• 제어로봇시스템학회:학술대회논문집
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• 제어로봇시스템학회 2002년도 ICCAS
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• pp.52.2-52
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• 2002

In this paper we consider an approach to a formal linearization for time-variant nonlinear systems. A time-variant nonlinear sysetm is assumed to be described by a time-variant nonlinear differential equation. For this system, we introduce a coordinate transformation function which is composed of the Chebyshev polynomials. Using Chebyshev expansion to the state variable and Laguerre expansion to the time variable, the time-variant nonlinear sysetm is transformed into the time-variant linear one with respect to the above transformation function. As an application, we synthesize a time-variant nonlinear observer. Numerical experiments are included to demonstrate the validity of...

• 대한수학회보
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• 제57권4호
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• pp.1061-1073
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• 2020

In this paper, the expressions of meromorphic solutions of the following nonlinear complex differential equation of the form $$f^n+Qd(z,f)=\sum\limits_{i=1}^{3}pi(z)e^{{\alpha}_i(z)}$$ are studied by using Nevanlinna theory, where n ≥ 5 is an integer, Q_d(z, f) is a differential polynomial in f of degree d ≤ n - 4 with rational functions as its coefficients, p₁(z), p₂(z), p₃(z) are non-vanishing rational functions, and α₁(z), α₂(z), α₃(z) are nonconstant polynomials such that α'₁(z), α'₂(z), α'₃(z) are distinct each other. Moreover, examples are given to illustrate the accuracy of the condition.

• Journal of applied mathematics & informatics
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• 제31권1_2호
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• pp.165-177
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• 2013

In this paper, we apply Homotopy perturbation transform method (HPTM) for solving singular fourth order parabolic partial differential equations with variable coefficients. This method is the combination of the Laplace transform method and Homotopy perturbation method. The nonlinear terms can be easily handled by the use of He's polynomials. The aim of using the Laplace transform is to overcome the deficiency that is mainly caused by unsatisfied conditions in other semi-analytical methods such as Homotopy perturbation method (HPM), Variational iteration method (VIM) and Adomain Decomposition method (ADM). The proposed scheme finds the solutions without any discretization or restrictive assumptions and avoids the round-off errors. The comparison shows a precise agreement between the results and introduces this method as an applicable one which it needs fewer computations and is much easier and more convenient than others, so it can be widely used in engineering too.

• Structural Engineering and Mechanics
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• 제43권2호
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• pp.211-223
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• 2012

This paper deals with large deformation post-buckling of a linear-elastic and hygrothermal beam with axially nonmovable pinned-pinned ends and subjected to a significant increase in swelling by an alternative method. Analytical approximate solutions for the geometrically nonlinear problem are presented. The solution for the limiting case of a string is also obtained. By coupling of the well-known Maclaurin series expansion and orthogonal Chebyshev polynomials, the governing differential equation with sinusoidal nonlinearity can be reduced to form a cubic-nonlinear equation, and supplementary condition with cosinoidal nonlinearity can also be simplified to be a polynomial integral equation. Analytical approximations to the resulting boundary condition problem are established by combining the Newton's method with the method of harmonic balance. Two approximate formulae for load along axis, potential strain for free hygrothermal expansion and periodic solution are established for small as well as large angle of rotation at the end of the beam. Illustrative examples are selected and compared to "reference" solution obtained by the shooting method to substantiate the accuracy and correctness of the approximate analytical approach.

• Nonlinear Functional Analysis and Applications
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• 제29권2호
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• pp.527-543
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• 2024

In the realm of differential games and bioeconomic modeling, where intricate systems and multifaceted interactions abound, we explore the precision and efficiency of the Chebyshev Tau method (CTM). We begin with the Weierstrass Approximation Theorem, employing Chebyshev polynomials to pave the way for solving intricate bioeconomic differential games. Our case study revolves around a three-player bioeconomic differential game, unveiling a unique open-loop Nash equilibrium using Hamiltonians and the FilippovCesari existence theorem. We then transition to numerical implementation, employing CTM to resolve a Three-Point Boundary Value Problem (TPBVP) with varying degrees of approximation.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제26권4호
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• pp.185-223
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• 2022

The classical equation of Jonathan Homer Lane and Robert Emden, a nonlinear second-order ordinary differential equation, models the isothermal spherical clouded gases under the influence of the mutual attractive interaction between the gases' molecules. In this paper, the Adomian decomposition method (ADM) is presented to obtain highly accurate and reliable analytical solutions of a class of generalised Lane-Emden equations with strong nonlinearities. The nonlinear term f(y(x)) of the proposed problem is given by the integer powers of a continuous real-valued function h(y(x)), that is, f(y(x)) = h^m(y(x)), for integer m ≥ 0, real x > 0. In the end, numerical comparisons are presented between the analytical results obtained using the ADM and numerical solutions using the eighth-order nested second derivative two-step Runge-Kutta method (NSDTSRKM) to illustrate the reliability, accuracy, effectiveness and convenience of the proposed methods. The special cases h(y) = sin y(x), cos y(x); h(y) = sinh y(x), cosh y(x) are considered explicitly using both methods. Interestingly, in each of these methods, a unified result is presented for an integer power of any continuous real-valued function - compared with the case by case computations for the nonlinear functions f(y). The results presented in this paper are a generalisation of several published results. Several examples are given to illustrate the proposed methods. Tables of expansion coefficients of the series solutions of some special Lane-Emden type equations are presented. Comparisons of the two results indicate that both methods are reliably and accurately efficient in solving a class of singular strongly nonlinear ordinary differential equations.

