Mohammed M . Salih | Ninevah University​ (original) (raw)

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Papers by Mohammed M . Salih

Research paper thumbnail of Trigonometrically-Fitted Fifth Order Four-Step Predictor-Corrector Method for Solving Linear Ordinary Differential Equations with Oscillatory Solutions

Malaysian Journal of Mathematical Sciences

In this paper, we proposed a trigonometrically-fitted fifth order four-step predictor-corrector m... more In this paper, we proposed a trigonometrically-fitted fifth order four-step predictor-corrector method based on the four-step Adams-Bashforth method as predictor and five-step Adams-Moulton method as corrector to solve linear ordinary differential equations with oscillatory solutions. This method is constructed which exactly integrate initial value problems whose solutions can be expressed as linear combinations of the set functions {sin(υx),cos(υx)} with υ ∈ R, where v represents an approximation of the frequency of the problem. The frequency will be used in the method to raise the accuracy of the solution. Stability of the proposed method is examined and the corresponding region of stability is depicted. The new fifth algebraic order trigonometrically-fitted predictor-corrector method is applied to solve the initial value problems whose solutions involved trigonometric functions. Numerical results presented proved that the prospective method is more efficient than the widely used ...

Research paper thumbnail of Construction of General Implicit-Block Method with Three-Points for Solving Seventh-Order Ordinary Differential Equations

Symmetry

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...

Research paper thumbnail of Fifth Order Runge-Kutta-Nyström Methods for Solving Linear Second Order Oscillatory Problems

Far East Journal of Applied Mathematics, 2016

In this paper, order conditions for Runge-Kutta-Nyström (RKN) method for solving second order lin... more In this paper, order conditions for Runge-Kutta-Nyström (RKN) method for solving second order linear ordinary differential equations (LODEs) are derived up to order six. Based on the order conditions, a new fifth order four-stage RKN method which is specially designed for the integration of second order linear ordinary differential equations (LODEs) is constructed with first same as last (FSAL) property. Then we phase-fitted the method such that it has zero phase-lag and zero dissipation. Phase-lag or dispersion error is the angle between the true and the approximated solutions, whereas dissipation is the distance of the computed solution from the standard cyclic solution. A set of test problems are used to validate the method and numerical results show that the phase-fitted method produced smaller global error compared to the original method as well as other existing RKN method in the scientific literature.

Research paper thumbnail of Runge-Kutta Methods of Higher Order for Solving Stiff Problems

Our purpose in this research is the development of higher order Runge-Kutta methods for solving s... more Our purpose in this research is the development of higher order Runge-Kutta methods for solving stiff systems. We have developed methods of order five, six, and seven. We studied their stability Region and applications for solving stiff systems. Then we developed the corresponding implicit forms of these methods and we analyzed their stability and implementation for solving stiff systems.

Research paper thumbnail of Goeken-Johnson Sixth-Order Runge-Kutta Method

Research paper thumbnail of Using predictor-corrector methods for solving fuzzy differential equations

In this paper we studied the Fuzzy Differential Equations and we used Predictor-Corrector methods... more In this paper we studied the Fuzzy Differential Equations and we used Predictor-Corrector methods for solving these equations (these methods are Multi-step methods), we used two methods Adams-Bashforth-Moulton three-step method and Adams-Bashforth-Moulton four-step method, and we compared the solution of each of method with the exact solution, and then compared the solution of each of them with the others

Research paper thumbnail of Phase Fitted Classical Runge-Kutta Method of Order Four for Solving Oscillatory Problems

Far East Journal of Mathematical Sciences, 2015

In this paper, we derive a new phase fitted Runge-Kutta method based on the existing classical Ru... more In this paper, we derive a new phase fitted Runge-Kutta method based on the existing classical Runge-Kutta method of order four to solve ordinary differential equations with oscillatory solutions. The new method has the property of zero phase-lag and zero dissipation. Phase-lag or dispersion error is the angle between the true and the approximated solution, whereas dissipation is the distance of the computed solution from the standard cyclic solution. A set of problems are tested upon over a large interval and the numerical results proved that the method is more accurate compared to the classical fourth order Runge-Kutta method.

Research paper thumbnail of Diagonally implicit Runge-Kutta method of order four with minimum phase-lag for solving first order linear ODEs

AIP Conference Proceedings, 2014

In this paper we derived a new diagonally implicit Runge-Kutta method of order four with minimum ... more In this paper we derived a new diagonally implicit Runge-Kutta method of order four with minimum phase-lag for solving first order linear ordinary differential equation. The stability polynomial of the method is obtained and the stability region is presented. A set of problems are tested upon and numerical results proved that the method is more accurate compared to other well known methods in the scientific literature.

Research paper thumbnail of Phase Fitted And Amplification Fitted Of Runge-Kutta-Fehlberg Method Of Order 4(5) For Solving Oscillatory Problems

Baghdad Science Journal, 2020

In this paper, the proposed phase fitted and amplification fitted of the Runge-Kutta-Fehlberg met... more In this paper, the proposed phase fitted and amplification fitted of the Runge-Kutta-Fehlberg method were derived on the basis of existing method of 4(5) order to solve ordinary differential equations with oscillatory solutions. The recent method has null phase-lag and zero dissipation properties. The phase-lag or dispersion error is the angle between the real solution and the approximate solution. While the dissipation is the distance of the numerical solution from the basic periodic solution. Many of problems are tested over a long interval, and the numerical results have shown that the present method is more precise than the 4(5) Runge-Kutta-Fehlberg method.

