scholarly journals Differential equation and inequalities of the generalized k-Bessel functions

2018 ◽  
Vol 2018 (1) ◽  
Author(s):  
Saiful R. Mondal ◽  
Mohamed S. Akel
Keyword(s):  
Differential Equation ◽  
Bessel Functions

On a singular eigenvalue problem

1968 ◽  
Vol 64 (2) ◽  
pp. 439-446 ◽  
Author(s):  
D. Naylor ◽  
S. C. R. Dennis
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Bessel Functions ◽  
Asymptotic Condition ◽  
Real Constant ◽  
Limit Circle ◽  
Expansion In Eigenfunctions ◽  
Image Position

Sears and Titchmarsh (1) have formulated an expansion in eigenfunctions which requires a knowledge of the s-zeros of the equationHere ka > 0 is supposed given and β is a real constant such that 0 ≤ β < π. The above equation is encountered when one seeks the eigenfunctions of the differential equationon the interval 0 < α ≤ r < ∞ subject to the condition of vanishing at r = α. Solutions of (2) are the Bessel functions J±is(kr) and every solution w of (2) is such that r−½w(r) belongs to L2 (α, ∞). Since the problem is of the limit circle type at infinity it is necessary to prescribe a suitable asymptotic condition there to make the eigenfunctions determinate. In the present instance this condition is

scholarly journals Spherical curves and quadratic relationships for special functions

1985 ◽  
Vol 27 (1) ◽  
pp. 111-124
Author(s):  
Even Mehlum ◽  
Jet Wimp
Keyword(s):  
Differential Equation ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Special Functions ◽  
Position Vector ◽  
Third Order ◽  
Generalized Hypergeometric Functions ◽  
Transcendental Functions ◽  
Horn Functions ◽  
Vector Differential Equation

AbstractWe show that the position vector of any 3-space curve lying on a sphere satisfies a third-order linear (vector) differential equation whose coefficients involve a single arbitrary function A(s). By making various identifications of A(s), we are led to nonlinear identities for a number of higher transcendental functions: Bessel functions, Horn functions, generalized hypergeometric functions, etc. These can be considered natural geometrical generalizations of sin2t + cos2t = 1. We conclude with some applications to the theory of splines.

A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order

1951 ◽  
Vol 47 (4) ◽  
pp. 699-712 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Differential Equation ◽  
Bessel Functions ◽  
Numerical Technique ◽  
Third Order ◽  
New Approach ◽  
The Third ◽  
Zeros Of Bessel Functions ◽  
Zeros Of Functions ◽  
Linear Differential ◽  
Image Position

In a recent paper (1) I described a method for the numerical evaluation of zeros of the Bessel functions Jn(z) and Yn(z), which was independent of computed values of these functions. The essence of the method was to regard the zeros ρ of the cylinder functionas a function of t and to solve numerically the third-order non-linear differential equation satisfied by ρ(t). It has since been successfully used to compute ten-decimal values of jn, s, yn, s, the sth positive zeros* of Jn(z), Yn(z) respectively, in the ranges n = 10 (1) 20, s = 1(1) 20. During the course of this work it was realized that the least satisfactory feature of the new method was the time taken for the evaluation of the first three or four zeros in comparison with that required for the higher zeros; the direct numerical technique for integrating the differential equation satisfied by ρ(t) becomes unwieldy for the small zeros and a different technique (described in the same paper) must be employed. It was also apparent that no mere refinement of the existing methods would remove this defect and that a new approach was required if it was to be eliminated. The outcome has been the development of the method to which the first part (§§ 2–6) of this paper is devoted.

The asymptotic solution of linear second order differential equations in a domain containing a turning point and a regular singularity

1957 ◽  
Vol 249 (971) ◽  
pp. 585-596 ◽  
Keyword(s):  
Differential Equation ◽  
Asymptotic Solution ◽  
Turning Point ◽  
Bessel Functions ◽  
Legendre Functions ◽  
Positive Real ◽  
Regular Singularity ◽  
Previous Theory ◽  
Second Order Differential Equations ◽  
Regular Functions

Asymptotic solutions of the differential equation d1 2wjdz2 = {u2z~2(z0—z) pi(z) +z ~2ql(z)} w, for large positive values of u are examined; P 1 (z) AND Q 1 (Z) are regular functions of the complex variable z in a domain in which P 1 (z) does not vanish. The point z = 0 is a regular singularity of the equation and a branch-cut extending from z = 0 is taken through the point Z=Z O which is assumed to lie on the positive real z axis. Asymptotic expansions for the solutions of the equation, valid uniformly with respect to z in domains including Z=0, Z 0+-iO are derived in terms of Bessel functions of large order. Expansions given by previous theory are not valid at all these points. The theory can be applied to the Legendre functions.

