scholarly journals An Automatic Quadrature Schemes and Error Estimates for Semibounded Weighted Hadamard Type Hypersingular Integrals

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Sirajo Lawan Bichi ◽  
Z. K. Eshkuvatov ◽  
N. M. A. Nik Long
Keyword(s):  
Density Function ◽  
Error Estimates ◽  
Chebyshev Polynomials ◽  
Rational Functions ◽  
Approximate Solutions ◽  
Numerical Results ◽  
Hypersingular Integrals ◽  
The Third ◽  
Automatic Quadrature ◽  
Smooth Density

The approximate solutions for the semibounded Hadamard type hypersingular integrals (HSIs) for smooth density function are investigated. The automatic quadrature schemes (AQSs) are constructed by approximating the density function using the third and fourth kinds of Chebyshev polynomials. Error estimates for the semibounded solutions are obtained in the class ofh(t)∈CN,α[-1,1]. Numerical results for the obtained quadrature schemes revealed that the proposed methods are highly accurate when the density functionh (t)is any polynomial or rational functions. The results are in line with the theoretical findings.

scholarly journals Generating Functions for Orthogonal Polynomials of A2, C2 and G2

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 354 ◽  
Author(s):  
Tomasz Czyżycki ◽  
Jiří Hrivnák ◽  
Jiří Patera
Keyword(s):  
Orthogonal Polynomials ◽  
Lie Algebras ◽  
Chebyshev Polynomials ◽  
Generating Functions ◽  
Bell Polynomials ◽  
The Third ◽  
Weight Lattice ◽  
Hybrid Character ◽  
Polynomial Family

The generating functions of fourteen families of generalized Chebyshev polynomials related to rank two Lie algebras A 2 , C 2 and G 2 are explicitly developed. There exist two classes of the orthogonal polynomials corresponding to the symmetric and antisymmetric orbit functions of each rank two algebra. The Lie algebras G 2 and C 2 admit two additional polynomial collections arising from their hybrid character functions. The admissible shift of the weight lattice permits the construction of a further four shifted polynomial classes of C 2 and directly generalizes formation of the classical univariate Chebyshev polynomials of the third and fourth kinds. Explicit evaluating formulas for each polynomial family are derived and linked to the incomplete exponential Bell polynomials.

scholarly journals Explicit computations of Serre’s obstruction for genus-3 curves and application to optimal curves

2010 ◽  
Vol 13 ◽  
pp. 192-207 ◽  
Author(s):  
Christophe Ritzenthaler
Keyword(s):  
Modular Form ◽  
Elliptic Curve ◽  
Complex Multiplication ◽  
Numerical Results ◽  
Siegel Modular Form ◽  
Square Root ◽  
The Third

AbstractLetkbe a field of characteristic other than 2. There can be an obstruction to a principally polarized abelian threefold (A,a) overk, which is a Jacobian over$\bar {k}$, being a Jacobian over k; this can be computed in terms of the rationality of the square root of the value of a certain Siegel modular form. We show how to do this explicitly for principally polarized abelian threefolds which are the third power of an elliptic curve with complex multiplication. We use our numerical results to prove or refute the existence of some optimal curves of genus 3.

The Combined Closed Form-Perturbation Approach to the Analysis of Mistuned Bladed Disks

1993 ◽  
Vol 115 (4) ◽  
pp. 771-780 ◽  
Author(s):  
M. P. Mignolet ◽  
C.-C. Lin
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Approximate Solutions ◽  
Forced Response ◽  
Perturbation Approach ◽  
Perturbation Techniques ◽  
Step Method ◽  
Bladed Disk

A two-step method is presented for the determination of reliable approximations of the probability density function of the forced response of a randomly mistuned bladed disk. Under the assumption of linearity, an integral representation of the probability density function of the blade amplitude is first derived. Then, deterministic perturbation techniques are employed to produce simple approximations of this function. The adequacy of the method is demonstrated by comparing several approximate solutions with simulation results.

