Abstract. We consider the existence of limit cycles for a predator-prey system with a functional ... more Abstract. We consider the existence of limit cycles for a predator-prey system with a functional response. The system has two or more parameters that represent the intrinsic rate of the predator population. A necessary and sufficient condition for the uniqueness of limit cycles in this system is presented. Such result will usually lead to a bifurcation curve. 2000 Mathematics Subject Classification. 92D40. 1. Introduction. The
In this paper we study a fuzzy predator-prey model with functional response arctan(ax). The fuzzy... more In this paper we study a fuzzy predator-prey model with functional response arctan(ax). The fuzzy derivatives are approximated using the generalized Hukuhara derivative. To execute the numerical simulation, we use the fuzzy Runge-Kutta method. The results obtained over time for the evolution and the population are presented numerically and graphically with some conclusions.
This article considers the numerical simulation of fuzzy two-point boundary value problems (FBVP)... more This article considers the numerical simulation of fuzzy two-point boundary value problems (FBVP) using general linear method (GLM). The author derived the method, which is a combination of a Runge-Kutta type method and multi-step method. It is originally designed to solve initial value problems. It requires fewer function evaluations than the traditional Runge-Kutta methods making it computationally more efficient in achieving the required accuracy. The author will utilize the combination of the GLM with initial value methods to solve the linear fuzzy BVP's and a shooting-like method for the nonlinear cases. Numerical testing and simulation of several examples, considered by other authors, will be presented to show the efficiency of the proposed method.
Following earlier works on second- and fourth-order problems, we develop an efficient method base... more Following earlier works on second- and fourth-order problems, we develop an efficient method based on the Adomian decomposition for computing the eigenelements of sixth-order Sturm-Liouville boundary value problems. Numerical examples show that the method proposed is easy to implement and produces accurate results.
We consider the numerical solution of the well known Falkner-Skan problem, which is third order n... more We consider the numerical solution of the well known Falkner-Skan problem, which is third order nonlinear boundary value problem. The approach we are going to follow is to first transform the third order boundary value problem on the semi-finite domain into a second order nonlinear boundary value problem on a finite domain through introducing a special transformation. The resulting two-point boundary value problem is then treated numerically using the sinc-collocation method which is known to converge exponentially. Numerical results will be presented for various values of the parameters representing various types of flows. Comparison with the work of others will also be done to show the accuracy of the Sinc method.
Chaos: An Interdisciplinary Journal of Nonlinear Science
We consider the numerical solution of a third-order Falkner-Skan-like boundary value problem aris... more We consider the numerical solution of a third-order Falkner-Skan-like boundary value problem arising in boundary layer theory. The problem is defined on a semi-infinite interval [0,∞) with a condition given at ∞. We first transform the problem into a second-order boundary value problem defined on a finite interval [γ,1]. To solve the resulting boundary value problem, we developed an iterative finite-difference scheme based on Newton's quasilinearization. At every step, the linearized differential equation is approximated using the finite-difference method. Numerical results will be presented to demonstrate the efficiency of the method and will be compared with other results presented in the literature.
We will consider index‐2 differential algebraic systems. Since they are usually harder to solve, ... more We will consider index‐2 differential algebraic systems. Since they are usually harder to solve, we will show how to reduce the index 2 problem to index 1 DAE which becomes easier to solve numerically. For the numerical treatment, we will treat the resulting index‐1 DAE using power series solutions coupled with pade' approximation for better convergence results. Numerical examples will be presented also.
Springer Proceedings in Mathematics & Statistics, 2013
We consider a singularly perturbed one-dimensional reaction-diffusion three-point boundary value ... more We consider a singularly perturbed one-dimensional reaction-diffusion three-point boundary value problem. To approximate the solution numerically, we employ an exponentially fitted finite uniform difference scheme defined on a piecewise uniform Shishkin mesh which is second order and uniformly convergent independent of the perturbation parameter. We will present some numerical examples to show the efficiency of the proposed method.
Advanced Studies in Contemporary Mathematics (Kyungshang)
We consider the homotopy perturbation method in the approximation of eigenvalues and eigenfunctio... more We consider the homotopy perturbation method in the approximation of eigenvalues and eigenfunctions for a class of two point boundary value problems. It is shown that the method is easy to use and competes well with other methods. Numerical examples are presented to show the efficiency of the method proposed.
