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, n∈ℕ0 [∇x(n)−q(n)x(σ(n))=0, n∈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))n≥0 and (q(n))n≥1 are sequences of nonnegative real numbers and (τ(n))n≥0, (σ(n))n≥1[{(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.
{"title":"Oscillation Tests for Linear Difference Equations with Non-Monotone Arguments","authors":"G. Chatzarakis, S. Grace, Irena JadloyskÁ","doi":"10.2478/tmmp-2021-0021","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0021","url":null,"abstract":"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, n∈ℕ0 [∇x(n)−q(n)x(σ(n))=0, n∈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))n≥0 and (q(n))n≥1 are sequences of nonnegative real numbers and (τ(n))n≥0, (σ(n))n≥1[{(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.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"79 1","pages":"81 - 100"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48096833","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"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","authors":"A. Berbache","doi":"10.2478/tmmp-2021-0019","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0019","url":null,"abstract":"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.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"79 1","pages":"47 - 58"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44183006","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Some new oscillation criteria are obtained for a class of thirdorder quasi-linear Emden-Fowler differential equations with unbounded neutral coefficients of the form (a(t)(z″(t))α)′+f(t)yλ(g(t))=0,[(a(t){(z(t))^alpha })' + f(t){y^lambda }(g(t)) = 0,] where z(t) = y(t) + p(t)y(σ(t)) and α, λ are ratios of odd positive integers. The established results generalize, improve and complement to known results.
{"title":"Oscillation Results for Third-Order Quasi-Linear Emden-Fowler Differential Equations with Unbounded Neutral Coefficients","authors":"G. Chatzarakis, R. Srinivasan, E. Thandapani","doi":"10.2478/tmmp-2021-0028","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0028","url":null,"abstract":"Abstract Some new oscillation criteria are obtained for a class of thirdorder quasi-linear Emden-Fowler differential equations with unbounded neutral coefficients of the form (a(t)(z″(t))α)′+f(t)yλ(g(t))=0,[(a(t){(z(t))^alpha })' + f(t){y^lambda }(g(t)) = 0,] where z(t) = y(t) + p(t)y(σ(t)) and α, λ are ratios of odd positive integers. The established results generalize, improve and complement to known results.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"80 1","pages":"1 - 14"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47589415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Oscillatory Behaviour of Second-Order Nonlinear Differential Equations with Mixed Neutral Terms","authors":"S. Grace, J. Graef, Tongxing Li, E. Tunç","doi":"10.2478/tmmp-2021-0023","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0023","url":null,"abstract":"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.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"79 1","pages":"119 - 134"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46611585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Properties of the Katugampola Fractional Operators","authors":"Barbara Łupińska","doi":"10.2478/tmmp-2021-0024","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0024","url":null,"abstract":"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.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"79 1","pages":"135 - 148"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47022183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Application of the Extended Fan Sub-Equation Method to Time Fractional Burgers-Fisher Equation","authors":"Djouaher Abbas, A. Kadem","doi":"10.2478/tmmp-2021-0016","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0016","url":null,"abstract":"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.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"79 1","pages":"1 - 12"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42859463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
G. Chatzarakis, A. George Maria Selvam, R. Janagaraj, G. Miliaras
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(ℓ)[α(ℓ)+β(ℓ)Δμu(ℓ)]η]+ϕ(ℓ)f[G(ℓ)]=0,ℓ∈Nℓ0+1−μ, [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 ℓ0>0, G(ℓ)=∑j=ℓ0ℓ−1+μ(ℓ−j−1)(−μ)u(j)[{ell _0} > 0,quad G(ell ) = sumlimits_{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.
摘要基于广义Riccati变换技术和一些不等式，研究了一类离散非线性分数阶导数方程Δ[γ(r)[α(r)+β(r)Δμu(r)]η]+ϕ(r)f[G(r)]=0, r∈N, r 0+1−μ， [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 }},]，其中，r 0>， G(r)=∑j= r 0, r (r)+ μ(r−j−1)(−μ)u(j) [{ell _0} > 0,quad G(ell ) = sumlimits_{j = {ell _0}}^{ell - 1 + mu } {{{(ell - j - 1)}^{( - mu )}}u(j)} ]， Δμ是阶μ的导数的Riemann-Liouville (R-L)差分算子，0 < μ≤1，η是奇数正整数的商。通过算例验证了理论结果的有效性。
{"title":"Oscillation Behaviour of Solutions for a Class of a Discrete Nonlinear Fractional-Order Derivatives","authors":"G. Chatzarakis, A. George Maria Selvam, R. Janagaraj, G. Miliaras","doi":"10.2478/tmmp-2021-0022","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0022","url":null,"abstract":"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(ℓ)[α(ℓ)+β(ℓ)Δμu(ℓ)]η]+ϕ(ℓ)f[G(ℓ)]=0,ℓ∈Nℓ0+1−μ, [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 ℓ0>0, G(ℓ)=∑j=ℓ0ℓ−1+μ(ℓ−j−1)(−μ)u(j)[{ell _0} > 0,quad G(ell ) = sumlimits_{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.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"79 1","pages":"101 - 118"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42530623","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In this paper, we analyse stability of survival of red blood cells in animal fractional order model with time delay. Results have been illustrated by numerical simulations.
