Pages: 47 - 58

Résumé

Abstract

In this paper, we deal with the discontinuous piecewise differential linear systems formed by two differential systems separated by a straight line when one of these two differential systems is a linear without equilibria and the other is a linear center. We are going to show that the maximum number of crossing limit cycles is one, and if exists, it is non algebraic and analytically given.

Mots clés

  • limit cycle
  • first integral
  • discontinuous piecewise linear differential system
Accès libre

Controllability of Nonlocal Impulsive Functional Differential Equations with Measure of Noncompactness in Banach Spaces

Publié en ligne: 01 Jan 2022
Pages: 59 - 80

Résumé

Abstract

This paper is concerned with the controllability of impulsive differential equations with nonlocal conditions. First, we establish a property of measure of noncompactness in the space of piecewise continuous functions. Then, by using this property and Darbo-Sadovskii’s Fixed Point Theorem, we get the controllability of nonlocal impulsive differential equations under compactness conditions, Lipschitz conditions and mixed-type conditions, respectively.

Mots clés

  • controllability
  • impulsive differential equations
  • nonlocal conditions
  • measure of non compactness
  • fixed point theorem
Accès libre

Oscillation Tests for Linear Difference Equations with Non-Monotone Arguments

Publié en ligne: 01 Jan 2022
Pages: 81 - 100

Résumé

Abstract

This paper presents sufficient conditions involving limsup for the oscillation of all solutions of linear difference equations with general deviating argument of the form Δx(n)+p(n)x(τ(n))=0,n0[x(n)q(n)x(σ(n))=0,n],\[\Delta x(n) + p(n)x(\tau (n)) = 0,\,n \in {_0}\quad [\nabla x(n) - q(n)x(\sigma (n)) = 0,\,n \in ],\ , where (p(n))n0 and (q(n))n1 are sequences of nonnegative real numbers and (τ(n))n0,(σ(n))n1\[{(\tau (n))_{n \ge 0}},\quad {(\sigma (n))_{n \ge 1}}\] are (not necessarily monotone) sequences of integers. The results obtained improve all well-known results existing in the literature and an example, numerically solved in MATLAB, illustrating the significance of these results is provided.

Mots clés

  • non-monotone argument
  • retarded argument
  • advanced argument
  • oscillation
  • Grönwall inequality
Accès libre

Oscillation Behaviour of Solutions for a Class of a Discrete Nonlinear Fractional-Order Derivatives

Publié en ligne: 01 Jan 2022
Pages: 101 - 118

Résumé

Abstract

Based on the generalized Riccati transformation technique and some inequality, we study some oscillation behaviour of solutions for a class of a discrete nonlinear fractional-order derivative equation Δ[γ()[α()+β()Δμu()]η]+ϕ()f[G()]=0,N0+1μ, \[\Delta [\gamma (\ell ){[\alpha (\ell ) + \beta (\ell ){\Delta ^\mu }u(\ell )]^\eta }] + \phi (\ell )f[G(\ell )] = 0,\ell \in {N_{{\ell _0} + 1 - \mu }},\] where 0>0,G()=j=01+μ(j1)(μ)u(j)\[{\ell _0} > 0,\quad G(\ell ) = \sum\limits_{j = {\ell _0}}^{\ell - 1 + \mu } {{{(\ell - j - 1)}^{( - \mu )}}u(j)} \] and Δμ is the Riemann-Liouville (R-L) difference operator of the derivative of order μ, 0 < μ ≤ 1 and η is a quotient of odd positive integers. Illustrative examples are given to show the validity of the theoretical results.

Mots clés

  • oscillation
  • Riemann-Liouville fractional derivatives
  • difference equations
Accès libre

Oscillatory Behaviour of Second-Order Nonlinear Differential Equations with Mixed Neutral Terms

Publié en ligne: 01 Jan 2022
Pages: 119 - 134

Résumé

Abstract

The authors examine the oscillation of second-order nonlinear differential equations with mixed nonlinear neutral terms. They present new oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are illustrated by some examples.

Mots clés

  • oscillation
  • second order
  • neutral differential equations
  • mixed neutral term
Accès libre

Properties of the Katugampola Fractional Operators

Publié en ligne: 01 Jan 2022
Pages: 135 - 148

Résumé

Abstract

In this work, there are considered higher order fractional operators defined in the sense of Katugampola. There are proved some fundamental properties of the Katugampola fractional operators of any arbitrary real order. Moreover, there are given conditions ensuring existence of the higher order Katugampola fractional derivative in space of the absolutely continuous functions.

Mots clés

  • fractional calculus
  • Katugampola fractional operators
  • higher order
  • existence
Accès libre

Existence of The Asymptotically Periodic Solution to the System of Nonlinear Neutral Difference Equations

Publié en ligne: 01 Jan 2022
Pages: 149 - 162

Résumé

Abstract

The system of nonlinear neutral difference equations with delays in the form { Δ(yi(n)+pi(n)yi(nτi))=ai(n)fi(yi+1(n))+gi(n),Δ(ym(n)+pm(n)ym(nτm))=am(n)fm(y1(n))+gm(n),\[\left\{ \begin{array}{l} \Delta ({y_i}(n) + {p_i}(n){y_i}(n - {\tau _i})) = {a_i}(n){f_i}({y_{i + 1}}(n)) + {g_i}(n),\\ \Delta ({y_m}(n) + {p_m}(n){y_m}(n - {\tau _m})) = {a_m}(n){f_m}({y_1}(n)) + {g_m}(n), \end{array} \right.\] for i = 1, . . . , m − 1, m ≥ 2, is studied. The sufficient conditions for the existence of an asymptotically periodic solution of the above system, are established. Here sequences (pi(n)), i = 1,..., m, are bounded away from -1. The presented results are illustrated by theoretical and numerical examples.

Mots clés

  • Difference equation
  • delays
  • neutral type
  • periodicity
  • asymptotic behaviour
Accès libre

Certain Singular Distributions and Fractals

Publié en ligne: 01 Jan 2022
Pages: 163 - 198

Résumé

Abstract

In the presented paper, the main attention is given to fractal sets whose elements have certain restrictions on using digits or combinations of digits in their own nega-P-representation. Topological, metric, and fractal properties of images of certain self-similar fractals under the action of some singular distributions, are investigated.

Mots clés

  • Fractal
  • Moran structure
  • Hausdorff dimension
  • range of values of a function
  • continuous function
Accès libre

Existence and Multiplicity of Positive Solutions for a Third-Order Two-Point Boundary Value Problem

Publié en ligne: 01 Jan 2022
Pages: 199 - 212

Résumé

Abstract

We study the existence and multiplicity of positive solutions for a third-order two-point boundary value problem by applying Krasnosel’skii’s fixed point theorem. To illustrate the applicability of the obtained results, we consider some examples.

Mots clés

  • Nonlinear boundary value problems
  • existence and multiplicity of positive solutions
  • Green’s function
  • Krasnosel’skii’s fixed point theorem

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