Note on the generalized Mehler transform

1964 ◽  
Vol 60 (1) ◽  
pp. 57-59 ◽  
Author(s):  
J. S. Lowndes
Keyword(s):  
Boundary Value Problems ◽  
Inversion Formula ◽  
Integral Transform ◽  
Legendre Function ◽  
Boundary Value ◽  
Diffusion Equations ◽  
Legendre Functions ◽  
Image Position ◽  
And Diffusion ◽  
Associated Legendre Function

1. The integral transform. One result of recent studies of boundary-value problems of the wave and diffusion equations involving wedge- or conically-shaped boundaries has been the interest shown in integrals in which the variable of integration appears as the order of Bessel or Legendre functions. An integral of this type occurs as the inversion formula for the generalized Mehler transform which is defined bywhere ψ(μ, k) = Γ(½ − k + iμ)Γ(½ − k − iμ) and is the associated Legendre function of the first kind. Oberhettinger and Higgins (4) have given a table of transform pairs corresponding to the above transform.

scholarly journals On an inversion formula

1983 ◽  
Vol 24 (2) ◽  
pp. 149-159
Author(s):  
D. Naylor
Keyword(s):  
Wave Equation ◽  
Boundary Value Problems ◽  
Inversion Formula ◽  
Space Form ◽  
Integral Transform ◽  
Boundary Value ◽  
Image Position

In this paper the author considers the problem of finding a formula of inversion for the integral transform defined by the equationwhere a >0, k > 0 and r-1f(r) εL (a, ∞). This transform appeared in connection with an earlier investigation [4] in which an attempt was made to devise an integral transform that could be adapted to the solution of certain boundary value problems involving the space form of the wave equation and the condition of radiation:

scholarly journals A Class of Integral Transforms

1964 ◽  
Vol 14 (1) ◽  
pp. 33-40 ◽  
Author(s):  
Jet Wimp
Keyword(s):  
Heat Conduction ◽  
Boundary Value Problems ◽  
Inversion Formula ◽  
Heat Conduction Equation ◽  
Boundary Value ◽  
Integral Transforms ◽  
Conduction Equation ◽  
New Class ◽  
Special Cases ◽  
Image Position

In this paper we discuss a new class of integral transforms and their inversion formula. The kernel in the transform is a G-function (for a treatment of this function, see ((1), 5.3) and integration is performed with respect to the argument of that function. In the inversion formula, the kernel is likewise a G-function, but there integration is performed with respect to a parameter. Known special cases of our results are the Kontorovitch-Lebedev transform pair ((2), v. 2; (3))and the generalised Mehler transform pair (7)These transforms are used in solving certain boundary value problems of the wave or heat conduction equation involving wedge or conically-shaped boundaries, and are extensively tabulated in (6).

scholarly journals On an integral transform

1979 ◽  
Vol 20 (1) ◽  
pp. 1-14 ◽  
Author(s):  
D. Naylor
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Helmholtz Equation ◽  
Boundary Value Problems ◽  
Integral Transform ◽  
Boundary Value ◽  
Partial Differential ◽  
Image Position

In this paper the author continues the search for a suitable integral transform that can be applied to certain boundary value problems involving the Helmholtz equation and the condition of radiation. The transform in question must be capable of eliminating the r-dependence appearing in the partial differential equation

scholarly journals Spatially Restricted Integrals in Gradiometric Boundary Value Problems

2009 ◽  
Vol 44 (4) ◽  
pp. 131-148 ◽  
Author(s):  
M. Eshagh
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Legendre Functions ◽  
Gravity Gradiometry ◽  
Satellite Gravity Gradiometry ◽  
Satellite Gravity ◽  
Earth’S Gravity Field ◽  
Associated Legendre Functions ◽  
Tensor Spherical Harmonics ◽  
Slepian Functions

Spatially Restricted Integrals in Gradiometric Boundary Value ProblemsThe spherical Slepian functions can be used to localize the solutions of the gradiometric boundary value problems on a sphere. These functions involve spatially restricted integral products of scalar, vector and tensor spherical harmonics. This paper formulates these integrals in terms of combinations of the Gaunt coefficients and integrals of associated Legendre functions. The presented formulas for these integrals are useful in recovering the Earth's gravity field locally from the satellite gravity gradiometry data.