• Kyungpook Mathematical Journal
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• 제28권2호
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• pp.185-196
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• 1988

We study existence of polynomial integrating factors and solutions F(x, y)=c of first order nonlinear differential equations. We characterize the homogeneous case, and give algorithms for finding existence of and a basis for polynomial solutions of linear difference and differential equations and rational solutions or linear differential equations with polynomial coefficients. We relate singularities to nature of the solution. Solution of differential equations in closed form to some degree might be called more an art than a science: The investigator can try a number of methods and for a number of classes of equations these methods always work. In particular integrating factors are tricky to find. An analogous but simpler situation exists for integrating inclosed form, where for instance there exists a criterion for when an exponential integral can be found in closed form. In this paper we make a beginning in several directions on these problems, for 2 variable ordinary differential equations. The case of exact differentials reduces immediately to quadrature. The next step is perhaps that of a polynomial integrating factor, our main study. Here we are able to provide necessary conditions based on related homogeneous equations which probably suffice to decide existence in most cases. As part of our investigations we provide complete algorithms for existence of and finding a basis for polynomial solutions of linear differential and difference equations with polynomial coefficients, also rational solutions for such differential equations. Our goal would be a method for decidability of whether any differential equation Mdx+Mdy=0 with polynomial M, N has algebraic solutions(or an undecidability proof). We reduce the question of all solutions algebraic to singularities but have not yet found a definite procedure to find their type. We begin with general results on the set of all polynomial solutions and integrating factors. Consider a differential equation Mdx+Ndy where M, N are nonreal polynomials in x, y with no common factor. When does there exist an integrating factor u which is (i) polynomial (ii) rational? In case (i) the solution F(x, y)=c will be a polynomial. We assume all functions here are complex analytic polynomial in some open set.

• Coupled systems mechanics
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• 제1권2호
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• pp.155-163
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• 2012

This paper presents analytical approximate solutions for the initial post-buckling deformation of the pipes in oil and gas wells. The governing differential equation with sinusoidal nonlinearity can be reduced to form a third-order-polynomial nonlinear equation, by coupling of the well-known Maclaurin series expansion and orthogonal Chebyshev polynomials. Analytical approximations to the resulting boundary condition problem are established by combining the Newton's method with the method of harmonic balance. The linearization is performed prior to proceeding with harmonic balancing thus resulting in a set of linear algebraic equations instead of one of non-linear algebraic equations, unlike the classical method of harmonic balance. We are hence able to establish analytical approximate solutions. The approximate formulae for load along axis, and periodic solution are established for derivative of the helix angle at the end of the pipe. Illustrative examples are selected and compared to "reference" solution obtained by the shooting method to substantiate the accuracy and correctness of the approximate analytical approach.

• 한국정보통신학회논문지
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• 제8권1호
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• pp.123-136
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• 2004

본 논문에서는 $GF({2^n})$상 곱셈의 복잡도와 규칙도를 GF(2)상의 다항식 곱셈을 표현하는 행렬식의 행과 열의 해밍 가중치를 이용하여 정의한다 차분공격에 강한 블록 암호 알고리즘을 만들기 위해서는 치환계층과 확산계층의 $GF({2^n})$상 곱셈의 복잡도와 규칙도가 높아야함을 실험을 통하여 보인다. 실험 결과를 활용하여 우리나라 표준인 128 비트 블록 암호 알고리즘인 SEED의 S 박스와 G 함수를 구성하는 방식을 제안한다. S 박스는 비 선형함수와 아핀변환으로 구성한다. 비 선형함수는 차분공격과 선형공격에 강한 특성을 가지며, '0'과 '1'을 제외하고 입력과 출력이 같은 고정점과 출력이 입력의 1의 보수가 되는 역고정점을 가지지 않는 $GF({2^8})$ 상의 역수로 구성한다. 아핀변환은 입력과 출력간의 상관을 최저로 하면서 고정점과 역고정점이 없도록 구성한다. G 함수는 4개의 S 박스 출력을 $GF({2^8}) 상의 4 {\times} 4$ 행렬식을 사용하여 선형변환한다. 선형변환 행렬식 성분은 높은 복잡도와 규칙도를 가지도록 구성한다 또한 MDS(Maximum Distance Separable) 코드를 생성하고, SAC(Strict Avalanche Criterion)를 만족하고, 고정점과 역고정점 및 출력이 입력의 2의 보수가 되는 약한 입력이 없도록 G 함수를 구성한다. 비선형함수와 아핀변환 및 G 함수의 원시다항식은 각기 다른 것을 사용한다. 본 논문에서 제안한 S 박스와 G 함수는 차분공격과 선형공격에 강하고, 약한 입력이 없으며, 확산 특성이 우수하므로 안전성이 높은 암호 방식의 구성 요소로 활용할 수 있다.