In this article, we consider the non-linear difference equation(f(z + 1)f(z)-1)(f(z)f(z-1)-1) =P(z, f(z))/Q(z, f(z)),where P(z, f(z)) and Q(z, f(z)) are relatively prime polynomials in f(z) with rational coefficients....In this article, we consider the non-linear difference equation(f(z + 1)f(z)-1)(f(z)f(z-1)-1) =P(z, f(z))/Q(z, f(z)),where P(z, f(z)) and Q(z, f(z)) are relatively prime polynomials in f(z) with rational coefficients. For the above equation, the order of growth, the exponents of convergence of zeros and poles of its transcendental meromorphic solution f(z), and the exponents of convergence of poles of difference △f(z) and divided difference △f(z)/f(z)are estimated. Furthermore, we study the forms of rational solutions of the above equation.展开更多
The intuitive fuzzy set has found important application in decision-making and machine learning.To enrich and utilize the intuitive fuzzy set,this study designed and developed a deep neural network-based glaucoma eye ...The intuitive fuzzy set has found important application in decision-making and machine learning.To enrich and utilize the intuitive fuzzy set,this study designed and developed a deep neural network-based glaucoma eye detection using fuzzy difference equations in the domain where the retinal images converge.Retinal image detections are categorized as normal eye recognition,suspected glaucomatous eye recognition,and glaucomatous eye recognition.Fuzzy degrees associated with weighted values are calculated to determine the level of concentration between the fuzzy partition and the retinal images.The proposed model was used to diagnose glaucoma using retinal images and involved utilizing the Convolutional Neural Network(CNN)and deep learning to identify the fuzzy weighted regularization between images.This methodology was used to clarify the input images and make them adequate for the process of glaucoma detection.The objective of this study was to propose a novel approach to the early diagnosis of glaucoma using the Fuzzy Expert System(FES)and Fuzzy differential equation(FDE).The intensities of the different regions in the images and their respective peak levels were determined.Once the peak regions were identified,the recurrence relationships among those peaks were then measured.Image partitioning was done due to varying degrees of similar and dissimilar concentrations in the image.Similar and dissimilar concentration levels and spatial frequency generated a threshold image from the combined fuzzy matrix and FDE.This distinguished between a normal and abnormal eye condition,thus detecting patients with glaucomatous eyes.展开更多
A finite difference/spectral scheme is proposed for the time fractional Ito equation.The mass conservation and stability of the numerical solution are deduced by the energy method in the L^(2)norm form.To reduce the c...A finite difference/spectral scheme is proposed for the time fractional Ito equation.The mass conservation and stability of the numerical solution are deduced by the energy method in the L^(2)norm form.To reduce the computation costs,the fast Fourier transform technic is applied to a pair of equivalent coupled differential equations.The effectiveness of the proposed algorithm is verified by the first numerical example.The mass conservation property and stability statement are confirmed by two other numerical examples.展开更多
The main purpose of this paper is to study the dynamic behavior of the rational difference equation of the fourth order Where α, β and γ are positive constants and the initial conditions y<sub>-3</sub>,...The main purpose of this paper is to study the dynamic behavior of the rational difference equation of the fourth order Where α, β and γ are positive constants and the initial conditions y<sub>-3</sub>, y<sub>-2</sub>, y<sub>-1</sub>, y<sub>0</sub> are arbitrary positive real numbers. Also, we obtain the solution of some special cases of this equation and investigate the existence of a periodic solutions of these equations. Finally, some numerical examples will be given to explicate our results. .展开更多
In this paper, we study the existence of the transcendental meromorphic solution of the delay differential equations , where a(z) is a rational function, and are polynomials in w(z) with rational c...In this paper, we study the existence of the transcendental meromorphic solution of the delay differential equations , where a(z) is a rational function, and are polynomials in w(z) with rational coefficients, k is a positive integer. Under the assumption when above equations own transcendental meromorphic solutions with minimal hyper-type, we derive the concrete conditions on the degree of the right side of them. Specially, when w(z)=0 is a root of , its multiplicity is at most k. Some examples are given here to illustrate that our results are accurate.展开更多
We present the numerical method for solution of some linear and non-linear parabolic equation. Using idea [1], we will present the explicit unconditional stable scheme which has no restriction on the step size ratio k...We present the numerical method for solution of some linear and non-linear parabolic equation. Using idea [1], we will present the explicit unconditional stable scheme which has no restriction on the step size ratio k/h2 where k and h are step sizes for space and time respectively. We will also present numerical results to justify the present scheme.展开更多
