Some recursion relations for solutions of a class of equations
This work presents some recursion relations for solutions of a class of singular partial differential equations. The equations consist of iterated differential operators each of which involves real parameters.
On the Cauchy problem for a class of iterated Fuchsian partial differential equations
Abstract In this paper, we consider the Cauchy problem with ramified data for a class of iterated Fuchsian partial differential equations. We give an explicit representation of the solution in terms of Gauss hypergeometric functions. Our results are illustrated through some examples.
The Stokes flow about a spindle
I. Statement of the problem for an axially symmetric body. The Stokes flow of a viscous, incompressible fluid about a body is defined by the assumption that inertial effects are negligible in comparison with those of viscosity, or, more precisely, that the Reynolds number of the flow is small. In the case in which the flow is two-dimensional or has radial symmetry, the introduction of a stream function and the Stokes assumption together with the no-slip boundary condition on the body and an appropriate assumption at infinity, yields a boundary value problem which has been solved in a number of instances. See Dryden, Murnaghan and Bateman [1], pp. 295-312 and the references of [2]. In the axi-symmetric case a body (or configuration of bodies) having an axis of symmetry is immersed in a flow which has a uniform velocity parallel to the axis of symmetry. It is then reasonable to suppose that the resulting flow pattern is identical in all planes through the axis of symmetry. If we introduce cylindrical coordinates (x, r, 6) into the flow space, where x(— <*> < x < oo)is measured along the axis of symmetry, r(0 < r < oo) normal to this axis, and 6(0 < d < 2x) is measured with respect to an arbitrary plane through the axis of symmetry, then 0 plays no further role in our analysis because of the axial symmetry. Now let the velocity of the fluid be u(x, r) = (ux{x, r), uT(x, r)) and introduce a stream function ^(x, r) by the equations