Yildiray Keskin

Reduced Differential Transform Method for Partial Differential Equations

Recently differential transform method (DTM) has been used to solve various partial differential equations. In this paper, an alternative approach called the reduced differential transform method (RDTM) is presented to overcome the demerit of complex calculation of differential transform method. Comparing the methodology with some known techniques shows that the present approach is effective and powerful. In addition, three test problems of mathematical physics are discussed to illustrate the effectiveness and the performance of the reduced differential transform method.

The solution of the Bagley–Torvik equation with the generalized Taylor collocation method

Abstract In this paper, the Bagley–Torvik equation, which has an important role in fractional calculus, is solved by generalizing the Taylor collocation method. The proposed method has a new algorithm for solving fractional differential equations. This new method has many advantages over variety of numerical approximations for solving fractional differential equations. To assess the effectiveness and preciseness of the method, results are compared with other numerical approaches. Since the Bagley–Torvik equation represents a general form of the fractional problems, its solution can give many ideas about the solution of similar problems in fractional differential equations.

Approximate Solutions of Generalized Pantograph Equations by the Differential Transform Method

Delay differential equations (DDEs) are a large and important class of dynamical systems. Many phenomena in applied sciences can be successfully modelled by these equations [1-4]. For example, in natural or control systems, a controller monitors the state of the system and make adjustments to the system based on its observations. Therefore a delay occurs between the observation and the control action. In particular, they become powerful models in explaining physical phenomena when ODEbased models fail. There are different kinds of DDEs studied by various methods [5-12] In this paper, we deal with the generalization of the pantograph equation which is studied in [5,6] given by

A new analytical approximate method for the solution of fractional differential equations

A new analytical approximate method for the solution of fractional differential equations is presented. This method, which does not require any symbolic computation, is important as a tool for scientists and engineers because it provides an iterative procedure for obtaining the solution of both linear and non-linear fractional differential equations. The effectiveness of the proposed method is illustrated with some examples.

Conformable variational iteration method, conformable fractional reduced differential transform method and conformable homotopy analysis method for non-linear fractional partial differential equations

In this paper, we introduce conformable variational iteration method (C-VIM), conformable fractional reduced differential transform method (CFRDTM) and conformable homotopy analysis method (C-HAM). Between these methods, the C-VIM is introduced for the first time for fractional partial differential equations (FPDEs). These methods are new versions of well-known VIM, RDTM and HAM. In addition, above-mentioned techniques are based on new defined conformable fractional derivative to solve linear and non-linear conformable FPDEs. Firstly, we present some basic definitions and general algorithm for proposal methods to solve linear and non-linear FPDEs. Secondly, to understand better, the presented new methods are supported by some examples. Finally, the obtained results are illustrated by the aid of graphics and the tables. The applications show that these new techniques C-VIM, CFRDTM and C-HAM are extremely reliable and highly accurate and it provides a significant improvement in solving linear and non-linear FPDEs.

Approximate solution of Kuramoto–Sivashinsky equation using reduced differential transform method

In this study, approximate solution of Kuramoto–Sivashinsky Equation, by the reduced differential transform method, are presented.We apply this method to an example.Thus, we have obtained numerical solution Kuramoto– Sivashinsky equation. Comparisons are made between the exact solution and the reduced differential transform method. The results show that this method is very effective and simple.

Reduced differential transform method for improved Boussinesq equation

In this paper, the approximate solution of improved Boussinesq equation was found through reduced differential transform method. The equation has been used in many mathematical, engineering problems and mathematical physics. It is known a complicated and time-consuming solution. These problems were overcome by RDTM. Algebraic equations which was obtained by transform been done with RDTM was solved with Maple 13 computer program and the results obtained by RDTM compared with the results of exact solution.

The Maple Program Procedures at Solution Systems of Differential Equation with Taylor Collocation Method

In this paper, a maple algorithm Taylor collocation method has been presented for numerically solving the systems of differential equation with variable coefficients under the mixed conditions. The solution is obtained in terms of Taylor polynomials. This method is based on taking the truncated Taylor series of the function in equations and then substituting their matrix forms in the given equation. Hence, the result of matrix equation can be solved and the unknown Taylor coefficients can be found approximately. The results obtained by Taylor collocation method will be compared with the results of differential transform method and Adomian decomposition method.