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... 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 derivatives are approximated using the generalized Hukuhara derivative. To execute the numerical simulation, we use the fuzzy Runge-Kutta... 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.
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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.... 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.
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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... 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.
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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... 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.
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... 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.
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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... 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.
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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... 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.
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.... 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 implicit Runge-Kutta method to solve periodic second order initial value problems. The systems involved will be solved using some type of... 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.
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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... 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
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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... 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.
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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... 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.
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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.... 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.
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We will consider the application of the Adomian decomposition method to approximate the solution of the Boussinesq equation. Both the well-posed and the ill-posed cases will be considered. The results obtained will be compared to the... more
We will consider the application of the Adomian decomposition method to approximate the solution of the Boussinesq equation. Both the well-posed and the ill-posed cases will be considered. The results obtained will be compared to the theoretical solution for single soliton wave. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006
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... 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,... 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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Research Interests: Mathematics, Applied Mathematics, Applied Mathematics and Computational Science, Finite differences, Second Order, and 6 moreFinite Difference, Numerical Analysis and Computational Mathematics, Boundary Value Problem, Electrical And Electronic Engineering, Singularly perturbed system, and Finite Difference Scheme
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... more
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.
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Simple and multiple shooting methods are proposed to numerically solving singular boundary value problems with a regular singularity at one end of the interval. The singularity is first removed using series solution in the vicinity of the... more
Simple and multiple shooting methods are proposed to numerically solving singular boundary value problems with a regular singularity at one end of the interval. The singularity is first removed using series solution in the vicinity of the singular point to produce a regular boundary value problem. Numerical examples with some comparison of the work of others are also included.
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Initial value solvers through shooting methods will be used to compute solutions to the primary two-point boundary value problem arising in modeling viscoelastic flow. We will show that the classical Runge-Kutta methods will have at least... more
Initial value solvers through shooting methods will be used to compute solutions to the primary two-point boundary value problem arising in modeling viscoelastic flow. We will show that the classical Runge-Kutta methods will have at least h -order of convergence due to the presence of the singularity. Comparison with the work of others will also be presented through some numerical
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... DOI: 10.1080/0020716042000272575 Basem S. Attili a * , Muhammed I. Syam ... 3–207 View all references and Rheinboldt [3]3. Rheinboldt WC 1986 Numerical Analysis of Parameterized Nonlinear Equations New York J. Wiley View all... more
... DOI: 10.1080/0020716042000272575 Basem S. Attili a * , Muhammed I. Syam ... 3–207 View all references and Rheinboldt [3]3. Rheinboldt WC 1986 Numerical Analysis of Parameterized Nonlinear Equations New York J. Wiley View all references used continuation methods for ...
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We will consider the Hilber–Hughes–Taylor-α (HHT-α ) method to solve periodic second-order initial value problems arising in, e.g. mechanics. We will consider the analysis of the method when applied to such problems. Second-order... more
We will consider the Hilber–Hughes–Taylor-α (HHT-α ) method to solve periodic second-order initial value problems arising in, e.g. mechanics. We will consider the analysis of the method when applied to such problems. Second-order convergence is theoretically demonstrated and numerically illustrated. In addition, we will consider the efficient implementation of an implicit fourth-order Runge–Kutta scheme. Numerical details and examples will also
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An iterative method that uses generalized divided differences to solve nonlinear operator equations is proposed. Local and semi local convergence of the proposed method is shown. Numerical examples are also presented to demonstrate the... more
An iterative method that uses generalized divided differences to solve nonlinear operator equations is proposed. Local and semi local convergence of the proposed method is shown. Numerical examples are also presented to demonstrate the efficiency of the method.
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We will consider the use of Predictor-corrector method to trace parameterized curves. Homotopy methods will be needed since a Newton like method cannot be used to solve the nonlinear systems involved. This is due to the fact that not much... more
We will consider the use of Predictor-corrector method to trace parameterized curves. Homotopy methods will be needed since a Newton like method cannot be used to solve the nonlinear systems involved. This is due to the fact that not much information is available about the zero point of the system. We will also consider systems which involve the presence of a natural parameter; in particular, tracing a parameter dependent curve which contains-a simple turning or bifurcation point at a critical value of the parameter.
