scholarly journals A New Approximate Analytical Method for ODEs

2014 ◽  
Vol 10 (1) ◽  
pp. 19-30
Author(s):  
Hossein Aminikhah
Keyword(s):  
Exact Solution ◽  
Nonlinear System ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Analytical Method ◽  
New Method ◽  
Good Stability ◽  
Linear And Nonlinear ◽  
Stability And Accuracy ◽  
Theoretical Considerations

Abstract In this paper, we propose a new algorithm for solving ordinary differential equations. We show the superiority of this algorithm by applying the new method for some famous ODEs. Theoretical considerations are discussed. The first He's polynomials have used to reach the exact solution of these problems. This method which has good stability and accuracy properties is useful in deal with linear and nonlinear system of ordinary differential equations.

scholarly journals Numerical solution of nonlinear system of integro-differential equations using Chebyshev wavelets

2015 ◽  
Vol 11 (2) ◽  
pp. 15-34
Author(s):  
H. Aminikhah ◽  
S. Hosseini
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Nonlinear System ◽  
Differential Equations ◽  
Numerical Solution ◽  
Chebyshev Wavelets ◽  
Linear And Nonlinear

Abstract This paper introduces an approach for obtaining the numerical solution of the linear and nonlinear integro-differential equations using Chebyshev wavelets approximations. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique and the results have been compared with the exact solution. Comparison of the approximate solution with exact solution shows that the used method is effectiveness and practical for classes of linear and nonlinear system of integro-differential equations.

scholarly journals On solvability of a nonlinear system of equations for the Fourier-Chebyshev coefficients in the problem of solving ordinary differential equations using Chebyshev series

2017 ◽  
pp. 169-174
Author(s):  
О.Б. Арушанян ◽  
С.Ф. Залеткин
Keyword(s):  
Nonlinear System ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Analytical Method ◽  
System Of Equations ◽  
Chebyshev Series ◽  
Nonlinear System Of Equations ◽  
Theoretical Substantiation ◽  
Solvability Theorem ◽  
Numerical Analytical Method

Сформулирована и доказана теорема о разрешимости нелинейной системы уравнений относительно приближенных значений коэффициентов Фурье-Чебышёва. Теорема является теоретическим обоснованием ранее предложенного численно-аналитического метода интегрирования обыкновенных дифференциальных уравнений с использованием рядов Чебышёва. A solvability theorem for a nonlinear system of equations with respect to approximate values of Fourier-Chebyshev coefficients is formulated and proved. This theorem is a theoretical substantiation for the previously proposed numerical-analytical method of solving ordinary differential equations using Chebyshev series.

scholarly journals Computatıonal Algorıthm for the Numerıcal Solutıon of Systems of Volterra Integro-Dıfferentıal Equatıons

2020 ◽  
pp. 66-76
Author(s):  
Falade Kazeem Iyanda ◽  
Tiamiyu Abd`gafar Tunde
Keyword(s):  
Exact Solution ◽  
Nonlinear System ◽  
Differential Equations ◽  
Numerical Solution ◽  
Analytical Approach ◽  
Computational Algorithm ◽  
Approximate Solutions ◽  
Numerical Approach ◽  
Linear And Nonlinear ◽  
Variational Iterative Method

In this paper, we employ variational iterative method (VIM) to develop a suitable Algorithm for the numerical solution of systems of Volterra integro-differential equations. The formulated algorithm is used to solve first and second order linear and nonlinear system of Volterra integrodifferential equations which demonstrated a good numerical approach to overcome lengthen computational and integral simplification involves. Moreover, the comparison of the exact solution with the approximated solutions are made and approximate solutions p(x) q(t) proved to converge to the exact solutions p(x) q(t) respectively. The results reveal that the formulated algorithm are simple, effective and faster than analytical approach of solving Volterra integro-differential equations.

scholarly journals Analytic and numerical solution for duffing equations

2016 ◽  
Vol 5 (2) ◽  
pp. 115 ◽  
Author(s):  
Majeed AL-Jawary ◽  
Sayl Abd- AL- Razaq
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Iterative Method ◽  
Numerical Solution ◽  
Ordinary Differential Equations ◽  
Numerical Solutions ◽  
Duffing Equations ◽  
Linear And Nonlinear ◽  
New Iterative Method

<p>Daftardar Gejji and Hossein Jafari have proposed a new iterative method for solving many of the linear and nonlinear equations namely (DJM). This method proved already the effectiveness in solved many of the ordinary differential equations, partial differential equations and integral equations. The main aim from this paper is to propose the Daftardar-Jafari method (DJM) to solve the Duffing equations and to find the exact solution and numerical solutions. The proposed (DJM) is very effective and reliable, and the solution is obtained in the series form with easily computed components. The software used for the calculations in this study was MATHEMATICA<sup>®</sup> 9.0.</p>

scholarly journals Modified differential transform method for solving linear and nonlinear pantograph type of differential and Volterra integro-differential equations with proportional delays

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Seyyedeh Roodabeh Moosavi Noori ◽  
Nasir Taghizadeh
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Differential Transform Method ◽  
Hybrid Technique ◽  
Transform Method ◽  
Proportional Delays ◽  
Differential Transform ◽  
Computational Work ◽  
Linear And Nonlinear ◽  
First Time

AbstractIn this study, a hybrid technique for improving the differential transform method (DTM), namely the modified differential transform method (MDTM) expressed as a combination of the differential transform method, Laplace transforms, and the Padé approximant (LPDTM) is employed for the first time to ascertain exact solutions of linear and nonlinear pantograph type of differential and Volterra integro-differential equations (DEs and VIDEs) with proportional delays. The advantage of this method is its simple and trusty procedure, it solves the equations straightforward and directly without requiring large computational work, perturbations or linearization, and enlarges the domain of convergence, and leads to the exact solution. Also, to validate the reliability and efficiency of the method, some examples and numerical results are provided.

scholarly journals The variational iteration method for solving n-th order fuzzy differential equations

Open Physics ◽  
2012 ◽  
Vol 10 (1) ◽  
Author(s):  
Hossein Jafari ◽  
Mohammad Saeidy ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equations ◽  
Nonlinear Equations ◽  
Analytical Method ◽  
Iteration Method ◽  
Initial Conditions ◽  
Variational Iteration Method ◽  
Linear Differential Equations ◽  
Variational Iteration ◽  
Linear And Nonlinear ◽  
Linear Differential

AbstractThe variational iteration method (VIM) proposed by Ji-Huan He is a new analytical method for solving linear and nonlinear equations. In this paper, the variational iteration method has been applied in solving nth-order fuzzy linear differential equations with fuzzy initial conditions. This method is illustrated by solving several examples.

Application of He’s Homotopy Perturbation Method to Stiff Systems of Ordinary Differential Equations

2008 ◽  
Vol 63 (1-2) ◽  
pp. 19-23 ◽  
Author(s):  
Mohammad Taghi Darvishi ◽  
Farzad Khani
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Perturbation Method ◽  
Ordinary Differential Equations ◽  
Homotopy Perturbation Method ◽  
Nonlinear Ordinary Differential Equations ◽  
Stiff Systems ◽  
Homotopy Perturbation ◽  
Linear And Nonlinear

We propose He’s homotopy perturbation method (HPM) to solve stiff systems of ordinary differential equations. This method is very simple to be implemented. HPM is employed to compute an approximation or analytical solution of the stiff systems of linear and nonlinear ordinary differential equations.

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