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.

The asymptotic solution of differential equations with a turning point and singularities

1957 ◽  
Vol 53 (2) ◽  
pp. 382-398 ◽  
Author(s):  
R. C. Thorne
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Asymptotic Solution ◽  
Complex Variable ◽  
Turning Point ◽  
Asymptotic Solutions ◽  
Regular Singularity ◽  
Image Position ◽  
Interior Points

ABSTRACTAsymptotic solutions of the differential equationfor large positive values of u, are examined; z is a complex variable in a domain Dz in which P1(z) and z2q(z) are regular and p1(z) does not vanish. In this paper it is shown that there exist Airy-type expansions of the solutions of this equation which are valid uniformly with respect to z in a domain in which z = 0 and z = z0 are interior points. If Dz is unbounded and the equation has a regular singularity at infinity, Airy-type expansions exist which are valid at z = 0, z = z0 and z = δ. If p(z) = constant + O (│z│-1) as │ z │ → ∞ in Dz, similar expansions also exist. The results given here are new.

scholarly journals Liouville-Green expansions of exponential form, with an application to modified Bessel functions

2019 ◽  
Vol 150 (3) ◽  
pp. 1289-1311 ◽  
Author(s):  
T. M. Dunster
Keyword(s):  
Error Bounds ◽  
Asymptotic Expansions ◽  
Turning Point ◽  
Bessel Functions ◽  
Alternative Form ◽  
Modified Bessel Functions ◽  
Exponential Form ◽  
Positive Order ◽  
Second Order Differential Equations ◽  
Computable Error Bounds

AbstractLinear second order differential equations of the form d2w/dz2 − {u2f(u, z) + g(z)}w = 0 are studied, where |u| → ∞ and z lies in a complex bounded or unbounded domain D. If f(u, z) and g(z) are meromorphic in D, and f(u, z) has no zeros, the classical Liouville-Green/WKBJ approximation provides asymptotic expansions involving the exponential function. The coefficients in these expansions either multiply the exponential or in an alternative form appear in the exponent. The latter case has applications to the simplification of turning point expansions as well as certain quantum mechanics problems, and new computable error bounds are derived. It is shown how these bounds can be sharpened to provide realistic error estimates, and this is illustrated by an application to modified Bessel functions of complex argument and large positive order. Explicit computable error bounds are also derived for asymptotic expansions for particular solutions of the nonhomogeneous equations of the form d2w/dz2 − {u2f(z) + g(z)}w = p(z).

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.

Asymptotic expansions for the coefficient functions that arise in turning-point problems

1987 ◽  
Vol 410 (1838) ◽  
pp. 35-60 ◽  
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Turning Point ◽  
Bessel Functions ◽  
Holomorphic Functions ◽  
Standard Theory ◽  
Large Parameter ◽  
Uniform Asymptotic Expansion ◽  
All Solutions ◽  
Linear Differential

We study the uniform asymptotic expansion for a large parameter u of solutions of second-order linear differential equations in domains of the complex plane which include a turning point. The standard theory of Olver demonstrates the existence of three such expansions, corresponding to solutions of the form Ai ( u 2/3 ζ e 2/3απi ) Ʃ n s = 0 A s (ζ)/ u 2 s + u -2 d / d ζ Ai ( u 2/3 ζ e 2/3απi ) Ʃ n -1 s =0 B s (ζ)/u 2 s + ϵ ( α ) n ( u , ζ) for α = 0, 1, 2, with bounds on ϵ ( α ) n . We proceed differently, by showing that the set of all solutions of the differential equation is of the form Ai ( u 2/3 ζ) A ( u , ζ) + u -2 (d/dζ) Ai ( u 2/3 ζ) B ( u , ζ), where Ai denotes any situation of Airy's equation. The coefficent functions A ( u , ζ) and B ( u , ζ) are the focus of our attention : we show that for sufficiently large u they are holomorphic functions of ζ in a domain including the turning point, and as functions of u are described asymptotically by descending power series in u 2 , with explicit error bounds. We apply our theory to Bessel functions.

scholarly journals Effect of higher approximation of Krylov-Bogoliubov-Mitropolskii's solution and matched asymptotic solution of a differential system with slowly varying coefficients and damping near to a turning point

2004 ◽  
Vol 26 (3) ◽  
pp. 182-192
Author(s):  
Roy K. C. ◽  
Shamsul Alam M.
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Asymptotic Solution ◽  
Small Parameter ◽  
Turning Point ◽  
Order Differential Equation ◽  
Higher Order ◽  
Varying Coefficients ◽  
Order Solution ◽  
Reduced Frequency

Second approximate solution of a second order differential equation with slowly varying coefficients and damping is obtained by Krylov-Bogoliubov-Mitropolskii method. The method is illustrated by an example. The second or higher order approximate solution is able to give better results than first approximate solution when the reduced frequency is many times larger than the small parameter. On the contrary, higher order solution diverges faster than the lower order solution when the reduced frequency becomes small (i .e., near to a turning point). In these situations matched asymptotic solution is important. An example is made to illustrate the matter.

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 Periodic solutions for periodic second-order differential equations with variable potentials

2018 ◽  
Vol 24 (2) ◽  
pp. 127-137
Author(s):  
Jaume Llibre ◽  
Ammar Makhlouf
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Second Order Differential Equation ◽  
Existence Of Periodic Solutions

Abstract We provide sufficient conditions for the existence of periodic solutions of the second-order differential equation with variable potentials {-(px^{\prime})^{\prime}(t)-r(t)p(t)x^{\prime}(t)+q(t)x(t)=f(t,x(t))} , where the functions {p(t)>0} , {q(t)} , {r(t)} and {f(t,x)} are {\mathcal{C}^{2}} and T-periodic in the variable t.

scholarly journals Interval oscillation criteria for certain forced second-order differential equations

2012 ◽  
Vol 28 (2) ◽  
pp. 337-344
Author(s):  
ERCAN TUNC ◽  
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Second Order ◽  
Half Line ◽  
Oscillation Criteria ◽  
Second Order Differential Equations ◽  
Interval Oscillation

By using generalized Riccati transformations and an inequality due to Hardy et al., several new interval oscillation criteria are established for the nonlinear damped differential equation... The new interval oscillation criteria are different from most known ones in the sense they are based on the information only on a sequence of subintervals of [t0, ∞), rather than on the whole half-line. Our results improve and extend the known some results in the literature.

Sign in / Sign up

Export Citation Format

Share Document