This paper deals with a new higher order compact difference scheme, which is, O(h4) using coupled approach on the 19-point 3D stencil for the solution of three dimensional nonlinear biharmonic equations. At each inter...This paper deals with a new higher order compact difference scheme, which is, O(h4) using coupled approach on the 19-point 3D stencil for the solution of three dimensional nonlinear biharmonic equations. At each internal grid point, the solution u(x,y,z) and its Laplacian Δ4u are obtained. The resulting stencil algo-rithm is presented and hence this new algorithm can be easily incorporated to solve many problems. The present discretization allows us to use the Dirichlet boundary conditions only and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. Convergence analysis for a model problem is briefly discussed. The method is tested on three problems and compares very favourably with the corresponding second order approximation which we also discuss using coupled approach.展开更多
In this paper, we study the propagation and its failure to propagate (pinning) of a travelling wave in a Nagumo type equation, an equation that describes impulse propagation in nerve axons that also models population ...In this paper, we study the propagation and its failure to propagate (pinning) of a travelling wave in a Nagumo type equation, an equation that describes impulse propagation in nerve axons that also models population growth with Allee effect. An analytical solution is derived for the traveling wave and the work is extended to a discrete formulation with a piecewise linear reaction function. We propose an operator splitting numerical scheme to solve the equation and demonstrate that the wave either propagates or gets pinned based on how the spatial mesh is chosen.展开更多
This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either...This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.展开更多
We applied a spatial high-order finite-difference-time-domain (HO-FDTD) scheme to solve 2D Maxwell’s equations in order to develop a fluid model employed to study the production of terahertz radiation by the filament...We applied a spatial high-order finite-difference-time-domain (HO-FDTD) scheme to solve 2D Maxwell’s equations in order to develop a fluid model employed to study the production of terahertz radiation by the filamentation of two femtosecond lasers in air plasma. We examined the performance of the applied scheme, in this context, we implemented the developed model to study selected phenomena in terahertz radiation production, such as the excitation energy and conversion efficiency of the produced THz radiation, in addition to the influence of the pulse chirping on properties of the produced radiation. The obtained numerical results have clarified that the applied HO-FDTD scheme is precisely accurate to solve Maxwell’s equations and sufficiently valid to study the production of terahertz radiation by the filamentation of two femtosecond lasers in air plasma.展开更多
The aim of this paper is to give an appropriate numerical method to solve Allen-Cahn equation, with Dirichlet or Neumann boundary condition. The time discretization involves an explicit scheme for the nonlinear part o...The aim of this paper is to give an appropriate numerical method to solve Allen-Cahn equation, with Dirichlet or Neumann boundary condition. The time discretization involves an explicit scheme for the nonlinear part of the operator and an implicit Euler discretization of the linear part. Finite difference schemes are used for the spatial part. This finally leads to the numerical solution of a sparse linear system that can be solved efficiently.展开更多
In this paper, we use our method to solve the extended Lotka-Volterra equation and discrete KdV equation. With the help of Maple, we obtain a number of exact solutions to the two equations including soliton solutions ...In this paper, we use our method to solve the extended Lotka-Volterra equation and discrete KdV equation. With the help of Maple, we obtain a number of exact solutions to the two equations including soliton solutions presented by hyperbolic functions of sinh and cosh, periodic solutions presented by trigonometric functions of sin and cos, and rational solutions. This method can be used to solve some other nonlinear difference-differential equations.展开更多
Using Nevanlinna theory of the value distribution of meromorphic functions and Wiman-Valiron theory of entire functions, we investigate the problem of growth order of solutions of a type of systems of difference equat...Using Nevanlinna theory of the value distribution of meromorphic functions and Wiman-Valiron theory of entire functions, we investigate the problem of growth order of solutions of a type of systems of difference equations, and extend some results of the growth order of solutions of systems of differential equations to systems of difference equations.展开更多
Some oscillation criteria are established by Raccati transformation techniques for the following second-order nonlinear neutral difference equation △(pn(△(Xn + CnXn-τ))^γ) + qnX^Bn-σ = 0, n :0, 1, 2...wh...Some oscillation criteria are established by Raccati transformation techniques for the following second-order nonlinear neutral difference equation △(pn(△(Xn + CnXn-τ))^γ) + qnX^Bn-σ = 0, n :0, 1, 2...which extend and include several oscillation criteria in [11], and also correct a theorem and its proof in [10].展开更多
Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference...Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form {j=1∑nαj(z)f1(λj1)(z+cj)=R2(z,f2(z)),j=1∑nβj(z)f2(λj2)(z+cj)=R1(z,f1(z)). where λij (j = 1, 2,…, n; i = 1, 2) are finite non-negative integers, and cj (j = 1, 2,… , n) are distinct, nonzero complex numbers, αj(z), βj(z) (j = 1,2,… ,n) are small functions relative to fi(z) (i =1, 2) respectively, Ri(z, f(z)) (i = 1, 2) are rational in fi(z) (i =1, 2) with coefficients which are small functions of fi(z) (i = 1, 2) respectively.展开更多
We consider the existence, the growth, poles, zeros, fixed points and the Borel exceptional value of solutions for the following difference equations relating to Gamma function y(z + 1) -y(z) = R(z) and y(z ...We consider the existence, the growth, poles, zeros, fixed points and the Borel exceptional value of solutions for the following difference equations relating to Gamma function y(z + 1) -y(z) = R(z) and y(z + 1) = P (z)y(z).展开更多
In this article, we study the complex oscillation problems of entire solutions to homogeneous and nonhomogeneous linear difference equations, and obtain some relations of the exponent of convergence of zeros and the o...In this article, we study the complex oscillation problems of entire solutions to homogeneous and nonhomogeneous linear difference equations, and obtain some relations of the exponent of convergence of zeros and the order of growth of entire solutions to complex linear difference equations.展开更多
This article studies the existence of analytic invariant curves for a nonlinear second order difference equation which was modeled from macroeconomics of the business cycle. The author not only discusses the case of t...This article studies the existence of analytic invariant curves for a nonlinear second order difference equation which was modeled from macroeconomics of the business cycle. The author not only discusses the case of the eigenvalue off the unit circle S^1 and the case on S^1 with the Diophantine condition but also considers the case of the eigenvalue at a root of the unity, which obviously violates the Diophantine condition.展开更多
In this article, the existence of finite order entire solutions of nonlinear difference equations fn+ Pd(z, f) = p1 eα1 z+ p2 eα2 z are studied, where n ≥ 2 is an integer, Pd(z, f) is a difference polynomial ...In this article, the existence of finite order entire solutions of nonlinear difference equations fn+ Pd(z, f) = p1 eα1 z+ p2 eα2 z are studied, where n ≥ 2 is an integer, Pd(z, f) is a difference polynomial in f of degree d(≤ n-2), p1, p2 are small meromorphic functions of ez, and α1, α2 are nonzero constants. Some necessary conditions are given to guarantee that the above equation has an entire solution of finite order. As its applications, we also find some type of nonlinear difference equations having no finite order entire solutions.展开更多
基金supported by the National Natural Science Foundation of China(11371225)National Natural Science Foundation of Guangdong Province(2016A030313686)
文摘In this article, we consider the non-linear difference equation(f(z + 1)f(z)-1)(f(z)f(z-1)-1) =P(z, f(z))/Q(z, f(z)),where P(z, f(z)) and Q(z, f(z)) are relatively prime polynomials in f(z) with rational coefficients. For the above equation, the order of growth, the exponents of convergence of zeros and poles of its transcendental meromorphic solution f(z), and the exponents of convergence of poles of difference △f(z) and divided difference △f(z)/f(z)are estimated. Furthermore, we study the forms of rational solutions of the above equation.
基金funding the publication of this research through the Researchers Supporting Program (RSPD2023R809),King Saud University,Riyadh,Saudi Arabia.
文摘The intuitive fuzzy set has found important application in decision-making and machine learning.To enrich and utilize the intuitive fuzzy set,this study designed and developed a deep neural network-based glaucoma eye detection using fuzzy difference equations in the domain where the retinal images converge.Retinal image detections are categorized as normal eye recognition,suspected glaucomatous eye recognition,and glaucomatous eye recognition.Fuzzy degrees associated with weighted values are calculated to determine the level of concentration between the fuzzy partition and the retinal images.The proposed model was used to diagnose glaucoma using retinal images and involved utilizing the Convolutional Neural Network(CNN)and deep learning to identify the fuzzy weighted regularization between images.This methodology was used to clarify the input images and make them adequate for the process of glaucoma detection.The objective of this study was to propose a novel approach to the early diagnosis of glaucoma using the Fuzzy Expert System(FES)and Fuzzy differential equation(FDE).The intensities of the different regions in the images and their respective peak levels were determined.Once the peak regions were identified,the recurrence relationships among those peaks were then measured.Image partitioning was done due to varying degrees of similar and dissimilar concentrations in the image.Similar and dissimilar concentration levels and spatial frequency generated a threshold image from the combined fuzzy matrix and FDE.This distinguished between a normal and abnormal eye condition,thus detecting patients with glaucomatous eyes.
