scholarly journals The extended variational iteration method for local fractional differential equation

2021 ◽  
pp. 54-54
Author(s):  
Yong-Ju Yang
Keyword(s):  
Differential Equation ◽  
Fractional Derivative ◽  
Lagrange Multipliers ◽  
Iteration Method ◽  
Sufficient Conditions ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
Sufficient Conditions For Convergence ◽  
Fractional Differential ◽  
First Time

An extended variational iteration method within the local fractional derivative is introduced for the first time., where two Lagrange multipliers are adopted. Moreover, the sufficient conditions for convergence of the new variational iteration method are also established..

scholarly journals Variational Iteration Method for the Solution of Differential Equation of Motion of the Mathematical Pendulum and Duffing-Harmonic Oscillator

2019 ◽  
pp. 101-109
Author(s):  
Muhammad Munib Khan
Keyword(s):  
Differential Equation ◽  
Harmonic Oscillator ◽  
Lagrange Multipliers ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Equation Of Motion ◽  
Mathematical Pendulum ◽  
Variational Iteration ◽  
Common Problems

In this work, the differential equation of motion of the undamped mathematical pendulum and Duffing-harmonic oscillator are discussed by using the variational iteration method. Additionally, common problems of pendulum are classified and Lagrange multipliers are obtained for each type of problem. Examples are given for illustration.

scholarly journals Solusi Persamaan Diferensial Fraksional Riccati Menggunakan Adomian Decomposition Method dan Variational Iteration Method

Matematika ◽  
2019 ◽  
Vol 18 (1) ◽  
Author(s):  
Muhamad Deni Johansyah ◽  
Herlina Napitupulu ◽  
Erwin Harahap ◽  
Ira Sumiati ◽  
Asep K. Supriatna
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Iteration Method ◽  
Decomposition Method ◽  
Adomian Decomposition Method ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
Adomian Decomposition ◽  
Fractional Differential

Abstrak. Pada umumnya orde dari persamaan diferensial adalah bilangan asli, namun orde pada persamaan diferensial dapat dibentuk menjadi orde pecahan yang disebut persamaan diferensial fraksional. Paper ini membahas persamaan diferensial fraksional Riccati dengan orde diantara nol dan satu, dan koefisien konstan. Metode numerik yang digunakan untuk mendapatkan solusi dari persamaan diferensial fraksional Riccati adalah Adomian Decomposition Method (ADM) dan Variational Iteration Method (VIM). Tujuan dari paper ini adalah untuk memperluas penerapan ADM dan VIM dalam menyelesaikan persamaan diferensial fraksional Riccati nonlinear dengan turunan Caputo. Perbandingan solusi yang diperoleh menunjukkan bahwa VIM adalah metode yang lebih sederhana untuk mencari solusi persamaan diferensial fraksional Riccati nonlinier dengan orde antara nol dan satu, kemudian hasil yang diperoleh disajikan dalam bentuk grafik.Kata kunci: diferensial, fraksional, riccati, adomian dekomposisiThe solution of Riccati Fractional Differential Equation using Adomian Decomposition methodAbstract. Generally, the order of differential equations is a natural numbers, but this order can be formed into fractional, called as fractional differential equations.  In this paper, the Riccati fractional differential equations with order between zero and one, and constant coefficient is discussed.  The numerical methods used to obtain solutions from Riccati fractional differential equations are the Adomian Decomposition Method (ADM) and Variational Iteration Method (VIM).  The aim of this paper is to expand the application of ADM and VIM in solving nonlinear Riccati fractional differential equations with Caputo derivatives.  The comparison of the obtained solutions shows that VIM is simpler method for finding solutions to Riccati nonlinear fractional differential equations with order between zero and one. The obtained results are presented graphically.Keywords: riccati, fractional, differential, adomian, decomposition

scholarly journals Solution of Space-Time-Fractional Problem by Shehu Variational Iteration Method

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Suleyman Cetinkaya ◽  
Ali Demir ◽  
Hulya Kodal Sevindir
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Iteration Method ◽  
Nonlinear Differential Equations ◽  
Variational Iteration Method ◽  
Space Time ◽  
Variational Iteration ◽  
Mathematical Problems ◽  
Fractional Differential ◽  
Time Fractional Differential Equations