Research paper thumbnail of Trigonometrically-Fitted Fifth Order Four-Step Predictor-Corrector Method for Solving Linear Ordinary Differential Equations with Oscillatory Solutions

Malaysian Journal of Mathematical Sciences

In this paper, we proposed a trigonometrically-fitted fifth order four-step predictor-corrector m... more In this paper, we proposed a trigonometrically-fitted fifth order four-step predictor-corrector method based on the four-step Adams-Bashforth method as predictor and five-step Adams-Moulton method as corrector to solve linear ordinary differential equations with oscillatory solutions. This method is constructed which exactly integrate initial value problems whose solutions can be expressed as linear combinations of the set functions {sin(υx),cos(υx)} with υ ∈ R, where v represents an approximation of the frequency of the problem. The frequency will be used in the method to raise the accuracy of the solution. Stability of the proposed method is examined and the corresponding region of stability is depicted. The new fifth algebraic order trigonometrically-fitted predictor-corrector method is applied to solve the initial value problems whose solutions involved trigonometric functions. Numerical results presented proved that the prospective method is more efficient than the widely used ...

Research paper thumbnail of Construction of General Implicit-Block Method with Three-Points for Solving Seventh-Order Ordinary Differential Equations

Symmetry

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...

Research paper thumbnail of Fifth Order Runge-Kutta-Nyström Methods for Solving Linear Second Order Oscillatory Problems

Far East Journal of Applied Mathematics, 2016

In this paper, order conditions for Runge-Kutta-Nyström (RKN) method for solving second order lin... more In this paper, order conditions for Runge-Kutta-Nyström (RKN) method for solving second order linear ordinary differential equations (LODEs) are derived up to order six. Based on the order conditions, a new fifth order four-stage RKN method which is specially designed for the integration of second order linear ordinary differential equations (LODEs) is constructed with first same as last (FSAL) property. Then we phase-fitted the method such that it has zero phase-lag and zero dissipation. Phase-lag or dispersion error is the angle between the true and the approximated solutions, whereas dissipation is the distance of the computed solution from the standard cyclic solution. A set of test problems are used to validate the method and numerical results show that the phase-fitted method produced smaller global error compared to the original method as well as other existing RKN method in the scientific literature.

Research paper thumbnail of Runge-Kutta Methods of Higher Order for Solving Stiff Problems

Our purpose in this research is the development of higher order Runge-Kutta methods for solving s... more Our purpose in this research is the development of higher order Runge-Kutta methods for solving stiff systems. We have developed methods of order five, six, and seven. We studied their stability Region and applications for solving stiff systems. Then we developed the corresponding implicit forms of these methods and we analyzed their stability and implementation for solving stiff systems.

Research paper thumbnail of Goeken-Johnson Sixth-Order Runge-Kutta Method

Research paper thumbnail of Using predictor-corrector methods for solving fuzzy differential equations

In this paper we studied the Fuzzy Differential Equations and we used Predictor-Corrector methods... more In this paper we studied the Fuzzy Differential Equations and we used Predictor-Corrector methods for solving these equations (these methods are Multi-step methods), we used two methods Adams-Bashforth-Moulton three-step method and Adams-Bashforth-Moulton four-step method, and we compared the solution of each of method with the exact solution, and then compared the solution of each of them with the others

Research paper thumbnail of Phase Fitted Classical Runge-Kutta Method of Order Four for Solving Oscillatory Problems

Far East Journal of Mathematical Sciences, 2015

In this paper, we derive a new phase fitted Runge-Kutta method based on the existing classical Ru... more In this paper, we derive a new phase fitted Runge-Kutta method based on the existing classical Runge-Kutta method of order four to solve ordinary differential equations with oscillatory solutions. The new method has the property of zero phase-lag and zero dissipation. Phase-lag or dispersion error is the angle between the true and the approximated solution, whereas dissipation is the distance of the computed solution from the standard cyclic solution. A set of problems are tested upon over a large interval and the numerical results proved that the method is more accurate compared to the classical fourth order Runge-Kutta method.

Research paper thumbnail of Diagonally implicit Runge-Kutta method of order four with minimum phase-lag for solving first order linear ODEs

AIP Conference Proceedings, 2014

In this paper we derived a new diagonally implicit Runge-Kutta method of order four with minimum ... more In this paper we derived a new diagonally implicit Runge-Kutta method of order four with minimum phase-lag for solving first order linear ordinary differential equation. The stability polynomial of the method is obtained and the stability region is presented. A set of problems are tested upon and numerical results proved that the method is more accurate compared to other well known methods in the scientific literature.

Research paper thumbnail of Phase Fitted And Amplification Fitted Of Runge-Kutta-Fehlberg Method Of Order 4(5) For Solving Oscillatory Problems

Baghdad Science Journal, 2020

In this paper, the proposed phase fitted and amplification fitted of the Runge-Kutta-Fehlberg met... more In this paper, the proposed phase fitted and amplification fitted of the Runge-Kutta-Fehlberg method were derived on the basis of existing method of 4(5) order to solve ordinary differential equations with oscillatory solutions. The recent method has null phase-lag and zero dissipation properties. The phase-lag or dispersion error is the angle between the real solution and the approximate solution. While the dissipation is the distance of the numerical solution from the basic periodic solution. Many of problems are tested over a long interval, and the numerical results have shown that the present method is more precise than the 4(5) Runge-Kutta-Fehlberg method.