Axisymmetric Helmholtz Research Articles

Autotransforms for Some Differential Equations and Their Physical Applications

General constructions of differential equations with autotransforms are presented. Differential operators connecting a linear case of the Grad – Shafranov equation and the axisymmetric Helmholtz equation are found. Infinite families of exact solutions to both equations are derived.

New Exact Axisymmetric Solutions to the Navier–Stokes Equations

Abstract Infinite-dimensional space of axisymmetric exact solutions to the Navier–Stokes equations with time-dependent viscosity ν ( t ) $\nu(t)$ is constructed. Inner transformations of the exact solutions are defined that produce an infinite sequence of new solutions from each known one. The solutions are analytic in the whole space ℝ3 and are described by elementary functions. The bifurcations of the instantaneous (for t = t 0 $t={t_{0}}$ ) phase portraits of the viscous fluid flows are studied for the new exact solutions. Backlund transforms between the axisymmetric Helmholtz equation and a linear case of the Grad–Shafranov equation are derived.

Fundamental solution of a multidimensional axisymmetric equation

For the axisymmetric Helmholtz equation it is shown that fundamental solutions constructed by the Gellerstedt method and by means of a transmutation operator are coinciding.

Fundamental solution of multidimensional axisymmetric Helmholtz equation

For axisymmetric Helmholtz equationfundamental solutions have been constructed. Those are written explicitly via confluent Horn functions. The obtained fundamental solutions have been proved to possess a power singularity .

The plane waves method for axisymmetric Helmholtz problems

The plane waves method is employed for the solution of Dirichlet and Neumann boundary value problems for the homogeneous Helmholtz equation in two- and three-dimensional domains possessing radial symmetry. The appropriate selection of collocation points and unitary direction vectors in the method leads to circulant and block circulant coefficient matrices in two and three dimensions, respectively. We propose efficient matrix decomposition algorithms which make use of fast Fourier transforms for the solution of the systems resulting from such a discretization. In conjunction with the method of particular solutions, the method is extended to the solution of inhomogeneous axisymmetric Helmholtz problems. Several numerical examples are presented.

The Method of Fundamental Solutions for Solving Exterior Axisymmetric Helmholtz Problems with High Wave-Number

In this paper, we investigate the method of fundamental solutions (MFS) for solving exterior Helmholtz problems with high wave-number in axisymmetric domains. Since the coefficient matrix in the linear system resulting from the MFS approximation has a block circulant structure, it can be solved by the matrix decomposition algorithm and fast Fourier transform for the fast computation of large-scale problems and meanwhile saving computer memory space. Several numerical examples are provided to demonstrate its applicability and efficacy in two and three dimensional domains.

The Method of Fundamental Solutions for Solving Exterior Axisymmetric Helmholtz Problems with High Wave-Number

AbstractIn this paper, we investigate the method of fundamental solutions (MFS) for solving exterior Helmholtz problems with high wave-number in axisymmetric domains. Since the coefficient matrix in the linear system resulting from the MFS approximation has a block circulant structure, it can be solved by the matrix decomposition algorithm and fast Fourier transform for the fast computation of large-scale problems and meanwhile saving computer memory space. Several numerical examples are provided to demonstrate its applicability and efficacy in two and three dimensional domains.

On the modeling of narrow gaps using the standard boundary element method.

Numerical methods based on the Helmholtz integral equation are well suited for solving acoustic scattering and diffraction problems at relatively low frequencies. However, it is well known that the standard method becomes degenerate if the objects that disturb the sound field are very thin. This paper makes use of a standard axisymmetric Helmholtz integral equation formulation and its boundary element method (BEM) implementation to study the behavior of the method on two test cases: a thin rigid disk of variable thickness and two rigid cylinders separated by a gap of variable width. Both problems give rise to the same kind of degeneracy in the method, and modified formulations have been proposed to overcome this difficulty. However, such techniques are better suited for the so-called thin-body problem than for the reciprocal narrow-gap problem, and only the first is usually dealt with in the literature. A simple integration technique that can extend the range of thicknesses/widths tractable by the otherwise unmodified standard formulation is presented and tested. This technique is valid for both cases. The modeling of acoustic transducers like sound intensity probes and condenser microphones has motivated this work, although the proposed technique has a wider range of applications.

Nonlocal Boundary Value Problems in a Vertical Half-Strip for a Generalized Axisymmetric Helmholtz Equation

Nonlocal Boundary Value Problems in a Vertical Half-Strip for a Generalized Axisymmetric Helmholtz Equation

Representation of elliptic by parabolic partial differential equations with an application to axially symmetric sound propagation

The ‘wide-angle’ approximation by Gilbert and White in the parabolic equation method of solution of the Helmholtz equation is rederived by a two-stage iterative process. The new approach is of more general applicability than the original, and is used to deduce a number of possible refinements for the solution of the axi-symmetric Helmholtz equation.

Boundary element analysis of axisymmetric resonant cavities

Two numerical techniques are applied to boundary element analysis of axisymmetric modes of a cylindrical resonant cavity. One of the techniques is based on a novel expression for the fundamental solution of the axisymmetric Helmholtz equation with double exponential formulas. The other is an integral equation method in which the fundamental solution of the Laplace equation is used by considering the axisymmetric Helmholtz equation as the axisymmetric Laplace equation with an inhomogeneous term. Axisymmetric modes of a cyclindrical resonant cavity are analyzed by these techniques, and the resonant wave numbers are obtained with high accuracy. These results show that the axisymmetric resonant cavities can be analyzed by the boundary element method. >

Triangular finite elements for the generalized Bessel equation of order m

AbstractA new family of triangular finite elements is described, useful for solving the axisymmetric vector Helmholtz equation, and a variety of scalar Helmholtz equation problems which lead to generalized Bessel equations of some order m. This family is similar in principle to the scalar axisymmetric Helmholtz elements derived earlier, but requires both reformulation of its describing equations and corresponding new universal element matrices, for successful computational implementation. The necessary formulation is given in this paper. Matrix elements to the sixth‐order inclusive have been calculated and extensively tested computationally.