Expansions for a particular class of exponential-logarithmic integrals

1942 ◽  
Vol 38 (1) ◽  
pp. 34-39
Author(s):  
William J. C. Orr
Keyword(s):  
Force Field ◽  
General Type ◽  
Special Method ◽  
Dynamic Data ◽  
Generalized Force ◽  
Image Position

In the course of the development of a special method of interpreting thermo-dynamic data in terms of a generalized force field for the mutual interactions of the molecules concerned, expansions were required for integrals of the following general type, which apparently have not hitherto been discussed. We require explicit expansions of F(α, s, n), in terms of the parameter α, for the integral values 0, 1, 2, 3, … of s and n, where

A note on Milne-Thomson's general solution of Navier–Stokes equations

1965 ◽  
Vol 61 (4) ◽  
pp. 915-916
Author(s):  
G. D. Nigam
Keyword(s):  
General Solution ◽  
Force Field ◽  
Velocity Vector ◽  
Stokes Equations ◽  
Scalar Potential ◽  
Compressible Fluids ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
External Force Field ◽  
Image Position

The Navier–Stokes equations governing the motion of viscous compressible fluids arewhere ρ is the density, q is the velocity vector, V is the scalar potential of the external force field, I is the idemfactor and Φ is the stress-tensorMilne-Thomson ((l)) has given a general solution of (1) and (2) in the form

scholarly journals XXVI.—On a Group of Linear Differential Equations of the 2nd Order, including Professor Chrystal's Seiche-equations

1906 ◽  
Vol 41 (3) ◽  
pp. 651-676 ◽  
Author(s):  
J. Halm
Keyword(s):  
Differential Equations ◽  
Stokes Equation ◽  
Mathematical Theory ◽  
General Type ◽  
The Other ◽  
Special Cases ◽  
Linear Differential ◽  
Image Position ◽  
Special Case ◽  
The Stokes Equation

It is readily seen that the two differential equationswhich play an important rôle in Professor Chrystal's mathematical theory of the Seiches, are special cases of the more general typeWith regard to the first, the Seiche-equation, this becomes at once apparent by writing a= − ½. Equation (2), on the other hand, which we may briefly call the Stokes equation [see Professor Chrystal's paper on “Some further Results in the Mathematical Theory of Seiches,” Proc. Roy. Soc. Edin., vol. xxv.] will be recognised as a special case (a = + 1) of the equationwhich is transformed into (3) by the substitution .

Equations for stochastic path integrals

1961 ◽  
Vol 57 (3) ◽  
pp. 568-573 ◽  
Author(s):  
M. S. Bartlett
Keyword(s):  
Gaussian Process ◽  
Autocorrelation Function ◽  
General Type ◽  
Path Integrals ◽  
Markovian Process ◽  
Diffusion Type ◽  
Riemann Sum ◽  
Distributional Properties ◽  
Image Position

Apart from the study of the integralwhere {X(u)} is a stationary Gaussian process with autocorrelation function ρ(t), by Kac and Siegert(1), most stochastic functionals of the general typehave been considered for {X(u)} either additive or Markovian (see, for example, (2), (3)), and in the Markovian case only for diffusion-type processes (Darling and Siegert (4)). More general approaches exist (e.g. (5), (6)), but seem less concerned with the investigation of specific problems. Some preliminary remarks here are therefore aimed at examining the structure of integrals of type (2), or such further extensions of the formal Riemann sum typethat would be expected to have well-behaved distributional properties for {X(t)} Markovian, and associated equations for studying these properties. As an example, the sumsay, is considered (i) for the normal linear Markovian process, (ii) for simple birth-and-death and emigration—immigration processes.

scholarly journals On some results concerning integral equations

1913 ◽  
Vol 32 ◽  
pp. 19-29
Author(s):  
Pierre Humbert
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Large Class ◽  
Integral Equations ◽  
Singular Integral ◽  
General Type ◽  
Singular Integral Equations ◽  
General Function ◽  
Linear Differential ◽  
Image Position

It is proposed in this paper to show now the well-known Laplace's transformation,which is of great help in finding the solution of linear differential equations, gives also interesting results concerning the theory of integral equations. In §2 we shall study its application to certain differential equations, and find a large class of equations which remain unchanged by this transformation. Then, (§3), taking instead of eazt, a more general function of the product zt, we shall find a solution for some homogeneous integral equations ; in § 4 we shall describe a method of solving a very general type of integral equation of the first kind, namely,a further extension to integral equations with the kernel ef(z)f(t) the object of §5. Then, studying an extension of Euler's transformation, we shall (§ 6) consider equations such aswhich will prove to be singular; and finally, in §7, we shall give other examples of singular integral equations.

Oscillation of Elliptic Equations in General Domains

1975 ◽  
Vol 27 (6) ◽  
pp. 1239-1245 ◽  
Author(s):  
E. S. Noussair
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Differential Operator ◽  
Euclidean Space ◽  
Unbounded Domain ◽  
Elliptic Equations ◽  
General Type ◽  
Elliptic Partial Differential Equation ◽  
Dimensional Euclidean Space ◽  
Image Position

Oscillation criteria will be obtained for the linear elliptic partial differential equationin an unbounded domain G of general type in n-dimensional Euclidean space En. The differential operator D is defined as usual by where each α (i), i = 1, … , n, is a non-negative integer.

Sign in / Sign up

Export Citation Format

Share Document