Accurate analog computer generation of Bessel functions for large ranges

SIMULATION ◽  
1967 ◽  
Vol 9 (5) ◽  
pp. 249-254 ◽  
Author(s):  
Arthur Hausner
Keyword(s):  
Differential Equation ◽  
Chebyshev Polynomials ◽  
Bessel Functions ◽  
Analog Computer ◽  
Integration Process ◽  
Computer Generation ◽  
Excellent Accuracy

The generation of Bessel functions Jn(x) is required when simulating many systems on an analog computer. Because of an indeterminate expression at the origin, however, they have not been amenable to an accurate generation for the range 0 ≤ x ≤ Xmax for large Xmax. This paper extends an idea of Van Remoortere to use an approximation for 0 ≤ x ≤ X 1 and solve a differential equation for X1 ≤ x ≤ Xmax, combining both phases by switching. The technique described here uses Chebyshev polynomials to minimize equipment in the approximation phase and generates the function 1/x by an integration process in the differential-equation phase to extend the range. The examples given for J0 and J9 indicate excellent accuracy for at least 0 ≤ x ≤ 100.

scholarly journals Application of the Bessel function to compute the air pollutant with the stratification of the atmospheric

2015 ◽  
Vol 18 (2) ◽  
pp. 14-20
Author(s):  
Dung Anh Tran ◽  
Hang Thi Chu ◽  
Long Ta Bui
Keyword(s):  
Differential Equation ◽  
Bessel Function ◽  
Bessel Functions ◽  
Analytic Solutions ◽  
Air Pollutant ◽  
Spherical Coordinates ◽  
Advection Diffusion Equation ◽  
Advection Diffusion ◽  
Bessel Differential Equation ◽  
Separation Of Variable

The Bessel differential equation with the Bessel function of solution has been applied. Bessel functions are the canonical solutions of Bessel's differential equation. Bessel's equation arises when finding separable solutions to Laplace's equation in cylindrical or spherical coordinates. Bessel functions are important for many problems of advection–diffusion progress and wave propagation. In this paper, authors present the analytic solutions of the atmospheric advection-diffusion equation with the stratification of the boundary condition. The solution has been found by applied the separation of variable method and Bessel’s equation.

scholarly journals Generating Functions for Bessel Functions

1959 ◽  
Vol 11 ◽  
pp. 148-155 ◽  
Author(s):  
Louis Weisner
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Generating Function ◽  
Lie Group ◽  
Generating Functions ◽  
Bessel Functions ◽  
Partial Differential ◽  
Systematic Determination ◽  
Image Position

On replacing the parameter n in Bessel's differential equation1.1by the operator y(∂/∂y), the partial differential equation Lu = 0 is constructed, where1.2This operator annuls u(x, y) = v(x)yn if, and only if, v(x) satisfies (1.1) and hence is a cylindrical function of order n. Thus every generating function of a set of cylindrical functions is a solution of Lu = 0.It is shown in § 2 that the partial differential equation Lu = 0 is invariant under a three-parameter Lie group. This group is then applied to the systematic determination of generating functions for Bessel functions, following the methods employed in two previous papers (4; 5).

scholarly journals Use of strain-energy method for calculating rectangular beams in cases of lateral buckling

2019 ◽  
Vol 91 ◽  
pp. 02035
Author(s):  
Aleksandr Ishchenko ◽  
Ivan Zotov
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Iterative Method ◽  
Cross Section ◽  
Energy Method ◽  
Strain Energy ◽  
Bessel Functions ◽  
Critical Force ◽  
Lateral Buckling ◽  
Strain Energy Method

The paper deals with the lateral buckling problem of a freely supported wooden strip with a constant narrow cross section, loaded with a local force in the middle of the span. A differential equation is given for cases when the force is applied out of the gravity section center. Strain-energy method was used in the study of beam lateral buckling. In the case when the load is applied in the center of gravity, problem comes down to a generalized characteristic equation. The correlation between the magnitude of the critical force and the application point of the load was obtained. The linear approximating function was identified for the indicated dependence. The obtained results are compared with an analytical solution using the Bessel functions and a numerical iterative method.

An asymptotic expansion for Bessel functions derived with respect to their order

1961 ◽  
Vol 57 (2) ◽  
pp. 284-287
Author(s):  
A. C. Sim
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Bessel Functions ◽  
Integral Forms ◽  
Previous Discussion

Formulae for Bessel derivatives with respect to order have generally been obtained by differentiating integral forms of the Bessel functions. In this note an asymptotic expansion will be derived from the differential equation satisfied by the functions, and it will be obtained in a form which terminates when the order is half an odd integer. Previous discussion of these functions has been restricted to the order one half. (See, for example, Ansell and Fisher (1) and Oberhettinger(2).)

A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations

1950 ◽  
Vol 46 (4) ◽  
pp. 570-580 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Bessel Functions ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Zeros Of Bessel Functions ◽  
Zeros Of Solutions ◽  
Linear Differential ◽  
Homogeneous Linear ◽  
Homogeneous Linear Differential Equation

The zeros of solutions of the general second-order homogeneous linear differential equation are shown to satisfy a certain non-linear differential equation. The method here proposed for their determination is the numerical integration of this differential equation. It has the advantage of being independent of tabulated values of the actual functions whose zeros are being sought. As an example of the application of the method the Bessel functions Jn(x), Yn(x) are considered. Numerical techniques for integrating the differential equation for the zeros of these Bessel functions are described in detail.

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