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We consider simple and multiple shooting methods for simple turning points, perturbed bifurcation points, cubic turning points and cusps in boundary value problems for ordinary differential equations. We discretize the original problem... more
We consider simple and multiple shooting methods for simple turning points, perturbed bifurcation points, cubic turning points and cusps in boundary value problems for ordinary differential equations. We discretize the original problem using the shooting technique to obtain a finite dimensional problem. A direct method is used for the characetrization and computation of simple turning points. Some suitable extension of this direct method is employed for the computation of bifurcation points and cubic turning points and cusps. Enough numerical examples are solved to demonstrate that the method is efficient.
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We will consider the presence of a temperature gradient on the onset stability of the flow in a narrow gap between two rotating cylinders. The shooting method will be used to solve the eigenvalue problem representing the flow. We will... more
We will consider the presence of a temperature gradient on the onset stability of the flow in a narrow gap between two rotating cylinders. The shooting method will be used to solve the eigenvalue problem representing the flow. We will extend a method used by the author to compute the singularity. The resulting method is direct and produces an extended
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We will consider an extension of a direct method due to Griewank and Reddien for the characterization and computation of double singular points with corank 2. Singular points which satisfy certain type of symmetry will also be considered.... more
We will consider an extension of a direct method due to Griewank and Reddien for the characterization and computation of double singular points with corank 2. Singular points which satisfy certain type of symmetry will also be considered. The method used will produce an extended system which does not introduce the null vectors as variables, but gives a good idea
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We will consider the implementation of finite differences to the numerical computation of simple turning points, cubic turning points, simple bifurcation points and bifurcation points with corank 2 in B.V.Ps for ordinary differential... more
We will consider the implementation of finite differences to the numerical computation of simple turning points, cubic turning points, simple bifurcation points and bifurcation points with corank 2 in B.V.Ps for ordinary differential equations. The singularity will be characterized by an equation or a set of equations which will be augmented with the original system to produce a regular problem. Block elimination will be employed for saving on the amount of work while Richardson extrapolation will be used to produce accurate results. Numerical examples will be presented to show the efficiency of the above-described algorithm.
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A direct method for the numerical computation of pitchfork bifurcation points under certain symmetry conditions is presented. We will be interested in computing the critical parameter. The direct method presented produces a larger system... more
A direct method for the numerical computation of pitchfork bifurcation points under certain symmetry conditions is presented. We will be interested in computing the critical parameter. The direct method presented produces a larger system of full rank and hence solvable. The presence of symmetry will be of help since it will reduce the amount of work needed. Numerical experimentation will
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... ELSEVIER Applied Numerical Mathematics 25 (1997) 1-11 MATHEMATICS Tracing implicitly defined curves and the use of singular value decomposition Basem S. Attili K.EU.PM, PO Box 1927, Dhahran 31261, Saudi Arabia Abstract We consider... more
... ELSEVIER Applied Numerical Mathematics 25 (1997) 1-11 MATHEMATICS Tracing implicitly defined curves and the use of singular value decomposition Basem S. Attili K.EU.PM, PO Box 1927, Dhahran 31261, Saudi Arabia Abstract We consider tracing implicitly defined ...
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Simple and multiple shooting methods are proposed to numerically solving singular boundary value problems with a regular singularity at one end of the interval. The singularity is first removed using series solution in the vicinity of the... more
Simple and multiple shooting methods are proposed to numerically solving singular boundary value problems with a regular singularity at one end of the interval. The singularity is first removed using series solution in the vicinity of the singular point to produce a regular boundary value problem. Numerical examples with some comparison of the work of others are also included.
Research Interests:
Research Interests:
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... 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.
Research Interests:
We consider symmetry-breaking bifurcation points which arise in parameter-dependent nonlinear equations of the form f(x, λ) = 0. These types of bifurcation points are connected to pitchfork bifurcation points. A direct method is used to... more
We consider symmetry-breaking bifurcation points which arise in parameter-dependent nonlinear equations of the form f(x, λ) = 0. These types of bifurcation points are connected to pitchfork bifurcation points. A direct method is used to compute such points. Multiple shooting is used to discretise the two-point boundary-value problems to obtain a finite-dimensional problem.