• Title/Summary/Keyword: Legendre polynomial

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FUNCTION APPROXIMATION OVER TRIANGULAR DOMAIN USING CONSTRAINED Legendre POLYNOMIALS

  • Ahn, Young-Joon
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.9 no.2
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    • pp.99-106
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    • 2005
  • We present a relation between the orthogonality of the constrained Legendre polynomials over the triangular domain and the BB ($B{\acute{e}zier}\;-Bernstein$) coefficients of the polynomials using the equivalence of orthogonal complements. Using it we also show that the best constrained degree reduction of polynomials in BB form equals the best approximation of weighted Euclidean norm of coefficients of given polynomial in BB form from the coefficients of polynomials of lower degree in BB form.

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NUMERICAL EXPERIMENTS OF THE LEGENDRE POLYNOMIAL BY GENERALIZED DIFFERENTIAL TRANSFORM METHOD FOR SOLVING THE LAPLACE EQUATION

  • Amoupour, Ebrahim;Toroqi, Elyas Arsanjani;Najafi, Hashem Saberi
    • Communications of the Korean Mathematical Society
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    • v.33 no.2
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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.

Neighboring Optimal Control using Pseudospectral Legendre Method (Pseudospectral Legendre법을 이용한 근접 최적 제어)

  • 이대우;조겸래
    • Journal of the Korean Society for Precision Engineering
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    • v.21 no.7
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    • pp.76-82
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    • 2004
  • The solutions of neighboring optimal control are typically obtained using the sweep method or transition matrices. Due to the numerical integration, however, the gain matrix can become infinite as time go to final one in the transition matrices, and the Riccati solution can become infinite when the final time free. To overcome these disadvantages, this paper proposes the pseudospectral Legendre method which is to first discreteize the linear boundary value problem using the global orthogonal polynomial, then transforms into an algebraic equations. Because this method is not necessary to take any integration of transition matrix or Riccati equation, it can be usefully used in real-time operation. Finally, its performance is verified by the numerical example for the space vehicle's orbit transfer.

Modal Parameter Identification from Frequency Response Functions Using Legendre Polynomials (Legendre 다항식을 이용한 주파수 응답 함수의 곡선접합과 모드 매개변수 규명)

  • Park, Nam-Gyu;Jeon, Sang-Youn;Suh, Jeong-Min;Kim, Hyeong-Koo;Jang, Young-Ki;Kim, Kyu-Tae
    • Transactions of the Korean Society for Noise and Vibration Engineering
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    • v.16 no.7 s.112
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    • pp.769-776
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    • 2006
  • A measured frequency response function can be represented as a ratio of two polynomials. A curve-fitting of frequency responses with Legendre polynomialis suggested in the paper. And the suggested curve-fitting algorithm is based on the least-square error method. Since the Legendre polynomials satisfy the orthogonality condition, the curve-fitting with the polynomials results to more reliable curve-fitting than ordinary polynomial method. Though the proposed curve-fitting with Legendre polynomials cannot cover all frequency range of interest, example shows that the suggested method is quite applicable in a limited frequency band.

Prediction of Future Milk Yield with Random Regression Model Using Test-day Records in Holstein Cows

  • Park, Byoungho;Lee, Deukhwan
    • Asian-Australasian Journal of Animal Sciences
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    • v.19 no.7
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    • pp.915-921
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    • 2006
  • Various random regression models with different order of Legendre polynomials for permanent environmental and genetic effects were constructed to predict future milk yield of Holstein cows in Korea. A total of 257,908 test-day (TD) milk yield records from a total of 28,135 cows belonging to 1,090 herds were considered for estimating (co)variance of the random covariate coefficients using an expectation-maximization REML algorithm in an animal mixed model. The variances did not change much between the models, having different order of Legendre polynomial, but a decreasing trend was observed with increase in the order of Legendre polynomial in the model. The R-squared value of the model increased and the residual variance reduced with the increase in order of Legendre polynomial in the model. Therefore, a model with $5^{th}$ order of Legendre polynomial was considered for predicting future milk yield. For predicting the future milk yield of cows, 132,771 TD records from 28,135 cows were randomly selected from the above data by way of preceding partial TD record, and then future milk yields were estimated using incomplete records from each cow randomly retained. Results suggested that we could predict the next four months milk yield with an error deviation of 4 kg. The correlation of more than 70% between predicted and observed values was estimated for the next four months milk yield. Even using only 3 TD records of some cows, the average milk yield of Korean Holstein cows would be predicted with high accuracy if compared with observed milk yield. Persistency of each cow was estimated which might be useful for selecting the cows with higher persistency. The results of the present study suggested the use of a $5^{th}$ order Legendre polynomial to predict the future milk yield of each cow.