Legendre Collocation Method to Solve the Riccati Equations with Functional Arguments

2020 ◽  
Vol 17 (10) ◽  
pp. 2050011
Author(s):  
Şuayip Yüzbaşı ◽  
Gamze Yıldırım
Keyword(s):  
Approximate Solutions ◽  
Coefficient Matrix ◽  
Numerical Results ◽  
Matrix Form ◽  
Algebraic Equations ◽  
Mixed Condition ◽  
Collocation Points ◽  
Nonlinear Algebraic Equations ◽  
The Matrix ◽  
Legendre Collocation Method

In this study, a method for numerically solving Riccatti type differential equations with functional arguments under the mixed condition is presented. For the method, Legendre polynomials, the solution forms and the required expressions are written in the matrix form and the collocation points are defined. Then, by using the obtained matrix relations and the collocation points, the Riccati problem is reduced to a system of nonlinear algebraic equations. The condition in the problem is written in the matrix form and a new system of the nonlinear algebraic equations is found with the aid of the obtained matrix relation. This system is solved and thus the coefficient matrix is detected. This coefficient matrix is written in the solution form and hence approximate solution is obtained. In addition, by defining the residual function, an error problem is established and approximate solutions which give better numerical results are obtained. To demonstrate that the method is trustworthy and convenient, the presented method and error estimation technique are explicated by numerical examples. Consequently, the numerical results are shown more clearly with the aid of the tables and graphs and also the results are compared with the results of other methods.

scholarly journals Solving Non-Linear Fractional Variational Problems Using Jacobi Polynomials

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 224 ◽  
Author(s):  
Harendra Singh ◽  
Rajesh Pandey ◽  
Hari Srivastava
Keyword(s):  
Chebyshev Polynomials ◽  
Jacobi Polynomials ◽  
Ritz Method ◽  
Variational Problems ◽  
Algebraic Equations ◽  
The Third ◽  
Fractional Variational Problems ◽  
Nonlinear Algebraic Equations ◽  
Non Linear ◽  
Fourth Kind

The aim of this paper is to solve a class of non-linear fractional variational problems (NLFVPs) using the Ritz method and to perform a comparative study on the choice of different polynomials in the method. The Ritz method has allowed many researchers to solve different forms of fractional variational problems in recent years. The NLFVP is solved by applying the Ritz method using different orthogonal polynomials. Further, the approximate solution is obtained by solving a system of nonlinear algebraic equations. Error and convergence analysis of the discussed method is also provided. Numerical simulations are performed on illustrative examples to test the accuracy and applicability of the method. For comparison purposes, different polynomials such as 1) Shifted Legendre polynomials, 2) Shifted Chebyshev polynomials of the first kind, 3) Shifted Chebyshev polynomials of the third kind, 4) Shifted Chebyshev polynomials of the fourth kind, and 5) Gegenbauer polynomials are considered to perform the numerical investigations in the test examples. Further, the obtained results are presented in the form of tables and figures. The numerical results are also compared with some known methods from the literature.

Thermoplastic Instability for the Transient Contact Problem of Two Sliding Half-Planes

1986 ◽  
Vol 53 (3) ◽  
pp. 565-572 ◽  
Author(s):  
A. Azarkhin ◽  
J. R. Barber
Keyword(s):  
Steady State ◽  
Pressure Distribution ◽  
Half Plane ◽  
Initial Pressure ◽  
Approximate Solutions ◽  
Steady State Solution ◽  
Fourier Integral ◽  
Numerical Results ◽  
Initial Size ◽  
Transient Contact

We study the time dependent problem of a nonconducting half-plane sliding on the surface of a conductor with heat generation at the interface due to friction. The conducting half-plane is slightly rounded to give a Hertzian initial pressure distribution. Relationships are established for temperature and thermoelastic displacements due to a heat input of cosine type through the surface, and then these are used to obtain the solution in the form of a double Fourier integral. Numerical results show that, if the ratio of the initial size of the area of contact to that in the steady state is less than some critical value, the area of contact and the pressure distribution change smoothly toward the steady state solution. Otherwise the area of contact goes through bifurcation. The bifurcation accelerates the process. Numerical results are compared with previous approximate solutions.

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