Ibn AL-Haitham Journal For Pure and Applied Sciences
This paper sheds the light on the vital role that fractional ordinary differential equations(FrOD... more This paper sheds the light on the vital role that fractional ordinary differential equations(FrODEs) play in the mathematical modeling and in real life, particularly in the physical conditions. Furthermore, if the problem is handled directly by using numerical method, it is a far more powerful and efficient numerical method in terms of computational time, number of function evaluations, and precision. In this paper, we concentrate on the derivation of the direct numerical methods for solving fifth-order FrODEs in one, two, and three stages. Additionally, it is important to note that the RKM-numerical methods with two- and three-stages for solving fifth-order ODEs are convenient, for solving class's fifth-order FrODEs. Numerical examples have been analyzed to demonstrate the efficacy of the new methods in comparison to the analytical method. Therefore, the numerical compression is carried out to confirm the efficiency and precision of the modified numerical methods. Significantl...
In this study, the development of direct explicit numerical integrators of RK-type for solving a ... more In this study, the development of direct explicit numerical integrators of RK-type for solving a class of ninth-order ordinary differential equations (ODEs) to develop the computational efficiency of the methods. The objective of this paper is to generalize the numerical integrators of RK type for solving classes of ODEs of orders up to eighth-order ODEs. Using the approach of Taylor-expansion, we have derived the order-conditions (OCs) for RKM integrators. Based on these OCs, developed method of ninth-order with five-stage has been derived. The zero stability of RKM integrator is proven. Stability polynomial of RKM integrator for test problem has been introduced. The advantage and the efficiency of the proposed method have been shown clearly using the numerical results which are agree well with analytical solutions due to the proposed method is more efficient and accurate method.
In order to solve general seventh-order ordinary differential equations (ODEs), this study will d... more In order to solve general seventh-order ordinary differential equations (ODEs), this study will develop an implicit block method with three points of the form y(7)(ξ)=f(ξ,y(ξ),y′(ξ),y″(ξ),y‴(ξ),y(4)(ξ),y(5)(ξ),y(6)(ξ)) directly. The general implicit block method with Hermite interpolation in three points (GIBM3P) has been derived to solve general seventh-order initial value problems (IVPs) using the basic functions of Hermite interpolating polynomials. A block multi-step method is constructed to be suitable with the numerical approximation at three points. However, the construction of the new method has been presented while the numerical results of the implementations are used to prove the efficiency and the accuracy of the proposed method which compared with the RK and RKM numerical methods together to analytical method. We established the characteristics of the proposed method, including order and zero-stability. Applications of various IVP problems are also discussed, and the out...
Lane-Emden differential equations of order fractional has been studied. The numerical solution of... more Lane-Emden differential equations of order fractional has been studied. The numerical solution of this type is considered by the least square method. Some of examples are illustrated and a comparison between numerical and analytic methods has been introduced.
Recently, efficient direct numerical integrators of Runge-Kutta type (called RKD and RKT methods)... more Recently, efficient direct numerical integrators of Runge-Kutta type (called RKD and RKT methods) for solving third-order ordinary differential equations (ODEs) of the form y ′′′ = f (x, y) have been proposed. In this paper, we investigate the reliability of these RKD and RKT approaches, with focus on their stability and accuracy. We compare the stability regions of RKD and RKT methods. It is found that RKD stability region is adaptable, in the sense that its area can be controlled using a free parameter to get a more stable solution. To test the accuracy of RKD, we present some examples of this approach towards solving third-order ordinary differential equations. Simulation results show that the RKD approach, in addition to outperforming the existing RKT methods in terms of accuracy and time consumption, gives better control over stability region. AMS subject classification:
Ibn AL-Haitham Journal For Pure and Applied Sciences
This paper sheds the light on the vital role that fractional ordinary differential equations(FrOD... more This paper sheds the light on the vital role that fractional ordinary differential equations(FrODEs) play in the mathematical modeling and in real life, particularly in the physical conditions. Furthermore, if the problem is handled directly by using numerical method, it is a far more powerful and efficient numerical method in terms of computational time, number of function evaluations, and precision. In this paper, we concentrate on the derivation of the direct numerical methods for solving fifth-order FrODEs in one, two, and three stages. Additionally, it is important to note that the RKM-numerical methods with two- and three-stages for solving fifth-order ODEs are convenient, for solving class's fifth-order FrODEs. Numerical examples have been analyzed to demonstrate the efficacy of the new methods in comparison to the analytical method. Therefore, the numerical compression is carried out to confirm the efficiency and precision of the modified numerical methods. Significantl...
In this study, the development of direct explicit numerical integrators of RK-type for solving a ... more In this study, the development of direct explicit numerical integrators of RK-type for solving a class of ninth-order ordinary differential equations (ODEs) to develop the computational efficiency of the methods. The objective of this paper is to generalize the numerical integrators of RK type for solving classes of ODEs of orders up to eighth-order ODEs. Using the approach of Taylor-expansion, we have derived the order-conditions (OCs) for RKM integrators. Based on these OCs, developed method of ninth-order with five-stage has been derived. The zero stability of RKM integrator is proven. Stability polynomial of RKM integrator for test problem has been introduced. The advantage and the efficiency of the proposed method have been shown clearly using the numerical results which are agree well with analytical solutions due to the proposed method is more efficient and accurate method.
In order to solve general seventh-order ordinary differential equations (ODEs), this study will d... more In order to solve general seventh-order ordinary differential equations (ODEs), this study will develop an implicit block method with three points of the form y(7)(ξ)=f(ξ,y(ξ),y′(ξ),y″(ξ),y‴(ξ),y(4)(ξ),y(5)(ξ),y(6)(ξ)) directly. The general implicit block method with Hermite interpolation in three points (GIBM3P) has been derived to solve general seventh-order initial value problems (IVPs) using the basic functions of Hermite interpolating polynomials. A block multi-step method is constructed to be suitable with the numerical approximation at three points. However, the construction of the new method has been presented while the numerical results of the implementations are used to prove the efficiency and the accuracy of the proposed method which compared with the RK and RKM numerical methods together to analytical method. We established the characteristics of the proposed method, including order and zero-stability. Applications of various IVP problems are also discussed, and the out...
Lane-Emden differential equations of order fractional has been studied. The numerical solution of... more Lane-Emden differential equations of order fractional has been studied. The numerical solution of this type is considered by the least square method. Some of examples are illustrated and a comparison between numerical and analytic methods has been introduced.
Recently, efficient direct numerical integrators of Runge-Kutta type (called RKD and RKT methods)... more Recently, efficient direct numerical integrators of Runge-Kutta type (called RKD and RKT methods) for solving third-order ordinary differential equations (ODEs) of the form y ′′′ = f (x, y) have been proposed. In this paper, we investigate the reliability of these RKD and RKT approaches, with focus on their stability and accuracy. We compare the stability regions of RKD and RKT methods. It is found that RKD stability region is adaptable, in the sense that its area can be controlled using a free parameter to get a more stable solution. To test the accuracy of RKD, we present some examples of this approach towards solving third-order ordinary differential equations. Simulation results show that the RKD approach, in addition to outperforming the existing RKT methods in terms of accuracy and time consumption, gives better control over stability region. AMS subject classification:
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