• Journal of applied mathematics & informatics
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• 제41권1호
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• pp.71-82
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• 2023

In this article, we find the approximate solutions of Abel differential equation (ADE) with uncertainty using residual power series (RPS) method. This method helps to calculate the sequence of solutions of ADE. Finally, numerical illustrations demonstrate the applicability of the method.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제22권4호
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• pp.289-294
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• 2018

In this work, a recently developed semi-analytic technique, so called the residual power series method, is generalized to process higher-dimensional linear and nonlinear partial differential equations. The solutions obtained takes a form of an infinite power series which can, in turn, be expressed in a closed exact form. The results reveal that the proposed generalization is very effective, convenient and simple. This is achieved by handling the (m+1)-dimensional Burgers equation.

• 대한수학회보
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• 제51권1호
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• pp.77-82
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• 2014

We prove that if ${\mid}a_1{\mid}$ is large and ${\mid}a_0{\mid}$ is small enough, then every approximate zero of power series equation ${\sum}^{\infty}_{n=0}a_nx^n$=0 can be approximated by a true zero within a good error bound. Further, we obtain Hyers-Ulam stability of zeros of the polynomial equation of degree n, $a_nz^n$ + $a_{n-1}z^{n-1}$ + ${\cdots}$ + $a_1z$ + $a_0$ = 0 for a given integer n > 1.

• 대한수학회보
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• 제43권4호
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• pp.791-804
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• 2006

This paper is concerned with a second-order iterated differential equation of the form $c_0x'(Z)+c_1x'(z)+c_2x(z)=x(az+bx(z))+h(z)$ with the distinctive feature that the argument of the unknown function depends on the state. By constructing a convergent power series solution of an auxiliary equation, analytic solutions of the original equation are obtained.

• 대한수학회보
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• 제53권3호
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• pp.699-709
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• 2016

Our paper is devoted to the study of certain diophantine equations on the ring of polynomials over a finite field, which are intimately related to algebraic formal power series which have partial quotients of unbounded degree in their continued fraction expansion. In particular it is shown that there are Pisot formal power series with degree greater than 2, having infinitely many large partial quotients in their simple continued fraction expansions. This generalizes an earlier result of Baum and Sweet for algebraic formal power series.

• Journal of Advanced Research in Ocean Engineering
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• 제1권3호
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• pp.157-167
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• 2015

In order to provide simple and accurate wave theory in design of offshore structure, an analytical approximation is introduced in this paper. The solution is limited to flat bottom having a constant water depth. Water is considered as inviscid, incompressible and irrotational. The solution satisfies the continuity equation, bottom boundary condition and non-linear kinematic free surface boundary condition exactly. Error for dynamic condition is quite small. The solution is suitable in description of breaking waves. The solution is presented with closed form and dispersion relation is also presented with closed form. In the last century, there have been two main approaches to the nonlinear problems. One of these is perturbation method. Stokes wave and Cnoidal wave are based on the method. The other is numerical method. Dean's stream function theory is based on the method. In this paper, power series method was considered. The power series method can be applied to certain nonlinear differential equations (initial value problems). The series coefficients are specified by a nonlinear recurrence inherited from the differential equation. Because the non-linear wave problem is a boundary value problem, the power series method cannot be applied to the problem in general. But finite number of coefficients is necessary to describe the wave profile, truncated power series is enough. Therefore the power series method can be applied to the problem. In this case, the series coefficients are specified by a set of equations instead of recurrence. By using the set of equations, the nonlinear wave problem has been solved in this paper.

• Kyungpook Mathematical Journal
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• 제62권4호
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• pp.729-736
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• 2022

To help solving intractable nonlinear evolution equations (NLEEs) of waves in the field of fluid dynamics we develop an algorithm to find new high order solutions of the class of Abel, Bernoulli, Chini and Riccati equations of the form y' = ayⁿ + by + c, n > 1, with constant coefficients a, b, c. The role of this class of equations in NLEEs is explained in the introduction below. The basic algorithm to compute the coefficients of the power series solutions of the class, emerged long ago and is further developed in this paper. Practical application for hitherto unknown solutions is exemplified.

• 한국콘텐츠학회논문지
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• 제12권3호
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• pp.434-441
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• 2012

축대칭 함몰지형 위를 통과하는 파의 변형에 관한 확장형 완경사 방정식의 해석해를 유도하였다. 함몰지형내의 수심은 함몰지형의 중심으로부터의 거리의 멱에 비례하여 변화된다. 지배방정식은 변수분리법을 이용하여 상미분방정식으로 변환되었으며 Hunt(1979)의 근사식을 이용하여 계수들을 파속과 군속도의 항으로 이뤄진 양함수의 형태로 나타냈다. 확장형 완경사 방정식의 바닥의 곡률과 경사의 제곱으로 이뤄진 항은 멱급수형태로 변환하였다. 마지막으로 Frobenius 급수를 사용하여 확장형 완경사 방정식의 해석해를 유도하였다. 유도된 해석해는 FEM으로 도출된 수치해와 기존의 완경사 방정식의 해석해와 비교하여 검증하였다.

• Structural Engineering and Mechanics
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• 제43권5호
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• pp.561-582
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• 2012

Despite popularity of FEM in analysis of static and dynamic structural problems and the routine applicability of FE softwares, analytical methods based on simple mathematical relations is still largely sought by many researchers and practicing engineers around the world. Development of such analytical methods for analysis of free vibration of non-prismatic beams is also of primary concern. In this paper a new and simple method is proposed for determination of vibration frequencies of non-prismatic beams under variable axial forces. The governing differential equation is first obtained and, according to a harmonic vibration, is converted into a single variable equation in terms of location. Through repetitive integrations, integral equation for the weak form of governing equation is derived. The integration constants are determined using the boundary conditions applied to the problem. The mode shape functions are approximated by a power series. Substitution of the power series into the integral equation transforms it into a system of linear algebraic equations. Natural frequencies are determined using a non-trivial solution for system of equations. Presented method is formulated for beams having various end conditions and is extended for determination of the buckling load of non-prismatic beams. The efficiency and convergence rate of the current approach are investigated through comparison of the numerical results obtained to those obtained using available finite element software.

• 한국정보통신학회논문지
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• 제9권1호
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• pp.140-145
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• 2005

기존의 금속${\cdot}$유전체${\cdot}$금속 기판형태의 전력${\cdot}$접지층 사이의 전압표현식은 2차원 무한급수 형태로 표시된다. 계산시간 단축을 위해 Fourier 급수합 공식을 이용하여 2차원 무한급수를 1차원 무한급수로 변형시켰다. 이 식들을 $9‘{\times}4'$크기를 가지는 전력${\cdot}$접지층에 대한 전압 계산에 적용했다. 유도된 1차원 급수 계산식은 기존의 2차원 급수식에 비해 빠른 수렴성과 정확한 결과를 보였다. 이 결과는 반복적인 계산이 많이 필요한 전력${\cdot}$접지층 해석에 유용하게 적용될 수 있을 것이다.