Fractional variational iteration method for time-fractional non-linear functional partial differential equation having proportional delays

Dogan, Durgun Derya; Konuralp, Ali

doi:10.2298/tsci170612269d

Cited by 17 publications

(4 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The validity and application of the new technique are demonstrated through different examples. Dogan and Konuralp 30 applied fractional VIM to solve a nonlinear time-fractional PDE with time-taking proportions using the modified Riemann–Liouville fractional derivative. With the same data set and approximation order, the numerical solutions derived using this method are better than those obtained using the homotopy perturbation method and the differential transform method.…”

Section: Introductionmentioning

confidence: 99%

An application to formable transform: Novel numerical approach to study the nonlinear oscillator

Basit,

Tahir,

Shah

et al. 2023

Journal of Low Frequency Noise, Vibration and Active Control

View full text Add to dashboard Cite

Numerical methods in the area of nonlinear systems are extensively implemented for computing their approximate solutions because these systems are very difficult to tackle analytically. There are various numerical techniques available in the literature to find the solutions of nonlinear oscillators. Variational iteration method (VIM) is one of these approaches which is convenient to implement for these kinds of problems. In this work, our study aims to identify the numerical solution of nonlinear oscillator by making use of variational iteration method associated with Formable transformation. For the smooth utilization of this approach, we have to formulate the Lagrange multiplier through variational theory. Furthermore, we develop a new unified iterative scheme for the correction functional of VIM, considering the Formable transformation. Several new schemes of correction functional can be deduced from the newly proposed method considering the duality relation of Formable transform. In support of our primary finding, we discuss numerical example as application. A number of Physical applications of nonlinear oscillators are available in the field of vibrations and oscillations but in recent times nonlinear oscillators are used to describe complicated systems or to address mechanical, electrical, and other engineering phenomenon.

show abstract

Section: Introductionmentioning

confidence: 99%

An application to formable transform: Novel numerical approach to study the nonlinear oscillator

Basit,

Tahir,

Shah

et al. 2023

Journal of Low Frequency Noise, Vibration and Active Control

View full text Add to dashboard Cite

show abstract

“…In our study of nonlinear equations that always substitute some ansatzes into the Lagrange functional, we were able to accurately capture the soliton solutions and their evolution by applying the variational approximation method. As a result, the variational parameters can be determined by solving the Euler-Lagrange equations [37,38]. A variety of advantages are available to users of variational-based methods in comparison to analytical or numerical methods [39,40].…”

Section: Introductionmentioning

confidence: 99%

Extension of the Optimal Auxiliary Function Method to Solve the System of a Fractional-Order Whitham–Broer–Kaup Equation

Alsheekhhussain,

Moaddy,

Shah

et al. 2023

Fractal Fract

View full text Add to dashboard Cite

In this paper, we introduce and implement the optimal auxiliary function method to solve a system of fractional-order Whitham–Broer–Kaup equations, a class of nonlinear partial differential equations with broad applications in mathematical physics. This method provides a systematic and efficient approach to finding accurate solutions for complex systems of fractional-order equations. We give a full analysis using tables and figures to demonstrate the reliability and accuracy of our approach. We confirm the effectiveness of our suggested method in solving the considered equations using numerical simulations and comparisons, emphasizing its potential for applications in a variety of scientific and engineering areas.

show abstract

“…Additionally, numerous mathematical techniques have been developed to explore the approximate or exact solutions . Variational-based methods, such as Ritz technique [13], variational iteration method [14][15][16][17][18], and variational approximation method [19][20][21][22][23][24][25] et al, have been and continue to be useful and effective tools for nonlinear analysis. For example, the soliton solutions and their dynamics of lots of nonlinear equations were accurately captured by the variational approximation method [19][20][21][22][23][24][25], which always substitutes some ansatzes into the obtained Lagrange functional, and find the variational parameters by solving the corresponding Euler-Lagrange equations.…”

Section: Introductionmentioning

confidence: 99%

Variational Principles for Two Kinds of Coupled Nonlinear Equations in Shallow Water

et al. 2020

View full text Add to dashboard Cite

It is a very important but difficult task to seek explicit variational formulations for nonlinear and complex models because variational principles are theoretical bases for many methods to solve or analyze the nonlinear problem. By designing skillfully the trial-Lagrange functional, different groups of variational principles are successfully constructed for two kinds of coupled nonlinear equations in shallow water, i.e., the Broer-Kaup equations and the (2+1)-dimensional dispersive long-wave equations, respectively. Both of them contain many kinds of soliton solutions, which are always symmetric or anti-symmetric in space. Subsequently, the obtained variational principles are proved to be correct by minimizing the functionals with the calculus of variations. The established variational principles are firstly discovered, which can help to study the symmetries and find conserved quantities for the equations considered, and might find lots of applications in numerical simulation.

show abstract

Fractional variational iteration method for time-fractional non-linear functional partial differential equation having proportional delays

Cited by 17 publications

References 24 publications

An application to formable transform: Novel numerical approach to study the nonlinear oscillator

An application to formable transform: Novel numerical approach to study the nonlinear oscillator

Extension of the Optimal Auxiliary Function Method to Solve the System of a Fractional-Order Whitham–Broer–Kaup Equation

Variational Principles for Two Kinds of Coupled Nonlinear Equations in Shallow Water

Contact Info

Product

Resources

About