基金the National Natural Science Foundation of China(No.11701103)the Young Top-notch Talent Program of Guangdong Province of China(No.2017GC010379)+4 种基金the Natural Science Foundation of Guangdong Province of China(No.2022A1515012147)the Project of Science and Technology of Guangzhou of China(No.202102020704)the Opening Project of Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University of China(2021023)the Science and Technology Development Fund,Macao SAR(File No.0005/2019/A)the University of Macao of China(File Nos.MYRG2020-00035-FST,MYRG2018-00047-FST).
文摘A finite difference/spectral scheme is proposed for the time fractional Ito equation.The mass conservation and stability of the numerical solution are deduced by the energy method in the L^(2)norm form.To reduce the computation costs,the fast Fourier transform technic is applied to a pair of equivalent coupled differential equations.The effectiveness of the proposed algorithm is verified by the first numerical example.The mass conservation property and stability statement are confirmed by two other numerical examples.
文摘The main purpose of this paper is to study the dynamic behavior of the rational difference equation of the fourth order Where α, β and γ are positive constants and the initial conditions y<sub>-3</sub>, y<sub>-2</sub>, y<sub>-1</sub>, y<sub>0</sub> are arbitrary positive real numbers. Also, we obtain the solution of some special cases of this equation and investigate the existence of a periodic solutions of these equations. Finally, some numerical examples will be given to explicate our results. .
文摘In this paper, we study the existence of the transcendental meromorphic solution of the delay differential equations , where a(z) is a rational function, and are polynomials in w(z) with rational coefficients, k is a positive integer. Under the assumption when above equations own transcendental meromorphic solutions with minimal hyper-type, we derive the concrete conditions on the degree of the right side of them. Specially, when w(z)=0 is a root of , its multiplicity is at most k. Some examples are given here to illustrate that our results are accurate.
文摘We present the numerical method for solution of some linear and non-linear parabolic equation. Using idea [1], we will present the explicit unconditional stable scheme which has no restriction on the step size ratio k/h2 where k and h are step sizes for space and time respectively. We will also present numerical results to justify the present scheme.
文摘This paper deals with a new higher order compact difference scheme, which is, O(h4) using coupled approach on the 19-point 3D stencil for the solution of three dimensional nonlinear biharmonic equations. At each internal grid point, the solution u(x,y,z) and its Laplacian Δ4u are obtained. The resulting stencil algo-rithm is presented and hence this new algorithm can be easily incorporated to solve many problems. The present discretization allows us to use the Dirichlet boundary conditions only and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. Convergence analysis for a model problem is briefly discussed. The method is tested on three problems and compares very favourably with the corresponding second order approximation which we also discuss using coupled approach.
文摘In this paper, we study the propagation and its failure to propagate (pinning) of a travelling wave in a Nagumo type equation, an equation that describes impulse propagation in nerve axons that also models population growth with Allee effect. An analytical solution is derived for the traveling wave and the work is extended to a discrete formulation with a piecewise linear reaction function. We propose an operator splitting numerical scheme to solve the equation and demonstrate that the wave either propagates or gets pinned based on how the spatial mesh is chosen.
基金supported by the NSF under Grant DMS-2208391sponsored by the NSF under Grant DMS-1753581.
文摘This paper provides a study on the stability and time-step constraints of solving the linearized Korteweg-de Vries(KdV)equation,using implicit-explicit(IMEX)Runge-Kutta(RK)time integration methods combined with either finite difference(FD)or local discontinuous Galerkin(DG)spatial discretization.We analyze the stability of the fully discrete scheme,on a uniform mesh with periodic boundary conditions,using the Fourier method.For the linearized KdV equation,the IMEX schemes are stable under the standard Courant-Friedrichs-Lewy(CFL)conditionτ≤λh.Here,λis the CFL number,τis the time-step size,and h is the spatial mesh size.We study several IMEX schemes and characterize their CFL number as a function ofθ=d/h^(2)with d being the dispersion coefficient,which leads to several interesting observations.We also investigate the asymptotic behaviors of the CFL number for sufficiently refined meshes and derive the necessary conditions for the asymptotic stability of the IMEX-RK methods.Some numerical experiments are provided in the paper to illustrate the performance of IMEX methods under different time-step constraints.