In this study, we deal with the problem of constructing semianalytical solution of mathematical problems including space-time-fractional linear and nonlinear differential equations. The method, called Shehu Variational Iteration Method (SVIM), applied in this study is a combination of Shehu transform (ST) and variational iteration method (VIM). First, ST is utilized to reduce the time-fractional differential equation with fractional derivative in Liouville-Caputo sense into an integer-order differential equation. Later, VIM is implemented to construct the solution of reduced differential equation. The convergence analysis of this method and illustrated examples confirm that the proposed method is one of best procedures to tackle space-time-fractional differential equations.

scholarly journals Variational Iteration Method for the Solution of Differential Equation of Motion of the Mathematical Pendulum and Duffing-Harmonic Oscillator

2019 ◽  
pp. 101-109
Author(s):  
Muhammad Munib Khan
Keyword(s):  
Differential Equation ◽  
Harmonic Oscillator ◽  
Lagrange Multipliers ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Equation Of Motion ◽  
Mathematical Pendulum ◽  
Variational Iteration ◽  
Common Problems

In this work, the differential equation of motion of the undamped mathematical pendulum and Duffing-harmonic oscillator are discussed by using the variational iteration method. Additionally, common problems of pendulum are classified and Lagrange multipliers are obtained for each type of problem. Examples are given for illustration.

scholarly journals Variational Iteration Transform Method for Fractional Differential Equations with Local Fractional Derivative

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Yong-Ju Yang ◽  
Liu-Qing Hua
Keyword(s):  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Transform Method ◽  
Local Fractional Derivative ◽  
Variational Iteration ◽  
Differential Transform ◽  
Differential Operation ◽  
Fractional Differential

We propose the variational iteration transform method in the sense of local fractional derivative, which is derived from the coupling method of local fractional variational iteration method and differential transform method. The method reduces the integral calculation of the usual variational iteration computations to more easily handled differential operation. And the technique is more orderly and easier to analyze computing result as compared with the local fractional variational iteration method. Some examples are illustrated to show the feature of the presented technique.

scholarly journals Time- or Space-Dependent Coefficient Recovery in Parabolic Partial Differential Equation for Sensor Array in the Biological Computing

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
Guanglu Zhou ◽  
Boying Wu ◽  
Wen Ji ◽  
Seungmin Rho
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Inverse Problem ◽  
Iteration Method ◽  
Numerical Experiments ◽  
Sensor Array ◽  
Variational Iteration Method ◽  
Parabolic Partial Differential Equation ◽  
Partial Differential ◽  
Variational Iteration

This study presents numerical schemes for solving a parabolic partial differential equation with a time- or space-dependent coefficient subject to an extra measurement. Through the extra measurement, the inverse problem is transformed into an equivalent nonlinear equation which is much simpler to handle. By the variational iteration method, we obtain the exact solution and the unknown coefficients. The results of numerical experiments and stable experiments imply that the variational iteration method is very suitable to solve these inverse problems.

A modified variational iteration method for solving fractional Riccati differential equation by Adomian polynomials

2013 ◽  
Vol 16 (1) ◽  
Author(s):  
Hossein Jafari ◽  
Hale Tajadodi ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equation ◽  
Iteration Method ◽  
Variational Iteration Method ◽  
Approximate Solutions ◽  
Convergence Region ◽  
Riccati Differential Equation ◽  
Adomian Polynomials ◽  
Riccati Differential Equations ◽  
Variational Iteration ◽  
Modified Variational Iteration Method

AbstractIn this paper, we introduce a modified variational iteration method (MVIM) for solving Riccati differential equations. Also the fractional Riccati differential equation is solved by variational iteration method with considering Adomians polynomials for nonlinear terms. The main advantage of the MVIM is that it can enlarge the convergence region of iterative approximate solutions. Hence, the solutions obtained using the MVIM give good approximations for a larger interval. The numerical results show that the method is simple and effective.

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