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.

Reduced Differential Transform Method for Partial Differential Equations

In this paper, we will compare the differential transformation method DTM and Adomian decomposition method ADM to solve partial differential equations (PDEs). The definition and operations of differential transform method was introduced by Zhou [Zhou JK. Differential transformation and its application for electrical circuits. Wuuhahn, China: Huarjung University Press; 1986 [in Chinese]]. Adomian decomposition method which is given by Adomian [Adomian G. Convergent series solution of nonlinear equation. J Comput Appl Math 1984;11:113–7; Adomian G. Solutions of nonlinear PDE. Appl Math Lett 1989;11:121–3; Adomian G. Solving Frontier problems of physics. The decomposition method, Boston, 1994] for approximate solution of linear and nonlinear differential equations and to the solutions of various scientific models see [Abulwafa EM, Abdou MA, Mahmoud AA. The solution of nonlinear coagulation problem with mass loss. Chaos, Solitons & Fractals 2006;29:313–30; El-Danaf TS, Ramadan MA, Abd Alaal FEI. The use of Adomian decomposition method for solving the regularized long-wave equation. Chaos, Solitons & Fractals 2005;26:747–57; El-Sayed SM. The decomposition method for studying the Klein–Gordon equation. Chaos, Solitions & Fractals 2003;18:1025–30; Helal MA, Mehanna MS. A comparison between two different methods for solving KdV-Burgers equation. Chaos, Solitons & Fractals 2006;28:320–6, Hashim I, Noorani MSM, Ahmad R, Bakar SA, Ismail ES, Zakaria AM. Accuracy of the decomposition method applied to the Lorenz system. Chaos, Solitons & Fractals 2006;28:1149–58; Kaya D, El-Sayed SM. An application of the decomposition method for the generalized KdV and RLW equations, Chaos, Solitons & Fractals 2003;17:869–77; Lesnic D. Blow-up solutions obtained using the decomposition method. Chaos, Solitons & Fractals 2006;28:776–87; Wazwaz AM. Construction of solitary wave solutions and rational solutions for KdV equation by Adomian decomposition method. Chaos, Solitons & Fractals 2001;12:2283–93; Wazwaz AM. Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method. Chaos, Solitons & Fractals 2001;12:1549–56]. A distinctive practical feature of the differential transformation method DTM is ability to solve linear or nonlinear differential equations. Higher-order dimensional differential transformation are applied to a few some initial value problems to show that the solutions obtained by the proposed method DTM coincide with the approximate solution ADM and the analytic solutions.

Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems

Application of the differential transformation method to vibration analysis of pipes conveying fluid

Differential transform method for solving the linear and nonlinear Klein–Gordon equation

In this paper, we propose a reliable algorithm to develop exact and approximate solutions for the linear and nonlinear Schrodinger equations. The approach rest mainly on two-dimensional differential transform method which is one of the approximate methods. The method can easily be applied to many linear and nonlinear problems and is capable of reducing the size of computational work. Exact solutions can also be achieved by the known forms of the series solutions. Several illustrative examples are given to demonstrate the effectiveness of the present method.

Two-dimensional differential transform method for solving linear and non-linear Schrödinger equations

In this paper, we introduce a new approach on fractional integration, which generalizes the Riemann-Liouville fractional integral. We prove some properties for this new approach. We also establish some new integral inequalities using this new fractional integration.

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(k, s)-Riemann-Liouville fractional integral and applications

This paper presents the generalization of the differential transformation method to n-dimensional case in order to solve partial differential equations (PDEs). A distinctive practical feature of this method is its ability to solve especially nonlinear differential equations efficiently. We apply our results to a few initial boundary-value problems to illustrate the proposed method.

n-Dimensional differential transformation method for solving PDEs

In this paper, we obtain some new companion of Ostrowski type inequality and Cebysev type inequality for functions whose second derivativves absolute value are convex and second derivatives belongs to L_{p} spaces. Applications to composite quadrature rules, and special means are also given.

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On companion of Ostrowski type inequality and Cebysev type inequalty

In this paper, we establish several weighted inequalities for some differantiable mappings that are connected with the celebrated Hermite-Hadamard-Fejer type and Ostrowski type integral inequalities for s-convex functions. The results presented here would provide extensions of those given in earlier works.

ON THE WEIGHTED INTEGRAL INEQUALITIES FOR s-CONVEX FUNCTION

The main aim of the present paper is to introduce φh-convex stochastic processes and we investigate main properties of these mappings. Moreover, we prove the Hadamard-type inequalities for φh-convex stochastic processes. We also give some new general inequalities for φh-convex stochastic processes.

Mehmet Eyüp Kiriş

Papers

(k, s)-Riemann-Liouville fractional integral and applications

n-Dimensional differential transformation method for solving PDEs

On companion of Ostrowski type inequality and Cebysev type inequalty

ON THE WEIGHTED INTEGRAL INEQUALITIES FOR s-CONVEX FUNCTION

Hermite-Hadamard type inequality for φh-convex stochastic processes