NUMERICAL ANALYSIS OF LEGENDRE-GAUSS-RADAU AND LEGENDRE-GAUSS COLLOCATION METHODS

  • CHEN, DAOYONG;TIAN, HONGJIONG
    • Journal of applied mathematics & informatics
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    • v.33 no.5_6
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    • pp.657-670
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    • 2015
  • In this paper, we provide numerical analysis of so-called Legendre Gauss-Radau and Legendre-Gauss collocation methods for ordinary differential equations. After recasting these collocation methods as Runge-Kutta methods, we prove that the Legendre-Gauss collocation method is equivalent to the well-known Gauss method, while the Legendre-Gauss-Radau collocation method does not belong to the classes of Radau IA or Radau IIA methods in the Runge-Kutta literature. Making use of the well-established theory of Runge-Kutta methods, we study stability and accuracy of the Legendre-Gauss-Radau collocation method. Numerical experiments are conducted to confirm our theoretical results on the accuracy and numerical stability of the Legendre-Gauss-Radau collocation method, and compare Legendre-Gauss collocation method with the Gauss method.

Propagating and evanescent waves in a functionally graded nanoplate based on nonlocal theory

  • Cancan Liu;Jiangong Yu;Bo Zhang;Xiaoming Zhang;Xianhui Wang
    • Advances in nano research
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    • v.14 no.5
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    • pp.463-474
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    • 2023
  • The purpose of this paper is to present the analysis of propagating and evanescent waves in functionally graded (FG) nanoplates with the consideration of nonlocal effect. The analytical integration nonlocal stress expansion Legendre polynomial method is proposed to obtain complete dispersion curves in the complex domain. Unlike the traditional Legendre polynomial method that expanded the displacement, the presented polynomial method avoids employing the relationship between local stress and nonlocal stress to construct boundary conditions. In addition, the analytical expressions of numerical integrations are presented to improve the computational efficiency. The nonlocal effect, inhomogeneity of medium and their interactions on wave propagation are studied. It is found that the nonlocal effect and inhomogeneity of medium reduce the frequency bandwidth of complex evanescent Lamb waves, and make complex evanescent Lamb waves have a higher phase velocity at low attenuation. The occurrence of intersections of propagating Lamb wave in the nonlocal homogeneous plate needs to satisfy a smaller Poisson's ratio condition than that in the classical elastic theory. In addition, the inhomogeneity of medium enhances the nonlocal effect. The conclusions obtained can be applied to the design and dynamic response evaluation of composite nanostructures.

Relation between the Irreducible Polynomials that Generates the Same Binary Sequence Over Odd Characteristic Field

  • Ali, Md. Arshad;Kodera, Yuta;Park, Taehwan;Kusaka, Takuya;Nogmi, Yasuyuki;Kim, Howon
    • Journal of information and communication convergence engineering
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    • v.16 no.3
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    • pp.166-172
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    • 2018
  • A pseudo-random sequence generated by using a primitive polynomial, trace function, and Legendre symbol has been researched in our previous work. Our previous sequence has some interesting features such as period, autocorrelation, and linear complexity. A pseudo-random sequence widely used in cryptography. However, from the aspect of the practical use in cryptographic systems sequence needs to generate swiftly. Our previous sequence generated by utilizing a primitive polynomial, however, finding a primitive polynomial requires high calculating cost when the degree or the characteristic is large. It’s a shortcoming of our previous work. The main contribution of this work is to find some relation between the generated sequence and irreducible polynomials. The purpose of this relationship is to generate the same sequence without utilizing a primitive polynomial. From the experimental observation, it is found that there are (p - 1)/2 kinds of polynomial, which generates the same sequence. In addition, some of these polynomials are non-primitive polynomial. In this paper, these relationships between the sequence and the polynomials are shown by some examples. Furthermore, these relationships are proven theoretically also.

FRACTIONAL POLYNOMIAL METHOD FOR SOLVING FRACTIONAL ORDER POPULATION GROWTH MODEL

  • Krishnarajulu, Krishnaveni;Krithivasan, Kannan;Sevugan, Raja Balachandar
    • Communications of the Korean Mathematical Society
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    • v.31 no.4
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    • pp.869-878
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    • 2016
  • This paper presents an ecient fractional shifted Legendre polynomial method to solve the fractional Volterra's model for population growth model. The fractional derivatives are described based on the Caputo sense by using Riemann-Liouville fractional integral operator. The theoretical analysis, such as convergence analysis and error bound for the proposed technique has been demonstrated. In applications, the reliability of the technique is demonstrated by the error function based on the accuracy of the approximate solution. The numerical applications have provided the eciency of the method with dierent coecients of the population growth model. Finally, the obtained results reveal that the proposed technique is very convenient and quite accurate to such considered problems.

Guided viscoelastic wave in circumferential direction of orthotropic cylindrical curved plates

  • Yu, Jiangong;Ma, Zhijuan
    • Structural Engineering and Mechanics
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    • v.41 no.5
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    • pp.605-615
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    • 2012
  • In this paper, guided circumferential wave propagating in an orthotropic viscoelastic cylindrical curved plate subjected to traction-free conditions is investigated in the frame of the Kelvin-Voight viscoelastic theory. The obtained three wave equations are decoupled into two groups, Lamb-like wave and SH wave. They are separately solved by the Legendre polynomial series approach. The availability of the method is confirmed through the comparison with the published data of the SH wave for a viscoelastic flat plate. The dispersion curves and attenuation curves for the carbon fiber and prepreg cylindrical plates are illustrated and the viscous effect on dispersion curves is shown. The influences of the ratio of radius to thickness are analyzed.