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Real Functions, Dynamical Systems and their Applications

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Informacje o czasopiśmie
Format
Czasopismo
eISSN
1338-9750
Pierwsze wydanie
12 Nov 2012
Częstotliwość wydawania
3 razy w roku
Języki
Angielski

Wyszukiwanie

Tom 79 (2021): Zeszyt 2 (December 2021)

Informacje o czasopiśmie
Format
Czasopismo
eISSN
1338-9750
Pierwsze wydanie
12 Nov 2012
Częstotliwość wydawania
3 razy w roku
Języki
Angielski

Wyszukiwanie

12 Artykułów
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 1 - 12

Abstrakt

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.

Słowa kluczowe

  • extended Fan sub-equation method
  • time fractional Burgers-Fisher equation
  • solitary wave solution
Otwarty dostęp

Some Results Involving the Airy Functions and Airy Transforms

Data publikacji: 01 Jan 2022
Zakres stron: 13 - 32

Abstrakt

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.

Słowa kluczowe

  • Airy function
  • Airy transform
  • modified Bessel functions
  • Riccati differential equation
  • Parseval-Plancherel identity
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 33 - 46

Abstrakt

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.

Słowa kluczowe

  • Algebraic and non–algebraic limit cycle
  • planar polynomial differential system
  • first integral
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 47 - 58

Abstrakt

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.

Słowa kluczowe

  • limit cycle
  • first integral
  • discontinuous piecewise linear differential system
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 59 - 80

Abstrakt

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.

Słowa kluczowe

  • controllability
  • impulsive differential equations
  • nonlocal conditions
  • measure of non compactness
  • fixed point theorem
Otwarty dostęp

Oscillation Tests for Linear Difference Equations with Non-Monotone Arguments

Data publikacji: 01 Jan 2022
Zakres stron: 81 - 100

Abstrakt

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.

Słowa kluczowe

  • non-monotone argument
  • retarded argument
  • advanced argument
  • oscillation
  • Grönwall inequality
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 101 - 118

Abstrakt

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.

Słowa kluczowe

  • oscillation
  • Riemann-Liouville fractional derivatives
  • difference equations
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 119 - 134

Abstrakt

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.

Słowa kluczowe

  • oscillation
  • second order
  • neutral differential equations
  • mixed neutral term
Otwarty dostęp

Properties of the Katugampola Fractional Operators

Data publikacji: 01 Jan 2022
Zakres stron: 135 - 148

Abstrakt

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.

Słowa kluczowe

  • fractional calculus
  • Katugampola fractional operators
  • higher order
  • existence
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 149 - 162

Abstrakt

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.

Słowa kluczowe

  • Difference equation
  • delays
  • neutral type
  • periodicity
  • asymptotic behaviour
Otwarty dostęp

Certain Singular Distributions and Fractals

Data publikacji: 01 Jan 2022
Zakres stron: 163 - 198

Abstrakt

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.

Słowa kluczowe

  • Fractal
  • Moran structure
  • Hausdorff dimension
  • range of values of a function
  • continuous function
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 199 - 212

Abstrakt

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.

Słowa kluczowe

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

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

Data publikacji: 01 Jan 2022
Zakres stron: 1 - 12

Abstrakt

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.

Słowa kluczowe

  • extended Fan sub-equation method
  • time fractional Burgers-Fisher equation
  • solitary wave solution
Otwarty dostęp

Some Results Involving the Airy Functions and Airy Transforms

Data publikacji: 01 Jan 2022
Zakres stron: 13 - 32

Abstrakt

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.

Słowa kluczowe

  • Airy function
  • Airy transform
  • modified Bessel functions
  • Riccati differential equation
  • Parseval-Plancherel identity
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 33 - 46

Abstrakt

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.

Słowa kluczowe

  • Algebraic and non–algebraic limit cycle
  • planar polynomial differential system
  • first integral
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 47 - 58

Abstrakt

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.

Słowa kluczowe

  • limit cycle
  • first integral
  • discontinuous piecewise linear differential system
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 59 - 80

Abstrakt

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.

Słowa kluczowe

  • controllability
  • impulsive differential equations
  • nonlocal conditions
  • measure of non compactness
  • fixed point theorem
Otwarty dostęp

Oscillation Tests for Linear Difference Equations with Non-Monotone Arguments

Data publikacji: 01 Jan 2022
Zakres stron: 81 - 100

Abstrakt

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.

Słowa kluczowe

  • non-monotone argument
  • retarded argument
  • advanced argument
  • oscillation
  • Grönwall inequality
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 101 - 118

Abstrakt

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.

Słowa kluczowe

  • oscillation
  • Riemann-Liouville fractional derivatives
  • difference equations
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 119 - 134

Abstrakt

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.

Słowa kluczowe

  • oscillation
  • second order
  • neutral differential equations
  • mixed neutral term
Otwarty dostęp

Properties of the Katugampola Fractional Operators

Data publikacji: 01 Jan 2022
Zakres stron: 135 - 148

Abstrakt

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.

Słowa kluczowe

  • fractional calculus
  • Katugampola fractional operators
  • higher order
  • existence
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 149 - 162

Abstrakt

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.

Słowa kluczowe

  • Difference equation
  • delays
  • neutral type
  • periodicity
  • asymptotic behaviour
Otwarty dostęp

Certain Singular Distributions and Fractals

Data publikacji: 01 Jan 2022
Zakres stron: 163 - 198

Abstrakt

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.

Słowa kluczowe

  • Fractal
  • Moran structure
  • Hausdorff dimension
  • range of values of a function
  • continuous function
Otwarty dostęp

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

Data publikacji: 01 Jan 2022
Zakres stron: 199 - 212

Abstrakt

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.

Słowa kluczowe

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

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