Variational iteration algorithm I with an auxiliary parameter for the solution of differential equations of motion for simple and damped mass–spring systems

2019 ◽  
Vol 51 (1-2) ◽  
pp. 12-20 ◽  
Author(s):  
Hijaz Ahmad ◽  
Tufail A Khan
Keyword(s):  
Differential Equations ◽  
High Efficiency ◽  
Equations Of Motion ◽  
Iteration Algorithm ◽  
Auxiliary Parameter ◽  
Analytical Treatment ◽  
Variational Iteration ◽  
Vibration Problems ◽  
Mass Spring ◽  
Good Agreement

In this article, variational iteration algorithm I with an auxiliary parameter is used for simple and damped mass–spring systems that undergo forced vibrations and transverse vibration of uniform and variable beams with simply supported analytical treatment. In addition, common vibration problems are classified and the values of Lagrange multipliers are identified for each type of problem. Examples are given at the end, which shows that this modification demonstrated the high efficiency and attained very good agreement in illustrated models and might be promptly reached out to other nonlinear differential and partial differential equations.

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.

scholarly journals Variational Iteration Algorithm-I with an Auxiliary Parameter for Solving Fokker-Planck Equation

2019 ◽  
pp. 29-37 ◽  
Author(s):  
Hijaz Ahmad
Keyword(s):  
Large Class ◽  
Planck Equation ◽  
Suitable Method ◽  
Iteration Algorithm ◽  
Auxiliary Parameter ◽  
Variational Iteration ◽  
Fokker Planck Equation ◽  
Linear Problems ◽  
Fokker Planck Equations ◽  
Fokker Planck

In this paper, variational iteration algorithm-I with an auxiliary parameter is implemented to investigate Fokker-Planck equations. To show the accuracy and reliability of the technique comparisons are made between the variational iteration algorithm-I with an auxiliary parameter and classic variational iteration algorithm-I. The comparison shows that variational iteration algorithm-I with an auxiliary parameter is more powerful and suitable method for solving Fokker-Planck equations. Furthermore, the proposed algorithm can successfully be applied to a large class of nonlinear and linear problems.

Study on Free Oscillations of Systems with Even Nonlinearities by Means of the Method of Multiple Scales

2007 ◽  
Vol 10-12 ◽  
pp. 193-197
Author(s):  
J.M. Wen ◽  
Z.C. Cao
Keyword(s):  
Differential Equations ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Nonlinear Differential Equations ◽  
Approximate Solutions ◽  
Free Oscillations ◽  
Analytical Technique ◽  
Spring System ◽  
Mass Spring ◽  
Good Agreement

An analytical technique, namely the method of multiple scales, is applied to solve the differential equations of free oscillations with even nonlinearities in a mass-spring system. Unlike other perturbation methods, the method of multiple scales is effective in determining the transient response as well as determining the approximation to the frequency of the nonlinear system. In this paper, the periodic solutions of the even nonlinear differential equations have been obtained by using the method of multiple scales. Compared with the numerical examples, the approximate solutions are in good agreement with exact solutions. The numerical and analytical solutions have clearly shown that there exists the so-called drift phenomenon in the free oscillations of systems with even nonlinearities without any external excitation.

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 An Analytical Computational Algorithm for Solving a System of Multipantograph DDEs Using Laplace Variational Iteration Algorithm

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Mohamed S. M. Bahgat ◽  
A. M. Sebaq
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Computational Algorithm ◽  
Iteration Algorithm ◽  
Computational Results ◽  
Variational Iteration ◽  
Convergence Results ◽  
Delay Differential ◽  
High Degree ◽  
Symbolic Algorithm

In this research, an approximation symbolic algorithm is suggested to obtain an approximate solution of multipantograph system of type delay differential equations (DDEs) using a combination of Laplace transform and variational iteration algorithm (VIA). The corresponding convergence results are acquired, and an efficient algorithm for choosing a feasible Lagrange multiplier is designed in the solving process. The application of the Laplace variational iteration algorithm (LVIA) for the problems is clarified. With graphics and tables, LVIA approximates to a high degree of accuracy with a few numbers of iterates. Also, computational results of the considered examples imply that LVIA is accurate, simple, and appropriate for solving a system of multipantograph delay differential equations (SMPDDEs).

scholarly journals Variational Iteration Algorithm-I with an Auxiliary Parameter for Solving Boundary Value Problems

2020 ◽  
pp. 229-247 ◽  
Author(s):  
Hijaz Ahmad ◽  
Muhammad Rafiq ◽  
Clemente Cesarano ◽  
Hulya Durur
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Iteration Algorithm ◽  
Auxiliary Parameter ◽  
Variational Iteration

In this article, the variational iteration algorithm-I with an auxiliary parameter (VIA-I with AP) is elaborated to initial and boundary value problems. The effectiveness, absence of difficulty and accuracy of the proposed method is remarkable and its tractability is well suitable for the use of these type of problems. Some examples have been given to show the effectiveness and utilization of this technique. A comparison of variational iteration algorithm-I (VIA-I) along VIA-I with AP has been carried out. It can be seen that this technique is more appropriate than as VIA-I.

Theoretical and experimental research on slip and uplift of the timber-concrete composite beam

BioResources ◽  
2020 ◽  
Vol 15 (3) ◽  
pp. 7079-7099
Author(s):  
Jianying Chen ◽  
Guojing He ◽  
Xiaodong (Alice) Wang ◽  
Jiejun Wang ◽  
Jin Yi ◽  
...  
Keyword(s):  
Differential Equations ◽  
Deformation Theory ◽  
Concrete Slab ◽  
Theoretical Calculations ◽  
Structural Element ◽  
Structural Efficiency ◽  
Theoretical Investigations ◽  
New Type ◽  
Interlayer Slip ◽  
Good Agreement

Timber-concrete composite beams are a new type of structural element that is environmentally friendly. The structural efficiency of this kind of beam highly depends on the stiffness of the interlayer connection. The structural efficiency of the composite was evaluated by experimental and theoretical investigations performed on the relative horizontal slip and vertical uplift along the interlayer between composite’s timber and concrete slab. Differential equations were established based on a theoretical analysis of combination effects of interlayer slip and vertical uplift, by using deformation theory of elastics. Subsequently, the differential equations were solved and the magnitude of uplift force at the interlayer was obtained. It was concluded that the theoretical calculations were in good agreement with the results of experimentation.

Sign in / Sign up

Export Citation Format

Share Document