Fractional Differential Equations

This paper features a survey of some recent developments in asymptotic techniques, which are valid not only for weakly nonlinear equations, but also for strongly ones. Further, the obtained approximate analytical solutions are valid for the whole solution domain. The limitations of traditional perturbation methods are illustrated, various modied perturbation techniques are proposed, and some mathematical tools such as variational theory, homotopy technology, and iteration technique are introduced to overcome the shortcomings. In this paper the following categories of asymptotic methods are emphasized: (1) variational approaches, (2) parameter-expanding methods, (3) parameterized perturbation method, (4) homotopy perturbation method (5) iteration perturbation method, and ancient Chinese methods. The emphasis of this article is put mainly on the developments in this eld in China so the references, therefore, are not exhaustive.

Some asymptotic methods for strongly nonlinear equations

Variational iteration method-Some recent results and new interpretations

Variational iteration method has been favourably applied to various kinds of nonlinear problems. The main property of the method is in its flexibility and ability to solve nonlinear equations accurately and conveniently. In this paper recent trends and developments in the use of the method are reviewed. Major applications to nonlinear wave equation, nonlinear fractional differential equations, nonlinear oscillations and nonlinear problems arising in various engineering applications are surveyed. The confluence of modern mathematics and symbol computation has posed a challenge to developing technologies capable of handling strongly nonlinear equations which cannot be successfully dealt with by classical methods. Variational iteration method is uniquely qualified to address this challenge. The flexibility and adaptation provided by the method have made the method a strong candidate for approximate analytical solutions. This paper outlines the basic conceptual framework of variational iteration technique with application to nonlinear problems. Both achievements and limitations are discussed with direct reference to approximate solutions for nonlinear equations. A new iteration formulation is suggested to overcome the shortcoming. A very useful formulation for determining approximately the period of a nonlinear oscillator is suggested. Examples are given to illustrate the solution procedure.

Variational iteration method: New development and applications

This review is an elementary introduction to the concepts of the recently developed asymptotic methods and new developments. Particular attention is paid throughout the paper to giving an intuitive grasp for Lagrange multiplier, calculus of variations, optimization, variational iteration method, parameter-expansion method, exp-function method, homotopy perturbation method, and ancient Chinese mathematics as well. Subsequently, nanomechanics in textile engineering and E-innit y theory in high energy physics, Kleiber’s 3/4 law in biology, possible mechanism in spider-spinning process and fractal approach to carbon nanotube are briey introduced. Bubble-electrospinning for mass production of nanob ers is illustrated. There are in total more than 280 references.

An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering

The use of variational iteration method, differential transform method and adomian decomposition method for solving different types of nonlinear partial differential equations

Solution of different type of the partial differential equation by differential transform method and Adomian’s decomposition method

1Department of Mathematics, National Institute of Technology, Rourkela 769008, India 2Department of Applied Mathematics, Institute for Groundwater Studies, University of the Free State, Bloemfontein, South Africa 3Department of Mathematical Sciences, North-West University, Mafikeng Campus, Mmabatho 2735, South Africa 4Department of Mathematics, Yildiz Technical University, 34220 Istanbul, Turkey 5Department of Mathematics, Faculty of Art & Sciences, Celal Bayar University, Muradiye Campus, 45047 Manisa, Turkey 6Department of Mathematics and Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia

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Fractional Calculus and Its Applications in Applied Mathematics and Other Sciences

The aim of this work was to convert the Thiem and the Theis groundwater flow equation to the time-fractional groundwater flow model. We first derived the analytical solution of the Theim time-fractional groundwater flow equation in terms of the generalized Wright function. We presented some properties of the Laplace-Carson transform. We derived the analytical solution of the Theis-time-fractional groundwater flow equation (TFGFE) via the Laplace-Carson transform method. We introduced the generalized exponential integral, as solution of the TFGFE. This solution is in perfect agreement with the data observed from the pumping test performed by the Institute for Groundwater Study on one of its borehole settled on the test site of the University of the Free State. The test consisted of the pumping of the borehole at the constant discharge rate Q and monitoring the piezometric head for 350 minutes.

https://downloads.hindawi.com/journals/mpe/2013/543026.pdf

The Use of Fractional Order Derivative to Predict the Groundwater Flow

The implementation of the two-dimensional differential transform method (DTM), Adomian's decomposition method (ADM), and the variational iteration method (VIM) in the mathematical applications of partial differential equations is examined in this paper. The VIM has been found to be particularly valuable as a tool for the solution of differential equations in engineering, science, and applied mathematics. The three methods are compared and it is shown that the VIM is more efficient and effective than the ADM and the DTM, and also converges to its exact solution more rapidly. Numerical solutions of two examples are calculated and the results are presented in tables and figures.

Necdet Bildik

Papers

The use of variational iteration method, differential transform method and adomian decomposition method for solving different types of nonlinear partial differential equations

Solution of different type of the partial differential equation by differential transform method and Adomian’s decomposition method

Fractional Calculus and Its Applications in Applied Mathematics and Other Sciences

The Use of Fractional Order Derivative to Predict the Groundwater Flow

Two-dimensional differential transform method, Adomian's decomposition method, and variational iteration method for partial differential equations