scholarly journals Integral Representation of Polynomial Xn (x;a,b)

2021 ◽  
Vol 9 (8) ◽  
pp. 2452-2457
Author(s):  
P. G. Andhare
Keyword(s):  
Finite Difference ◽  
Integral Representation ◽  
Contour Integral ◽  
Double Integral ◽  
Difference Formula ◽  
Single Integral

Abstract: In the present paper we have obtained fine difference formula, contour integral representation, real integral representation, infinite single integral representation, finite single integral representation, finite double integral representation, finite double integral representation of polynomial ࢔ࢄ .(࢈ ,ࢇ ;࢞) Keywords: Finite difference, single integral representation, contour integral representation, simple generating relation, double integral representation

Diffraction by a perfectly conducting wedge in an anisotropic plasma

1965 ◽  
Vol 61 (3) ◽  
pp. 767-776 ◽  
Author(s):  
T. R. Faulkner
Keyword(s):  
Magnetic Field ◽  
Electromagnetic Wave ◽  
Difference Equation ◽  
Integral Representation ◽  
Surface Waves ◽  
Uniform Magnetic Field ◽  
Anisotropic Plasma ◽  
Contour Integral ◽  
Anisotropic Dielectric

SummaryThe problem considered is the diffraction of an electromagnetic wave by a perfectly conducting wedge embedded in a plasma on which a uniform magnetic field is impressed. The plasma is assumed to behave as an anisotropic dielectric and the problem is reduced, by employing a contour integral representation for the solution, to solving a difference equation. Surface waves are found to be excited on the wedge and expressions are given for their amplitudes.

scholarly journals THE NUMERICAL TREATMENT OF NONLINEAR FRACTAL–FRACTIONAL 2D EMDEN–FOWLER EQUATION UTILIZING 2D CHELYSHKOV POLYNOMIALS

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040042
Author(s):  
M. HOSSEININIA ◽  
M. H. HEYDARI ◽  
Z. AVAZZADEH
Keyword(s):  
Approximate Solution ◽  
Finite Difference ◽  
Test Problems ◽  
Operational Matrices ◽  
Algebraic Equations ◽  
Time Step ◽  
Discrete Method ◽  
Difference Formula ◽  
Fowler Equation ◽  
Chelyshkov Polynomials

This paper develops an effective semi-discrete method based on the 2D Chelyshkov polynomials (CPs) to provide an approximate solution of the fractal–fractional nonlinear Emden–Fowler equation. In this model, the fractal–fractional derivative in the concept of Atangana–Riemann–Liouville is considered. The proposed algorithm first discretizes the fractal–fractional differentiation by using the finite difference formula in the time direction. Then, it simplifies the original equation to the recurrent equations by expanding the unknown solution in terms of the 2D CPs and using the [Formula: see text]-weighted finite difference scheme. The differentiation operational matrices and the collocation method play an important role to obtaining a linear system of algebraic equations. Last, solving the obtained system provides an approximate solution in each time step. The validity of the formulated method is investigated through a sufficient number of test problems.

Proposing and Validation of a New Four-Point Finite-Difference Formula With Manipulator Application

2018 ◽  
Vol 14 (4) ◽  
pp. 1323-1333 ◽  
Author(s):  
Yang Shi ◽  
Binbin Qiu ◽  
Dechao Chen ◽  
Jian Li ◽  
Yunong Zhang
Keyword(s):  
Finite Difference ◽  
Difference Formula

THE EIGENVECTORS OF q-DEFORMED CREATION OPERATOR $a_q^+$ AND THEIR PROPERTIES

1995 ◽  
Vol 10 (08) ◽  
pp. 669-675
Author(s):  
GUOXIN JU ◽  
JINHE TAO ◽  
ZIXIN LIU ◽  
MIAN WANG
Keyword(s):  
Integral Representation ◽  
Creation Operator ◽  
Contour Integral ◽  
Root Of Unity

The eigenvectors of q-deformed creation operator [Formula: see text] are discussed for q being real or a root of unity by using the contour integral representation of δ function. The properties for the eigenvectors are also discussed. In the case of qp = 1, the eigenvectors may be normalizable.

h2- and h4-extrapolation in eigenvalue problems

1964 ◽  
Vol 60 (1) ◽  
pp. 67-74 ◽  
Author(s):  
S. C. R. Dennis
Keyword(s):  
Finite Difference ◽  
Eigenvalue Problems ◽  
Correction Term ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
General Level ◽  
End Points ◽  
Difference Formula ◽  
End Conditions ◽  
Image Position

Two recent papers have discussed eigenvalue problems relating to second-order, self-adjoint differential equations from the point of view of the deferred approach to the limit in the finite-difference treatment of the problem. In both cases the problem is made definite by considering the differential equationprimes denoting differentiation with respect to x, with two-point boundary conditionsand given at the ends of the interval (0, 1). The usual finite-difference approach is to divide the range (0, 1) into N equal strips of length h = 1/N, giving a set of N + 1 pivotal values φn as the analogue of a solution of (1), φn denoting the pivotal value at x = nh. In terms of central differences we then haveand retaining only second differences yields a finite-difference approximation φn = Un to (1), where the pivotal U-values satisfy the equationsdefined at all internal points, together with two equations holding at the end-points and approximately satisfying the end conditions (2). Here Λ is the corresponding approximation to the eigenvalue λ. A possible finite-difference treatment of the end conditions (2) would be to replace (1) at x = 0 by the central-difference formulaand use the corresponding result for the first derivative of φ, i.e.whereq(x) = λρ(x) – σ(x). Eliminating the external value φ–1 between these two and making use of (1) and (2) we obtain the equationwhere for convenience we write k0 = B0/A0. Similarly at x = 1 we obtainwithkN = B1/A1. If we neglect terms in h3 in these two they become what are usually taken to be the first approximation to the end conditions (2) to be used in conjunction with the set (4) (with the appropriate change φ = U, λ = Λ). This, however, results in a loss of accuracy at the end-points over the general level of accuracy of the set (4), which is O(h4), so there is some justification for retaining the terms in h3, e.g. if a difference correction method were being used they would subsequently be added as a correction term.

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