We will present an algorithmic approach to the implementation of a fourth order two stage implici... more We will present an algorithmic approach to the implementation of a fourth order two stage implicit Runge-Kutta method to solve periodic second order initial value problems. The systems involved will be solved using some type of factorization that usually involves both complex and real arithmetic. We will consider the real type case which will be efficient and leads to a system that is one fourth the size of similar systems using normal implicit Runge-Kutta method. We will present some numerical examples to show the efficiency of the method.
A numerical algorithm is proposed to solve a class of fourth order singularly perturbed two point... more A numerical algorithm is proposed to solve a class of fourth order singularly perturbed two point boundary value problems (BVP). The method starts by transforming the BVP into a system of two second order ordinary differential equations with appropriate boundary conditions. The interval over which the BVP is defined will be subdivided into three disjoint regions. The system will then
Journal of Dynamical Systems and Geometric Theories, 2008
Abstract We will consider the solution of Hessenberg index-2 differential algebraic equations(DAE... more Abstract We will consider the solution of Hessenberg index-2 differential algebraic equations(DAE) numerically. We first introduce a method to reduce the index from 2 to index 1 resulting in an easier problem to solve. For the numerical treatment, we will use power series solutions coupled with pade’ approximation for better convergence results. Numerical examples will be presented also.
We consider a predator prey system with the functional response of the form µ(x) = arctan(ax); a ... more We consider a predator prey system with the functional response of the form µ(x) = arctan(ax); a > 0. The main concern in this paper is the existence of limit cycles for such system. A necessary and sufficient condition for the nonexis- tence of limit cycles is given for such system.
Journal of Dynamical Systems and Geometric Theories, 2009
Abstract We will consider the numerical solution of the Hessenberg linear index-2 differential al... more Abstract We will consider the numerical solution of the Hessenberg linear index-2 differential algebraic equations (DAE’s) using the differential transform method. We will first present a method to reduce the index of the DAE to index-1. This results in a problem easier to solve. Numerical examples in addition to the comparison with the work of others will also be presented to show the efficiency of the method.
... in Simulation 36 (1994) 173184 The use of block elimination for the calculation of some types... more ... in Simulation 36 (1994) 173184 The use of block elimination for the calculation of some types of singularities efficiently Basem S. Attili, Yaqoub Shehadeh King Fahd University of Petroleum and Minerals, Mathematics Department, Dhahran 31261, Saudi Arabia Abstract We will ...
Abstract. We consider the existence of limit cycles for a predator-prey system with a functional ... more Abstract. We consider the existence of limit cycles for a predator-prey system with a functional response. The system has two or more parameters that represent the intrinsic rate of the predator population. A necessary and sufficient condition for the uniqueness of limit cycles in this system is presented. Such result will usually lead to a bifurcation curve. 2000 Mathematics Subject Classification. 92D40. 1. Introduction. The
In this paper we study a fuzzy predator-prey model with functional response arctan(ax). The fuzzy... more In this paper we study a fuzzy predator-prey model with functional response arctan(ax). The fuzzy derivatives are approximated using the generalized Hukuhara derivative. To execute the numerical simulation, we use the fuzzy Runge-Kutta method. The results obtained over time for the evolution and the population are presented numerically and graphically with some conclusions.
This article considers the numerical simulation of fuzzy two-point boundary value problems (FBVP)... more This article considers the numerical simulation of fuzzy two-point boundary value problems (FBVP) using general linear method (GLM). The author derived the method, which is a combination of a Runge-Kutta type method and multi-step method. It is originally designed to solve initial value problems. It requires fewer function evaluations than the traditional Runge-Kutta methods making it computationally more efficient in achieving the required accuracy. The author will utilize the combination of the GLM with initial value methods to solve the linear fuzzy BVP's and a shooting-like method for the nonlinear cases. Numerical testing and simulation of several examples, considered by other authors, will be presented to show the efficiency of the proposed method.
Following earlier works on second- and fourth-order problems, we develop an efficient method base... more Following earlier works on second- and fourth-order problems, we develop an efficient method based on the Adomian decomposition for computing the eigenelements of sixth-order Sturm-Liouville boundary value problems. Numerical examples show that the method proposed is easy to implement and produces accurate results.
We consider the numerical solution of the well known Falkner-Skan problem, which is third order n... more We consider the numerical solution of the well known Falkner-Skan problem, which is third order nonlinear boundary value problem. The approach we are going to follow is to first transform the third order boundary value problem on the semi-finite domain into a second order nonlinear boundary value problem on a finite domain through introducing a special transformation. The resulting two-point boundary value problem is then treated numerically using the sinc-collocation method which is known to converge exponentially. Numerical results will be presented for various values of the parameters representing various types of flows. Comparison with the work of others will also be done to show the accuracy of the Sinc method.
