scholarly journals Numerical Methods for Fractional-Order Fornberg-Whitham Equations in the Sense of Atangana-Baleanu Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Naveed Iqbal ◽  
Humaira Yasmin ◽  
Akbar Ali ◽  
Abdul Bariq ◽  
M. Mossa Al-Sawalha ◽  
...  
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Numerical Solution ◽  
Fractional Order ◽  
Nonlinear Differential Equations ◽  
Decomposition Technique ◽  
Whitham Equations ◽  
Linear And Nonlinear ◽  
The Future Method ◽  
Iteration Technique

In this paper, we investigate the numerical solution of the Fornberg-Whitham equations with the help of two powerful techniques: the modified decomposition technique and the modified variational iteration technique involving fractional-order derivatives with Mittag-Leffler kernel. To confirm and illustrate the accuracy of the proposed approach, we evaluated in terms of fractional order the projected models. Furthermore, the physical attitude of the results obtained has been acquired for the fractional-order different value graphs. The results demonstrated that the future method is easy to implement, highly methodical, and very effective in analyzing the behavior of complicated fractional-order linear and nonlinear differential equations existing in the related areas of applied science.

A Highly Accurate Collocation Method for Linear and Nonlinear Vibration Problems of Multi-Degree-Of-Freedom Systems Based on Barycentric Interpolation

2019 ◽  
Vol 20 (5) ◽  
pp. 543-550
Author(s):  
Meiling Zhuang ◽  
Changqing Miao ◽  
Caihong Wan
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Nonlinear Vibration ◽  
Collocation Method ◽  
Nonlinear Differential Equations ◽  
Degree Of Freedom ◽  
Barycentric Interpolation ◽  
Vibration Problems ◽  
Linear And Nonlinear ◽  
Linear Differential

AbstractA highly accurate collocation method based on barycentric interpolation (BICM) is proposed for solving linear and nonlinear vibration problems for multi-degree-of-freedom systems in this article. The mathematical model of the linear and nonlinear vibrations of multi-degree-of freedom systems is the initial value problem of the linear and nonlinear differential equations. The numerical solution of the linear differential equations can be directly solved by BICM. The numerical solution of nonlinear differential equations can be solved as following: Firstly, the nonlinear governing equation is converted to linear differential equation by assuming the initial function. Secondly, the linear differential equations are discretized into algebraic equations by using barycentric interpolation differential matrices. Thirdly, the numerical solution can be calculated by iteration method under given control precision. Finally, the numerical solution of calculation examples by using barycentric Lagrange interpolation iteration collocation method (BLIICM) and barycentric rational interpolation iteration collocation method (BRIICM) are compared with the analytical solution. Numerical results illustrate the advantages of proposed methodology are efficient, fast, simple formulations, and high precision. Comparing with BLIICM, BRIICM can still maintain its computational stability when dealing with a large number of nodes, especially the equidistant nodes.

NUMERICAL SOLUTION OF THE INVERSE PROBLEM FOR A SYSTEM OF DIFFERENTIAL EQUATIONS

2020 ◽  
pp. 106-110
Author(s):  
S.E. Kasenov ◽  
◽  
G.E. Kasenova ◽  
A.A. Sultangazin ◽  
B.D. Bakytbekova ◽  
...  
Keyword(s):  
Inverse Problem ◽  
Differential Equations ◽  
Numerical Solution ◽  
Direct Problem ◽  
Nonlinear Differential Equations ◽  
Direct And Inverse Problems ◽  
Additional Information ◽  
Several Variables ◽  
Gradient Of The Functional ◽  
Objective Functional

The article considers direct and inverse problems of a system of nonlinear differential equations. Such problems are often found in various fields of science, especially in medicine, chemistry and economics. One of the main methods for solving nonlinear differential equations is the numerical method. The initial direct problem is solved by the Rune-Kutta method with second accuracy and graphs of the numerical solution are shown. The inverse problem of finding the coefficients of a system of nonlinear differential equations with additional information on solving the direct problem is posed. The numerical solution of this inverse problem is reduced to minimizing the objective functional. One of the methods that is applicable to nonsmooth and noisy functionals, unconditional optimization of the functional of several variables, which does not use the gradient of the functional, is the Nelder-Mead method. The article presents the NellerMead algorithm. And also a numerical solution of the inverse problem is shown.

Linear and Nonlinear Differential Equations

1984 ◽  
Vol 52 (7) ◽  
pp. 670-670 ◽  
Author(s):  
I. D. Huntley ◽  
R. M. Johnson ◽  
Fred Brauer
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Linear And Nonlinear

scholarly journals Destroying synchrony in an array of the FitzHugh–Nagumo oscillators by external DC voltage source

2020 ◽  
Vol 25 (1) ◽  
Author(s):  
Elena Adomaitienė ◽  
Skaidra Bumelienė ◽  
Gytis Mykolaitis ◽  
Arūnas Tamaševičius
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Control Method ◽  
Nonlinear Differential Equations ◽  
Mean Field ◽  
Electrical Circuit ◽  
Voltage Source ◽  
Dc Voltage

A control method for desynchronizing an array of mean-field coupled FitzHugh–Nagumo-type oscillators is described. The technique is based on applying an adjustable DC voltage source to the coupling node. Both, numerical solution of corresponding nonlinear differential equations and hardware experiments with a nonlinear electrical circuit have been performed.

scholarly journals Modified Variational Iteration Algorithm-II: Convergence and Applications to Diffusion Models

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-14 ◽  
Author(s):  
Hijaz Ahmad ◽  
Tufail A. Khan ◽  
Predrag S. Stanimirović ◽  
Yu-Ming Chu ◽  
Imtiaz Ahmad
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Nonlinear Models ◽  
Nonlinear Differential Equations ◽  
Variational Iteration Method ◽  
Iteration Algorithm ◽  
Compact Finite Difference Method ◽  
Variational Iteration ◽  
Linear And Nonlinear ◽  
The Stability

Variational iteration method has been extensively employed to deal with linear and nonlinear differential equations of integer and fractional order. The key property of the technique is its ability and flexibility to investigate linear and nonlinear models conveniently and accurately. The current study presents an improved algorithm to the variational iteration algorithm-II (VIA-II) for the numerical treatment of diffusion as well as convection-diffusion equations. This newly introduced modification is termed as the modified variational iteration algorithm-II (MVIA-II). The convergence of the MVIA-II is studied in the case of solving nonlinear equations. The main advantage of the MVIA-II improvement is an auxiliary parameter which makes sure a fast convergence of the standard VIA-II iteration algorithm. In order to verify the stability, accuracy, and computational speed of the method, the obtained solutions are compared numerically and graphically with the exact ones as well as with the results obtained by the previously proposed compact finite difference method and second kind Chebyshev wavelets. The comparison revealed that the modified version yields accurate results, converges rapidly, and offers better robustness in comparison with other methods used in the literature. Moreover, the basic idea depicted in this study is relied upon the possibility of the MVIA-II being utilized to handle nonlinear differential equations that arise in different fields of physical and biological sciences. A strong motivation for such applications is the fact that any discretization, transformation, or any assumptions are not required for this proposed algorithm in finding appropriate numerical solutions.

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