A Numerical Algorithm and a Variational Iteration Technique for Solving Higher Order Fuzzy Integro-differential Equations

2014 ◽  
Vol 133 (4) ◽  
pp. 421-431 ◽  
Author(s):  
S. Narayanamoorthy ◽  
K. Murugan
Keyword(s):  
Differential Equations ◽  
Numerical Algorithm ◽  
Higher Order ◽  
Variational Iteration ◽  
Iteration Technique

scholarly journals Higher-Order Root-Finding Algorithms and Their Basins of Attraction

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Amir Naseem ◽  
M. A. Rehman ◽  
Thabet Abdeljawad
Keyword(s):  
Iterative Methods ◽  
Nonlinear Equations ◽  
Basins Of Attraction ◽  
Higher Order ◽  
Complex Polynomials ◽  
Convergence Criteria ◽  
Root Finding ◽  
Variational Iteration ◽  
Order Root ◽  
Iteration Technique

In this paper, we proposed and analyzed three new root-finding algorithms for solving nonlinear equations in one variable. We derive these algorithms with the help of variational iteration technique. We discuss the convergence criteria of these newly developed algorithms. The dominance of the proposed algorithms is illustrated by solving several test examples and comparing them with other well-known existing iterative methods in the literature. In the end, we present the basins of attraction using some complex polynomials of different degrees to observe the fractal behavior and dynamical aspects of the proposed algorithms.

A general numerical algorithm for nonlinear differential equations by the variational iteration method

2020 ◽  
Vol 30 (11) ◽  
pp. 4797-4810 ◽  
Author(s):  
Ji-Huan He ◽  
Habibolla Latifizadeh
Keyword(s):  
Differential Equations ◽  
Numerical Algorithm ◽  
Iteration Method ◽  
Solution Process ◽  
Nonlinear Problems ◽  
Variational Iteration Method ◽  
Iteration Algorithm ◽  
Content Type ◽  
Variational Iteration ◽  
Laplace Transform Technique

Purpose The purpose of this paper is to suggest a general numerical algorithm for nonlinear problems by the variational iteration method (VIM). Design/methodology/approach Firstly, the Laplace transform technique is used to reconstruct the variational iteration algorithm-II. Secondly, its convergence is strictly proved. Thirdly, the numerical steps for the algorithm is given. Finally, some examples are given to show the solution process and the effectiveness of the method. Findings No variational theory is needed to construct the numerical algorithm, and the incorporation of the Laplace method into the VIM makes the solution process much simpler. Originality/value A universal iteration formulation is suggested for nonlinear problems. The VIM cleans up the numerical road to differential equations.

scholarly journals Approximating non linear higher order ODEs by a three point block algorithm

PLoS ONE ◽  
2021 ◽  
Vol 16 (2) ◽  
pp. e0246904
Author(s):  
Ahmad Fadly Nurullah Rasedee ◽  
Mohammad Hasan Abdul Sathar ◽  
Khairil Iskandar Othman ◽  
Siti Raihana Hamzah ◽  
Norizarina Ishak
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Numerical Algorithm ◽  
Computational Cost ◽  
Real Life ◽  
Higher Order ◽  
Multistep Method ◽  
Stability And Convergence ◽  
Block Algorithm ◽  
Life Problems

Differential equations are commonly used to model various types of real life applications. The complexity of these models may often hinder the ability to acquire an analytical solution. To overcome this drawback, numerical methods were introduced to approximate the solutions. Initially when developing a numerical algorithm, researchers focused on the key aspect which is accuracy of the method. As numerical methods becomes more and more robust, accuracy alone is not sufficient hence begins the pursuit of efficiency which warrants the need for reducing computational cost. The current research proposes a numerical algorithm for solving initial value higher order ordinary differential equations (ODEs). The proposed algorithm is derived as a three point block multistep method, developed in an Adams type formulae (3PBCS) and will be used to solve various types of ODEs and systems of ODEs. Type of ODEs that are selected varies from linear to nonlinear, artificial and real life problems. Results will illustrate the accuracy and efficiency of the proposed three point block method. Order, stability and convergence of the method are also presented in the study.

scholarly journals Application of Variational Iteration Method to Fractional Hyperbolic Partial Differential Equations

2009 ◽  
Vol 2009 ◽  
pp. 1-10 ◽  
Author(s):  
Fadime Dal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential ◽  
Variational Iteration ◽  
Gauss Elimination ◽  
Gauss Elimination Method ◽  
Iteration Technique

The solution of the fractional hyperbolic partial differential equation is obtained by means of the variational iteration method. Our numerical results are compared with those obtained by the modified Gauss elimination method. Our results reveal that the technique introduced here is very effective, convenient, and quite accurate to one-dimensional fractional hyperbolic partial differential equations. Application of variational iteration technique to this problem has shown the rapid convergence of the sequence constructed by this method to the exact solution.

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