Chaos: An Interdisciplinary Journal of Nonlinear Science
We consider the numerical solution of a third-order Falkner-Skan-like boundary value problem aris... more We consider the numerical solution of a third-order Falkner-Skan-like boundary value problem arising in boundary layer theory. The problem is defined on a semi-infinite interval [0,∞) with a condition given at ∞. We first transform the problem into a second-order boundary value problem defined on a finite interval [γ,1]. To solve the resulting boundary value problem, we developed an iterative finite-difference scheme based on Newton's quasilinearization. At every step, the linearized differential equation is approximated using the finite-difference method. Numerical results will be presented to demonstrate the efficiency of the method and will be compared with other results presented in the literature.
We will consider index‐2 differential algebraic systems. Since they are usually harder to solve, ... more We will consider index‐2 differential algebraic systems. Since they are usually harder to solve, we will show how to reduce the index 2 problem to index 1 DAE which becomes easier to solve numerically. For the numerical treatment, we will treat the resulting index‐1 DAE using power series solutions coupled with pade' approximation for better convergence results. Numerical examples will be presented also.
Springer Proceedings in Mathematics & Statistics, 2013
We consider a singularly perturbed one-dimensional reaction-diffusion three-point boundary value ... more We consider a singularly perturbed one-dimensional reaction-diffusion three-point boundary value problem. To approximate the solution numerically, we employ an exponentially fitted finite uniform difference scheme defined on a piecewise uniform Shishkin mesh which is second order and uniformly convergent independent of the perturbation parameter. We will present some numerical examples to show the efficiency of the proposed method.
Advanced Studies in Contemporary Mathematics (Kyungshang)
We consider the homotopy perturbation method in the approximation of eigenvalues and eigenfunctio... more We consider the homotopy perturbation method in the approximation of eigenvalues and eigenfunctions for a class of two point boundary value problems. It is shown that the method is easy to use and competes well with other methods. Numerical examples are presented to show the efficiency of the method proposed.
We will present an algorithmic approach to the implementation of a fourth order two stage implici... more We will present an algorithmic approach to the implementation of a fourth order two stage implicit Runge-Kutta method to solve periodic second order initial value problems. The systems involved will be solved using some type of factorization that usually involves both complex and real arithmetic. We will consider the real type case which will be efficient and leads to a system that is one fourth the size of similar systems using normal implicit Runge-Kutta method. We will present some numerical examples to show the efficiency of the method.
A numerical algorithm is proposed to solve a class of fourth order singularly perturbed two point... more A numerical algorithm is proposed to solve a class of fourth order singularly perturbed two point boundary value problems (BVP). The method starts by transforming the BVP into a system of two second order ordinary differential equations with appropriate boundary conditions. The interval over which the BVP is defined will be subdivided into three disjoint regions. The system will then
Journal of Dynamical Systems and Geometric Theories, 2008
Abstract We will consider the solution of Hessenberg index-2 differential algebraic equations(DAE... more Abstract We will consider the solution of Hessenberg index-2 differential algebraic equations(DAE) numerically. We first introduce a method to reduce the index from 2 to index 1 resulting in an easier problem to solve. For the numerical treatment, we will use power series solutions coupled with pade’ approximation for better convergence results. Numerical examples will be presented also.
We consider a predator prey system with the functional response of the form µ(x) = arctan(ax); a ... more We consider a predator prey system with the functional response of the form µ(x) = arctan(ax); a > 0. The main concern in this paper is the existence of limit cycles for such system. A necessary and sufficient condition for the nonexis- tence of limit cycles is given for such system.
Journal of Dynamical Systems and Geometric Theories, 2009
Abstract We will consider the numerical solution of the Hessenberg linear index-2 differential al... more Abstract We will consider the numerical solution of the Hessenberg linear index-2 differential algebraic equations (DAE’s) using the differential transform method. We will first present a method to reduce the index of the DAE to index-1. This results in a problem easier to solve. Numerical examples in addition to the comparison with the work of others will also be presented to show the efficiency of the method.
... in Simulation 36 (1994) 173184 The use of block elimination for the calculation of some types... more ... in Simulation 36 (1994) 173184 The use of block elimination for the calculation of some types of singularities efficiently Basem S. Attili, Yaqoub Shehadeh King Fahd University of Petroleum and Minerals, Mathematics Department, Dhahran 31261, Saudi Arabia Abstract We will ...
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