• Journal of applied mathematics & informatics
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• 제26권5_6호
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• pp.1057-1069
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• 2008

We consider singularly perturbed Boundary Value Problems (BVPs) for third and fourth order Ordinary Differential Equations(ODEs) of convection-diffusion type with discontinuous source term and a small positive parameter multiplying the highest derivative. Because of the type of Boundary Conditions(BCs) imposed on these equations these problems can be transformed into weakly coupled systems. In this system, the first equation does not have the small parameter but the second contains it. In this paper a computational method named as 'An asymptotic finite element method' for solving these systems is presented. In this method we first find an zero order asymptotic approximation to the solution and then the system is decoupled by replacing the first component of the solution by this approximation in the second equation. Then the second equation is independently solved by a fitted mesh Finite Element Method (FEM). Numerical experiments support our theoritical results.

• 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 Mechanical Science and Technology
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• 제19권spc1호
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• pp.364-370
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• 2005

A stable walking pattern generation method for a biped robot is presented in this paper. In general, the ZMP (zero moment point) equations, which are expressed as differential equations, are solved to obtain a stable walking pattern. However, the number of differential equations is less than that of unknown coordinates in the ZMP equations. It is impossible to integrate the ZMP equations directly since one or more constraint equations are involved in the ZMP equations. To overcome this difficulty, DAE (differential and algebraic equation) solution method is employed. The proposed method has enough flexibility for various kinematic structures. Walking simulation for a virtual biped robot is performed to demonstrate the effectiveness and validity of the proposed method. The method can be applied to the biped robot for stable walking pattern generation.

• Wind and Structures
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• 제28권6호
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• pp.347-354
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• 2019

This study provides an approximate analytical solution to the fractional vibration problem of thin plate governing anomalous motion of plate subjected to in-plane loads. The method of variable separable is employed to transform the fractional partial differential equations under consideration into a fractional ordinary differential equation in temporal variable and a bi-harmonic plate equation in spatial variable. The technique of conformable fractional derivative is utilized to solve the resulting fractional differential equation and the approach of finite sine integral transform method is used to solve the accompanying bi-harmonic plate equation. The deflection field which measures the transverse displacement of the plate is expressed in terms of product of Bessel and trigonometric functions via the temporal and spatial variables respectively. The obtained solution reduces to the solution of the free vibration problem of thin plate in literature. This work shows that conformable fractional derivative is an efficient mathematical tool for tracking analytical solution of fractional partial differential equation governing anomalous vibration of thin plates.

• Journal of applied mathematics & informatics
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• 제35권1_2호
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• pp.165-180
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• 2017

In this study, Taylor matrix algorithm is designed for the approximate solution of linear and non-linear differential equation systems. The algorithm is essentially based on the expansion of the functions in differential equation systems to Taylor series and substituting the matrix forms of these expansions into the given equation systems. Using the Mathematica program, the matrix equations are solved and the unknown Taylor coefficients are found approximately. The presented numerical approach is discussed on samples from various linear and non-linear differential equation systems as well as stiff systems. The computational data are then compared with those of some earlier numerical or exact results. As a result, this comparison demonstrates that the proposed method is accurate and reliable.

• 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.

• 대한수학회논문집
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• 제26권4호
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• pp.709-720
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• 2011

We obtain special type of differential equations which their solution are random variable with known continuous density function. Stochastic differential equations (SDE) of continuous distributions are determined by the Fokker-Planck theorem. We approximate solution of differential equation with numerical methods such as: the Euler-Maruyama and ten stages explicit Runge-Kutta method, and analysis error prediction statistically. Numerical results, show the performance of the Rung-Kutta method with respect to the Euler-Maruyama. The exponential two parameters, exponential, normal, uniform, beta, gamma and Parreto distributions are considered in this paper.

• Structural Engineering and Mechanics
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• 제24권1호
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• pp.51-62
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• 2006

Differential transform method (DTM) for free vibration analysis of both ends simply supported beam resting on elastic foundation is suggested. The fourth order partial differential equation for free vibration of the beam resting on elastic foundation subjected to bending moment, shear and axial compressive load is obtained by using Winkler hypothesis and small displacement theory. It is assumed that the material is linear-elastic, and that axial load and modulus of subgrade reaction to be constant. In the analysis, shear and axial load effects are considered. The frequency factors of the beam are calculated by using DTM due to the values of relative stiffness; the results are presented in graphs and tables.

• Journal of applied mathematics & informatics
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• 제29권1_2호
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• pp.55-60
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• 2011

The aim of the paper is to apply Homotopy Perturbation Method (HPM) for the solution of a nonlinear fractional differential equation. Finally, the solution obtained by the Homotopy perturbation method has been numerically evaluated and presented in the form of tables and then compared with those obtained by truncated series method. A good agreement of the results is observed.

• Journal of applied mathematics & informatics
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• 제24권1_2호
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• pp.167-178
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• 2007

This paper deals with the solutions of linear inhomogeneous time-fractional partial differential equations in applied mathematics and fluid mechanics. The fractional derivatives are described in the Caputo sense. The fractional Green function method is used to obtain solutions for time-fractional wave equation, linearized time-fractional Burgers equation, and linear time-fractional KdV equation. The new approach introduces a promising tool for solving fractional partial differential equations.