摘要在本文中，我们分析了具有时间延迟的动物分数阶模型中红细胞存活的稳定性。数值模拟已经说明了结果。
{"title":"A Fractional Order Delay Differential Model for Survival of Red Blood Cells in an Animal: Stability Analysis","authors":"Santqshi Panigrahi, Sunita Chand","doi":"10.2478/tmmp-2021-0034","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0034","url":null,"abstract":"Abstract In this paper, we analyse stability of survival of red blood cells in animal fractional order model with time delay. Results have been illustrated by numerical simulations.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"80 1","pages":"135 - 144"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42708322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Finite volume (FV) numerical schemes for the approximation of Affine Morphological Scale Space (AMSS) model are proposed. For the scheme parameter θ, 0 ≤ θ ≤ 1 the numerical schemes of Crank-Nicolson type were derived. The explicit (θ = 0), semi-implicit, fully-implicit (θ = 1) and Crank-Nicolson (θ = 0.5) schemes were studied. Stability estimates for explicit and implicit schemes were derived. On several numerical experiments the properties and comparison of the numerical schemes are presented.
{"title":"Finite Volume Schemes for the Affine Morphological Scale Space (Amss) Model","authors":"A. Handlovicová, K. Mikula","doi":"10.2478/tmmp-2021-0031","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0031","url":null,"abstract":"Abstract Finite volume (FV) numerical schemes for the approximation of Affine Morphological Scale Space (AMSS) model are proposed. For the scheme parameter θ, 0 ≤ θ ≤ 1 the numerical schemes of Crank-Nicolson type were derived. The explicit (θ = 0), semi-implicit, fully-implicit (θ = 1) and Crank-Nicolson (θ = 0.5) schemes were studied. Stability estimates for explicit and implicit schemes were derived. On several numerical experiments the properties and comparison of the numerical schemes are presented.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"80 1","pages":"53 - 70"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45498734","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. E. Hajaji, A. Serghini, S. Melliani, J. E. Ghordaf, K. Hilal
Abstract In this paper, we develop a new numerical algorithm for solving a time dependent convection-diffusion equation with Dirichlet’s type boundary conditions. The method comprises the horizontal method of lines for time integration and (θ-method, θ ∈ [1/2, 1] (θ = 1 corresponds to the backward Euler method and θ = 1/2 corresponds to the Crank-Nicolson method) to discretize in temporal direction and the quintic spline collocation method. The convergence analysis of proposed method is discussed in detail, and it justified that the approximate solution converges to the exact solution of orders O(Δt + h3) for the backward Euler method and O(Δt2 + h3) for the Crank-Nicolson method, where Δt and h are mesh sizes in the time and space directions, respectively. It is also shown that the proposed method is unconditionally stable. This scheme is applied on some test examples, the numerical results illustrate the efficiency of the method and confirm the theoretical behaviour of the rates of convergence. Results shown by this method are in good agreement with the known exact solutions. The produced results are also more accurate than some available results given in the literature.
{"title":"A Quintic Spline Collocation Method for Solving Time-Dependent Convection-Diffusion Problems","authors":"A. E. Hajaji, A. Serghini, S. Melliani, J. E. Ghordaf, K. Hilal","doi":"10.2478/tmmp-2021-0029","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0029","url":null,"abstract":"Abstract In this paper, we develop a new numerical algorithm for solving a time dependent convection-diffusion equation with Dirichlet’s type boundary conditions. The method comprises the horizontal method of lines for time integration and (θ-method, θ ∈ [1/2, 1] (θ = 1 corresponds to the backward Euler method and θ = 1/2 corresponds to the Crank-Nicolson method) to discretize in temporal direction and the quintic spline collocation method. The convergence analysis of proposed method is discussed in detail, and it justified that the approximate solution converges to the exact solution of orders O(Δt + h3) for the backward Euler method and O(Δt2 + h3) for the Crank-Nicolson method, where Δt and h are mesh sizes in the time and space directions, respectively. It is also shown that the proposed method is unconditionally stable. This scheme is applied on some test examples, the numerical results illustrate the efficiency of the method and confirm the theoretical behaviour of the rates of convergence. Results shown by this method are in good agreement with the known exact solutions. The produced results are also more accurate than some available results given in the literature.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"80 1","pages":"15 - 34"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44877200","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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