scholarly journals Positive solutions and eigenvalues of conjugate boundary value problems

1999 ◽  
Vol 42 (2) ◽  
pp. 349-374 ◽  
Author(s):  
Ravi P. Agarwal ◽  
Martin Bohner ◽  
Patricia J. Y. Wong
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solution ◽  
Positive Solutions ◽  
Lower Bounds ◽  
Boundary Value ◽  
Upper And Lower Bounds ◽  
Special Cases ◽  
Image Position

We consider the following boundary value problemwhere λ > 0 and 1 ≤ p ≤ n – 1 is fixed. The values of λ are characterized so that the boundary value problem has a positive solution. Further, for the case λ = 1 we offer criteria for the existence of two positive solutions of the boundary value problem. Upper and lower bounds for these positive solutions are also established for special cases. Several examples are included to dwell upon the importance of the results obtained.

scholarly journals On the Reduction of Ferrers' Associated Legendre Function

1929 ◽  
Vol 1 (4) ◽  
pp. 241-243
Author(s):  
Hrishikesh Sircar
Keyword(s):  
Finite Number ◽  
Infinite Series ◽  
Infinite Number ◽  
Legendre Function ◽  
Legendre Functions ◽  
Associated Legendre Functions ◽  
Reduced Order ◽  
Zonal Harmonics ◽  
Integral Degree ◽  
Associated Legendre Function

Introduction. In the present paper a formula will be obtained to express a Ferrers' Associated Legendre Function of any integral degree and order as a sum of a finite number of Associated Legendre Functions of an order reduced by an even number. When the order is reduced by unity, an infinite series of the functions of reduced order is required. Thus a Ferrers' function can be expressed as the sum of a finite or infinite number of zonal harmonics according as the order of the function is even or odd.

Iterative bounds for the stable solutions of convex non-linear boundary value problems

1977 ◽  
Vol 76 (2) ◽  
pp. 81-94 ◽  
Author(s):  
J. W. Mooney ◽  
G. F. Roach
Keyword(s):  
Boundary Value Problems ◽  
Differential Expression ◽  
Boundary Value ◽  
Linear Boundary ◽  
Non Linear ◽  
Unstable Solutions ◽  
Neumann Type ◽  
Thermal Combustion ◽  
Stable Solutions ◽  
Image Position

SynopsisWe consider a class of convex non-linear boundary value problems of the formwhere L is a linear, uniformly elliptic, self-adjoint differential expression, f is a given non-linear function, B is a boundary differential expression of either Dirichlet or Neumann type and D is a bounded open domain with boundary ∂D. Particular problems of this class arise in the process of thermal combustion [8].In this paper we show that stable solutions of this class can be bounded from below (above) by a monotonically increasing (decreasing) sequence of Newton (Picard) iterates. The possibility of using these schemes to construct unstable solutions is also considered.

On the convolution operators arising in the study of abstract initial boundary value problems

1996 ◽  
Vol 126 (3) ◽  
pp. 515-539 ◽  
Author(s):  
I. Alonso-Mallo ◽  
C. Palencia
Keyword(s):  
Banach Space ◽  
Boundary Value Problems ◽  
Infinitesimal Generator ◽  
Boundary Value ◽  
Continuous Mapping ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Convolution Operators ◽  
Image Position ◽  
Initial Boundary Value

We consider convolution operators arising in the study of abstract initial boundary value problems. These operators are of the formwhere {S(t)}t ≧0 is a C0-semigroup in a Banach space X,, with infinitesimal generator A0,: D(A0), ⊂ X, → X, and K(z): Y → X is a linxear, continuous mapping defined in another Banach space Y., We study the continuity of T between the spaces Lp([0, + ∞), Y), and Lq([0, + ∞), X), 1 ≦ p, q, ≦ + ∞. We give several examples of the applicability of the results to some familiar initial boundary value problems, including both parabolic and hyperbolic cases.

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