• Journal of the Korean Mathematical Society
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• v.48 no.3
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• pp.499-512
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• 2011

The purpose of this paper is twofold. The first is to establish a uniqueness theorem for entire function sharing two polynomials with its ${\kappa}$-th derivative, by using the theory of normal families. Meanwhile, the theorem generalizes some related results of Rubel and Yang and of Li and Yi. Several examples are provided to show the conditions are necessary. The second is to generalize the Br$\"{u}$-ck conjecture with the idea of sharing polynomial.

• Structural Engineering and Mechanics
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• v.43 no.2
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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.

• Kyungpook Mathematical Journal
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• v.28 no.2
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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.

• Structural Engineering and Mechanics
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• v.75 no.6
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• pp.737-746
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• 2020

This paper attempts to investigate the free vibration of functionally graded material beams with nonuniform width based on the nonlocal elasticity theory. The theoretical formulations are established following the Euler-Bernoulli beam theory, and the governing equations of motion of the system are derived from the minimum total potential energy principle using the nonlocal elasticity theory. In addition, the Differential Quadrature Method (DQM) is applied, along with the Chebyshev-Gauss-Lobatto polynomials, in order to determine the weighting coefficient matrices. Furthermore, the effects of the nonlocal parameter, cross-section area of the functionally graded material (FGM) beam and various boundary conditions on the natural frequencies are examined. It is observed that the nonlocal parameter and boundary conditions significantly influence the natural frequencies of the functionally graded material beam cross-section. The results obtained, using the Differential Quadrature Method (DQM) under various boundary conditions, are found in good agreement with analytical and numerical results available in the literature.

• Bulletin of the Korean Mathematical Society
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• v.59 no.5
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• pp.1145-1166
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• 2022

In this paper, for a transcendental meromorphic function f and α ∈ ℂ, we have exhaustively studied the nature and form of solutions of a new type of non-linear differential equation of the following form which has never been investigated earlier: $$f^n+{\alpha}f^{n-2}f^{\prime}+P_d(z,f)={\sum\limits_{i=1}^{k}}{p_i(z)e^{{\alpha}_i(z)},$$ where P_d(z, f) is a differential polynomial of f, p_i's and α_i's are non-vanishing rational functions and non-constant polynomials, respectively. When α = 0, we have pointed out a major lacuna in a recent result of Xue [17] and rectifying the result, presented the corrected form of the same equation at a large extent. In addition, our main result is also an improvement of a recent result of Chen-Lian [2] by rectifying a gap in the proof of the theorem of the same paper. The case α ≠ 0 has also been manipulated to determine the form of the solutions. We also illustrate a handful number of examples for showing the accuracy of our results.

• Communications of the Korean Mathematical Society
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• v.31 no.2
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• pp.311-327
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• 2016

We obtain similar types of conclusions as that of $Br{\ddot{u}}ck$ [1] for two differential polynomials which in turn radically improve and generalize several existing results. Moreover a number of examples have been exhibited to justify the necessity or sharpness of some conditions used in the paper. At last we pose an open problem for future research.

• Communications of the Korean Mathematical Society
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• v.31 no.2
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• pp.295-309
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• 2016

The paper deals with the problem of meromorphic functions sharing a small function with its differential polynomials and improves the results of Liu and Gu [9], Lahiri and Sarkar [8], Zhang [13] and Zhang and Yang [14] and also answer some open questions posed by Kit-Wing Yu [16]. In this paper we provide some examples to show that the conditions in our results are the best possible.

• Communications of the Korean Mathematical Society
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• v.33 no.3
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• pp.833-851
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• 2018

In this article, we prove some normality criteria for a family of meromorphic functions having zeros with some multiplicity. Our main result involves sharing of a holomorphic function by certain differential polynomials. Our results generalize some of the results of Fang and Zalcman [4] and Chen et al. [2] to a great extent.

• 제어로봇시스템학회:학술대회논문집
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• 2002.10a
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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...

• Bulletin of the Korean Mathematical Society
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• v.44 no.4
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• pp.623-629
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• 2007

In this paper, we study the uniqueness of entire functions and prove the following result: Let f and g be two nonconstant entire functions, n, m be positive integers. If $f^n(f^m-1)f#\;and\;g^n(g^m-1)g#$ share 1 IM and n>4m+11, then $f{\equiv}g$. The result improves the result of Fang-Fang.