• Journal of applied mathematics & informatics
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• 제30권3_4호
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• pp.555-560
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• 2012

In this paper, differential transform method (DTM) is considered to obtain solution to the generalized Fisher's equation. This method is easy to apply and because of high level of accuracy can be used to solve other linear and nonlinear problems. Furthermore, is capable of reducing the size of computational work. In the present work, the generalization of the two-dimensional transform method that is based on generalized Taylor's formula is applied to solve the generalized Fisher equation and numerical example demonstrates the accuracy of the present method.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제23권4호
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• pp.283-300
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• 2019

In this paper, we propose a new semi-analytic approach based on the generalized Taylor series for solving nonlinear differential equations of fractional order. Assuming the solution is expanded as the generalized Taylor series, the coefficients of the series can be computed by solving the corresponding recursive relation of the coefficients which is generated by the given problem. This method is called the generalized differential transform method(GDTM). In several literatures the standard GDTM was applied in each sub-domain to obtain an accurate approximation. As noticed in [19], however, a direct application of the GDTM in each sub-domain loses a term of memory which causes an inaccurate approximation. In this work, we derive a new recursive relation of the coefficients that reflects an effect of memory. Several illustrative examples are demonstrated to show the effectiveness of the proposed method. It is shown that the proposed method is robust and accurate for solving nonlinear differential equations of fractional order.

• Korean Journal of Mathematics
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• 제29권4호
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• pp.749-763
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• 2021

This research article deals with the Mittag-Leffler-Hyers-Ulam stability of linear and impulsive fractional order differential equation which involves the Caputo derivative. The application of the generalized fractional Fourier transform method and fixed point theorem, evaluates the existence, uniqueness and stability of solution that are acquired for the proposed non-linear problems on Lizorkin space. Finally, examples are introduced to validate the outcomes of main result.

• 대한수학회논문집
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• 제33권2호
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• pp.639-650
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• 2018

Finding a solution for the Legendre equation is difficult. Especially if it is as a part of the Laplace equation solving in the electric fields. In this paper, first a problem of the generalized differential transform method (GDTM) is solved by the Sturm-Liouville equation, then the Legendre equation is solved by using it. To continue, the approximate solution is compared with the nth-degree Legendre polynomial for obtaining the inner and outer potential of a sphere. This approximate is more accurate than the previous solutions, and is closer to an ideal potential in the intervals.

• Journal of applied mathematics & informatics
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• 제39권3_4호
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• pp.267-275
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• 2021

In this study, we investigate the generalized Hyers-Ulam Stability of partial differential equation of the form y_t - ky_xx = 0.

• Advances in aircraft and spacecraft science
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• 제1권2호
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• pp.125-143
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• 2014

This paper provides a new technique for solving the static analysis of arbitrarily shaped composite plates by using Strong Formulation Finite Element Method (SFEM). Several papers in literature by the authors have presented the proposed technique as an extension of the classic Generalized Differential Quadrature (GDQ) procedure. The present methodology joins the high accuracy of the strong formulation with the versatility of the well-known Finite Element Method (FEM). The continuity conditions among the elements is carried out by the compatibility or continuity conditions. The mapping technique is used to transform both the governing differential equations and the compatibility conditions between two adjacent sub-domains into the regular master element in the computational space. The numerical implementation of the global algebraic system obtained by the technique at issue is easy and straightforward. The main novelty of this paper is the application of the stress and strain recovery once the displacement parameters are evaluated. Computer investigations concerning a large number of composite plates have been carried out. SFEM results are compared with those presented in literature and a perfect agreement is observed.

• Ocean Systems Engineering
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• 제6권4호
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• pp.363-375
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• 2016

Large numbers of submarine pipelines are laid as the world now is attaching great importance to offshore oil exploitation. Free spanning of submarine pipelines may be caused by seabed unevenness, change of topology, artificial supports, etc. By combining Iwan's wake oscillator model with the differential equation which describes the vibration behavior of free-span submarine pipelines, the pipe-fluid coupling equation is developed and solved in order to study the effect of both internal and external fluid on the vibration behavior of free-span submarine pipelines. Through generalized integral transform technique (GITT), the governing equation describing the transverse displacement is transformed into a system of second-order ordinary differential equations (ODEs) in temporal variable, eliminating the spatial variable. The MATHEMATICA built-in function NDSolve is then used to numerically solve the transformed ODE system. The good convergence of the eigenfunction expansions proved that this method is applicable for predicting the dynamic response of free-span pipelines subjected to both internal flow and external current.

• 소음진동
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• 제9권1호
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• pp.103-112
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• 1999

A new modeling and analysis technique for one-dimensional distributed parameter systems is presented. First. discretized equations of motion in Laplace domain are derived by applying discretization methods for partial differential equations of a one-dimensional structure with respect to spatial coordinate. Secondly. the z and inverse z transformations are applied to the discretized equations of motion for obtaining a dynamic matrix for a uniform element. Four different discretization methods are tested with an example. Finally, taking infinite on the number of step for a uniform element leads to an exact dynamic matrix for the uniform element. A generalized modal analysis procedure for eigenvalue analysis and modal expansion is also presented. The resulting element dynamic matrix is tested with a numerical example. Another application example is provided to demonstrate the applicability of the proposed method.

• Structural Engineering and Mechanics
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• 제75권3호
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• pp.357-367
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• 2020

The objective of this article is investigation of dynamic response of thick multilayer functionally graded (FG) beam under generalized dynamic forces. The plane stress problem is exploited to describe the constitutive equation of thick FG beam to get realistic and accurate response. Applied dynamic forces are assumed to be sinusoidal harmonic, sinusoidal pulse or triangle in time domain and point load. Equations of motion of deep FG beam are derived based on the Hamilton principle from kinematic relations and constitutive equations of plane stress problem. The numerical finite element procedure is adopted to discretize the space domain of structure and transform partial differential equations of motion to ordinary differential equations in time domain. Numerical time integration method is used to solve the system of equations in time domain and find the time responses. Numerical parametric studies are performed to illustrate effects of force type, graduation parameter, geometrical and stacking sequence of layers on the time response of deep multilayer FG beams.

• 한국전산구조공학회논문집
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• 제13권1호
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• pp.37-49
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• 2000

본 논문은 구조물의 동역학 및 열탄성 연성문제 해석을 위한 통합된 유한요소법을 개발하는데 초점을 두고있다. 첫째로, 열전도 방정식에 열변위라는 물리량을 도입하여 동역학의 운동 방정식과 유사하도록 유도한 후, 변분법과 일반좌표계를 이용하여 시간영역에서 정식화하였다. 둘째로, 두 방정식에 라플라스 변환을 동시에 도입하고, 공간변수만을 갖는 형상함수와 가중잔여법을 적용하여 유한요소식을 변환영역에서 표현하였다. 연성된 방정식을 문제의 특성에 따라서 분류하였고 정식화 과정을 검증하였다. 또한 수치해석 알고리듬이 갖는 수치 역 변환의 정성적인 경향에 대하여 검토하였다.