文摘We applied a spatial high-order finite-difference-time-domain (HO-FDTD) scheme to solve 2D Maxwell’s equations in order to develop a fluid model employed to study the production of terahertz radiation by the filamentation of two femtosecond lasers in air plasma. We examined the performance of the applied scheme, in this context, we implemented the developed model to study selected phenomena in terahertz radiation production, such as the excitation energy and conversion efficiency of the produced THz radiation, in addition to the influence of the pulse chirping on properties of the produced radiation. The obtained numerical results have clarified that the applied HO-FDTD scheme is precisely accurate to solve Maxwell’s equations and sufficiently valid to study the production of terahertz radiation by the filamentation of two femtosecond lasers in air plasma.
文摘The aim of this paper is to give an appropriate numerical method to solve Allen-Cahn equation, with Dirichlet or Neumann boundary condition. The time discretization involves an explicit scheme for the nonlinear part of the operator and an implicit Euler discretization of the linear part. Finite difference schemes are used for the spatial part. This finally leads to the numerical solution of a sparse linear system that can be solved efficiently.
文摘In this paper, we use our method to solve the extended Lotka-Volterra equation and discrete KdV equation. With the help of Maple, we obtain a number of exact solutions to the two equations including soliton solutions presented by hyperbolic functions of sinh and cosh, periodic solutions presented by trigonometric functions of sin and cos, and rational solutions. This method can be used to solve some other nonlinear difference-differential equations.
基金supported by the Natural Science Foundation of China (10471065)the Natural Science Foundation of Guangdong Province (04010474)
文摘Using Nevanlinna theory of the value distribution of meromorphic functions and Wiman-Valiron theory of entire functions, we investigate the problem of growth order of solutions of a type of systems of difference equations, and extend some results of the growth order of solutions of systems of differential equations to systems of difference equations.
基金revised September 27,2005.Research support by Natural Science Foundation of China(10271043)
文摘Some oscillation criteria are established by Raccati transformation techniques for the following second-order nonlinear neutral difference equation △(pn(△(Xn + CnXn-τ))^γ) + qnX^Bn-σ = 0, n :0, 1, 2...which extend and include several oscillation criteria in [11], and also correct a theorem and its proof in [10].
基金supported by the National Natural Science Foundation of China(10471067)NSF of Guangdong Province(04010474)
文摘Applying Nevanlinna theory of the value distribution of meromorphic functions, we mainly study the growth and some other properties of meromorphic solutions of the type of system of complex differential and difference equations of the following form {j=1∑nαj(z)f1(λj1)(z+cj)=R2(z,f2(z)),j=1∑nβj(z)f2(λj2)(z+cj)=R1(z,f1(z)). where λij (j = 1, 2,…, n; i = 1, 2) are finite non-negative integers, and cj (j = 1, 2,… , n) are distinct, nonzero complex numbers, αj(z), βj(z) (j = 1,2,… ,n) are small functions relative to fi(z) (i =1, 2) respectively, Ri(z, f(z)) (i = 1, 2) are rational in fi(z) (i =1, 2) with coefficients which are small functions of fi(z) (i = 1, 2) respectively.
基金supported by the National Natural Science Foundation of China (10871076 11026096)
文摘We consider the existence, the growth, poles, zeros, fixed points and the Borel exceptional value of solutions for the following difference equations relating to Gamma function y(z + 1) -y(z) = R(z) and y(z + 1) = P (z)y(z).
基金supported by the National Natural Science Foundation of China (11171119 and 10871076)
文摘In this article, we study the complex oscillation problems of entire solutions to homogeneous and nonhomogeneous linear difference equations, and obtain some relations of the exponent of convergence of zeros and the order of growth of entire solutions to complex linear difference equations.
基金supported by the Fund of Educational Reform Project of Guangxi Province of China (200710961)the Scientific Research Foundation of the Education Department of Guangxi Province of China (200707MS112)+1 种基金the Natural Science Fund of Hechi University (2006N001)the fund of Key discipline of applied mathematics of Hechi University (200725)
文摘This article studies the existence of analytic invariant curves for a nonlinear second order difference equation which was modeled from macroeconomics of the business cycle. The author not only discusses the case of the eigenvalue off the unit circle S^1 and the case on S^1 with the Diophantine condition but also considers the case of the eigenvalue at a root of the unity, which obviously violates the Diophantine condition.
基金supported by the National Natural Science Foundation of China(11661044)
文摘In this article, the existence of finite order entire solutions of nonlinear difference equations fn+ Pd(z, f) = p1 eα1 z+ p2 eα2 z are studied, where n ≥ 2 is an integer, Pd(z, f) is a difference polynomial in f of degree d(≤ n-2), p1, p2 are small meromorphic functions of ez, and α1, α2 are nonzero constants. Some necessary conditions are given to guarantee that the above equation has an entire solution of finite order. As its applications, we also find some type of nonlinear difference equations having no finite order entire solutions.