Papers by Dr. Tamer Abassy
International Journal of Applied Mathematical Research, 2021
In this paper, piecewise analytic method (PAM) is used for solving highly nonlinear 2nd order di... more In this paper, piecewise analytic method (PAM) is used for solving highly nonlinear 2nd order differential equation (pendulum equations) which is a big problem for engineers and scientists. PAM is used for showing the nonlinear dynamics of the solution with and without linearizion. The error and accuracy of the solution are controlled easily according to our needs.
International Journal of Applied Mathematical Research, 2020
Even though Runge-Kutta (RK) method is the most used by scientists and engineers, it is not the m... more Even though Runge-Kutta (RK) method is the most used by scientists and engineers, it is not the most powerful method. In this paper, a comparative study between Piecewise Analytic Method (PAM) and RK methods is achieved. The result of comparative study shows that PAM is more powerful and gives results better than RK Methods. PAM can be considered as a new step in the evolution of solving nonlinear differential equations.
International Journal of Applied Mathematical Research, 2019
In this paper, a new method is introduced for engineers and scientists which can be used for solv... more In this paper, a new method is introduced for engineers and scientists which can be used for solving highly nonlinear differential equations. The method is called Piecewise Analytic Method (PAM). PAM is used to solve problems which other methods can't solve. The paper also shows how the accuracy and error can be controlled according to the needs.
International Journal of Applied Mathematical Research, 2012
Mathematical and Computer Modelling, 2012
A new application of modified variational iteration method (MVIM) for non-linear nonhomogeneous d... more A new application of modified variational iteration method (MVIM) for non-linear nonhomogeneous differential equations is introduced. The modified form introduces a change in the formulation of variational iteration relation and provides a qualitative improvement over standard variational iteration method (VIM). This paper is an extension of the previous work in Abassy (2010) [33], Abassy et al. (2007) [31,29,32,30]. Some illustrative numerical examples and Mathematica program codes are introduced.
The purpose of this paper is introducing the piecewise analytic method (PAM). PAM is a new method... more The purpose of this paper is introducing the piecewise analytic method (PAM). PAM is a new method for solving di�erential equations. PAM gives an approximate analytic solution which is very accurate. The error in PAM approaches to zero as the interval size approaches zero or the convergence order goes to in�nity. The piecewise analytic method can be made to be of any order of accuracy. Comparison between PAM and Runge-Kutta enhances the use of PAM.
International Journal of Nonlinear Sciences and Numerical Simulation, 2004
Adomian Decomposition methods (ADM) is am approximation method which can be adapted to solve nonl... more Adomian Decomposition methods (ADM) is am approximation method which can be adapted to solve nonlinear partial differential equations. In this paper, we solve the KdV and the modified KdV (mKdV) equations using ADM-Pade technique, which gives the approximate solution ...
International Journal of Applied Mathematical Research, 2012
A new method for solving differential equations is introduced. The new method is called piecewise... more A new method for solving differential equations is introduced. The new method is called piecewise analytic method (PAM). The PAM is based on dividing the solution interval into subintervals. PAM gives an approximate analytic solution which is very accurate and can be applied to each subinterval successively. The piecewise analytic method can be made to be of any order of accuracy. PAM gives very interesting results especially when the solution contains singular points.
Journal of Computational and Applied Mathematics, 2007
The variational iteration method (VIM) attracted much attention in the past few years as a promis... more The variational iteration method (VIM) attracted much attention in the past few years as a promising method for solving nonlinear differential equations. It is shown in this paper that the application of VIM to a special kind of nonlinear differential equations leads to calculation of unneeded terms and more time consumed in repeated calculations for series solutions. A modified VIM is introduced to eliminate the shortcomings; and its effectiveness is illustrated by some examples.
Journal of Computational and Applied Mathematics, 2009
A linearized implicit finite difference method is devised for K (n, n). The stability and accurac... more A linearized implicit finite difference method is devised for K (n, n). The stability and accuracy of the proposed methods are discussed. A compacton wave solution of the equation K (n, n) is used to examine the accuracy and efficiency of the proposed methods and study the effect of the added artificial dissipation term to solve the K (n, n) equation using finite difference method. The dynamics of waves having various initial wavepackets are discussed.
Computers & Mathematics with Applications, 2007
In this paper, the variational iteration method (VIM) is reintroduced with Laplace transforms and... more In this paper, the variational iteration method (VIM) is reintroduced with Laplace transforms and the Padé technique treatment to obtain closed form solutions of nonlinear equations. Some examples, including the coupled Burger's equation, compacton k(n, n) equation, Zakharov-Kuznetsov Zk(n, n) equation, and KdV and mKdV equations are given to show the effectiveness of the coupled VIM-Laplace-Padé and VIM-Padé techniques.
Computers & Mathematics with Applications, 2010
The modified variational iteration method is applied for analytical treatment of nonlinear homoge... more The modified variational iteration method is applied for analytical treatment of nonlinear homogeneous initial value problem. The modified variational iteration method accelerates the convergence of the power series solution and reduces the size of work. A comparison between modified variational iteration method (MVIM) and variational iteration method (VIM) was made. The comparison enhances the use of the modified variational iteration method if we wish to obtain an approximate power series solution that converges faster to the closed form solution. The method is very simple and easy.
Computers & Mathematics with Applications, 2010
In this paper a new treatment for the Adomian decomposition method (ADM) is introduced. The new t... more In this paper a new treatment for the Adomian decomposition method (ADM) is introduced. The new treatment is called the improved Adomian decomposition method (IADM) which improves the results obtained from the known Adomian decomposition method. The improved Adomian decomposition method is applied for the analytic treatment of nonlinear initial value problems. The improved method accelerates the convergence of the series solution, and provides the exact power series solution. It solves the drawbacks in the standard Adomian decomposition method.
Chaos, Solitons & Fractals, 2007
ADMPadé technique is a combination of Adomian decomposition method (ADM) and Padé approximants. ... more ADMPadé technique is a combination of Adomian decomposition method (ADM) and Padé approximants. It is an approximate method, which can be adapted to solve nonlinear partial differential equations. In this paper, we solve Burgers' and Boussinesq equation using ADMPadé ...
The Adomian Decomposition Method (ADM) shows a very good ability in dealing with nonlinear differ... more The Adomian Decomposition Method (ADM) shows a very good ability in dealing with nonlinear differential equations but it is still have some drawbacks. In this paper, it is shown how we can improve the efficiency of the method by using Improved Adomian Decomposition Method (IADM) and linking the method with other techniques like Padé approximants and Laplace transform. The results shows efficiency of the new treatment with a special type of Nonlinear Partial Differential Equations (NPDE) like compacton equations K(n,n), B(n,n) and Zakharov-Kuznetsov ZK(n,n). The technique is powerful with these equations. It gives the exact solution directly.
Journal of the Franklin Institute, 2011
... 1 Improved Adomian Decomposition Method (Solving Non-linear Non-homogenous Initial Value Prob... more ... 1 Improved Adomian Decomposition Method (Solving Non-linear Non-homogenous Initial Value Problem) Tamer A. Abassy E-mail: [email protected], Department of Scientific Computing, Faculty of Computers and Informatics ... (28) MA Abdou, Adomian decomposition ...
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Papers by Dr. Tamer Abassy