Convolution of Rayleigh functions with respect to the Bessel index

Варламов, В.В.

doi:10.1016/j.jmaa.2004.12.055

Cited by 7 publications

(6 citation statements)

References 12 publications

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“…v being the index of the Bessel function J v (x) whose zeros are λ v,n . Varlamov [26,27] succeeded in treating the Rayleigh functions σ l (v) as special functions by presenting some references for functions of the Rayleigh type as well as in their own right (see the references cited by Varlamov [26,27]).…”

Section: Wwwmn-journalcom 3 Convolutions Of the Rayleigh Functionsmentioning

confidence: 99%

Some applications of the Gamma and polygamma functions involving convolutions of the Rayleigh functions, multiple Euler sums and log‐sine integrals

Choi

Srivástava

2009

Mathematische Nachrichten

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The Gamma function and its nth logarithmic derivatives (that is, the polygamma or the psi-functions) have found many interesting and useful applications in a variety of subjects in pure and applied mathematics. Here we mainly apply these functions to treat convolutions of the Rayleigh functions by recalling a general identity expressing a certain class of series as psi-functions and to evaluate a class of log-sine integrals in an algorithmic way. We also evaluate some Euler sums and give much simpler psi-function expressions for some known parameterized multiple sums.

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Section: Wwwmn-journalcom 3 Convolutions Of the Rayleigh Functionsmentioning

confidence: 99%

Some applications of the Gamma and polygamma functions involving convolutions of the Rayleigh functions, multiple Euler sums and log‐sine integrals

Choi

Srivástava

2009

Mathematische Nachrichten

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“…Therefore it can be identified with a continuous function on R. Also, R(x) R(0) for x ∈ R, where R(0) 0.1429 (see [32]). …”

Section: Corollarymentioning

confidence: 99%

“…Below we shall assume only that the source term belongs to L 2 ( ) with respect to (r, ) and prove that the constructed mild solutions belong to H s ( ), s < 3 2 . To this end we shall employ a new special function introduced in [32], namely, a convolution of Rayleigh functions with respect to the Bessel index.…”

Section: Introductionmentioning

confidence: 99%

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Forced nonlinear oscillations of elastic membranes

Варламов¹,

Balogh²

2006

Nonlinear Analysis: Real World Applications

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“…It turns out that different types of convolutions are needed for constructing global-in-time solutions of semi-linear equations in circular domains [18,19]. These are convolutions with respect to the Bessel function index The study of the family of functions (1.4) was initiated in [19], where R 1 (m) was considered and its representation in terms of the ψ -function was derived, namely…”

Section: Introductionmentioning

confidence: 99%

Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains

Варламов¹

2007

Journal of Mathematical Analysis and Applications

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Rayleigh functions σ l (ν) are defined as series in inverse powers of the Bessel function zeros λ ν,n = 0, σ l (ν) = ∞ n=1 1 λ 2l ν, n , where l = 1, 2, . . . ; ν is the index of the Bessel function J ν (x) and n = 1, 2, . . . is the number of the zeros. Convolutions of Rayleigh functions with respect to the Bessel index, R l (m) = ∞ k=−∞ σ l |m − k| σ l |k| for l = 1, 2, . . . ; m = 0, ±1, ±2, . . . , are needed for constructing global-in-time solutions of semi-linear evolution equations in circular domains [V. Varlamov, On the spatially two-dimensional Boussinesq equation in a circular domain, Nonlinear Anal. 46 (2001) 699-725; V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413-424]. The study of this new family of special functions was initiated in [V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413-424], where the properties of R 1 (m) were investigated.In the present work a general representation of R l (m) in terms of σ l (ν) is deduced. On the basis of this a representation for the function R 2 (m) is obtained in terms of the ψ-function. An asymptotic expansion is computed for R 2 (m) as |m| → ∞. Such asymptotics are needed for establishing function spaces for solutions of semi-linear equations in bounded ✩ domains with periodicity conditions in one coordinate. As an example of application of R l (m) a forced Boussinesq equation u tt − 2bΔu t = −αΔ 2 u + Δu + βΔ u 2 + f with α, b = const > 0 and β = const ∈ R is considered in a unit disc with homogeneous boundary and initial data. Construction of its global-in-time solutions involves the use of the functions R 1 (m) and R 2 (m) which are responsible for the nonlinear smoothing effect.

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Convolution of Rayleigh functions with respect to the Bessel index

Cited by 7 publications

References 12 publications

Some applications of the Gamma and polygamma functions involving convolutions of the Rayleigh functions, multiple Euler sums and log‐sine integrals

Some applications of the Gamma and polygamma functions involving convolutions of the Rayleigh functions, multiple Euler sums and log‐sine integrals

Forced nonlinear oscillations of elastic membranes

Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains

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