scholarly journals Variational Iteration Method for Analytical Solution of the Lane-Emden Type Equation with Singular Initial and Boundary Conditions

2019 ◽  
pp. 127-142 ◽  
Author(s):  
Muhammad Nadeem ◽  
Hijaz Ahmad
Keyword(s):  
Exact Solution ◽  
Analytical Solution ◽  
Iteration Method ◽  
Initial Value Problems ◽  
Type Equation ◽  
Variational Iteration Method ◽  
Mathematical Physics ◽  
Initial Value ◽  
Emden Equation ◽  
Variational Iteration

In this paper, a well-known equation used in astrophysics and mathematical physics called the Lane-Emden equation is to be solved by a variational iteration method. The main purpose of this approach is to solve the singular initial value problems and also boundary value problem of Lane-Emden type equations. This technique overcomes its singularity at origin rapidly. It gives the approximate and exact solution with easily computable terms. The approach is illustrated with some examples to show its reliability and compactness.

scholarly journals Approximate Solution of Linear Fuzzy Initial Value Problems Using Modified Variaional Iteration Method

2021 ◽  
Vol 24 (4) ◽  
pp. 32-39
Author(s):  
Hussein M. Sagban ◽  
◽  
Fadhel S. Fadhel ◽  
Keyword(s):  
Approximate Solution ◽  
Rate Of Convergence ◽  
Initial Value Problem ◽  
Iteration Method ◽  
Initial Value Problems ◽  
Initial Conditions ◽  
Variational Iteration Method ◽  
Initial Value ◽  
Variational Iteration ◽  
Modified Variational Iteration Method

The main objective of this paper is to solve fuzzy initial value problems, in which the fuzziness occurs in the initial conditions. The proposed approach, namely the modified variational iteration method, will be used to find the solution of fuzzy initial value problem approximately and to increase the rate of convergence of the variational iteration method. From the obtained results, as it is expected, the approximate results of the proposed method are more accurate than those results obtained without using the modified variational iteration method.

scholarly journals Variational Iteration Method for Singular Perturbation Initial Value Problems with Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Yongxiang Zhao ◽  
Aiguo Xiao ◽  
Li Li ◽  
Chengjian Zhang
Keyword(s):  
Singular Perturbation ◽  
Lagrange Multipliers ◽  
Iteration Method ◽  
Initial Value Problems ◽  
Variational Iteration Method ◽  
Variational Theory ◽  
Initial Value ◽  
Numerical Examples ◽  
Variational Iteration ◽  
Convergence Results

The variational iteration method (VIM) is applied to solve singular perturbation initial value problems with delays (SPIVPDs). Some convergence results of VIM for solving SPIVPDs are given. The obtained sequence of iterates is based on the use of general Lagrange multipliers; the multipliers in the functionals can be identified by the variational theory. Moreover, the numerical examples show the efficiency of the method.

scholarly journals Modified variational iteration method for variant Boussinesq equation

2015 ◽  
Vol 19 (4) ◽  
pp. 1195-1199 ◽  
Author(s):  
Jun-Feng Lu
Keyword(s):  
Exact Solutions ◽  
Iteration Method ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
Boussinesq Equation ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Initial Value ◽  
Variational Iteration ◽  
Modified Variational Iteration Method

In this paper, we solve the variant Boussinesq equation by the modified variational iteration method. The approximate solutions to the initial value problems of the variant Boussinesq equation are provided, and compared with the exact solutions. Numerical experiments show that the modified variational iteration method is efficient for solving the variant Boussinesq equation.

scholarly journals The variational iteration method for solving the Volterra integro-differential forms of the Lane-Emden and the Emden-Fowler problems with initial and boundary value conditions

2015 ◽  
Vol 5 (1) ◽  
Author(s):  
Abdul-Majid Wazwaz ◽  
Suheil A. Khuri
Keyword(s):  
Boundary Value Problems ◽  
Lagrange Multiplier ◽  
Iteration Method ◽  
Initial Value Problems ◽  
Boundary Value ◽  
Differential Forms ◽  
Variational Iteration Method ◽  
Initial Value ◽  
Shape Factors ◽  
Variational Iteration

AbstractIn this paper, the variational iteration method (VIM) is used to examine the Volterra integro-differential forms of the singular Lane–Emden and the Emden–Fowler initial value problems and boundary value problems arising in physics, astrophysics and stellar structures. The Volterra integro-differential forms of the Lane–Emden and the Emden–Fowler equations overcome the singularity behavior at the origin x = 0. The Lagrange multiplier, needed for the VIM, is λ = −1 for the various cases of the specified equations having distinct shape factors. We illustrate our work by analyzing few initial value problems and boundary value problems to emphasize the convergence of the acquired results.

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