Higher-order Parabolic Equation Research Articles

Solutions in Besov spaces of a class of abstract parabolic equations of higher order in time

Solutions in Besov spaces of a class of abstract parabolic equations of higher order in time

Existence of solutions for degenerate quasilinear parabolic equations of higher order

In the paper, existence results for degenerate parabolic boundary value problems of higher order are proved. The weak solution is sought in a suitable weighted Sobolev space by using the generalized degree theory.

Analysis and model/data comparison of broadband acoustic propagation at the Atlantic Generating Station (AGS) site

An investigation of broadband acoustic propagation in a shallow-water environment is performed by comparing experimental acoustic data [Badiey et al., J. Acoust. Soc. Am. 96, 3593–3604 (1994)] with results from a frequency-domain, high-order parabolic equation technique. The principal quantity examined, transmission loss versus frequency, is determined for fixed source/receiver positions over short ranges (≈0.5 km) corresponding to experimental configurations. High-resolution geoacoustic data from extensive coring of the site, along with CTD and detailed bathymetric data, are used to construct accurate range-dependent input for PE simulations and range-varying wave number spectra calculations. Generally very good agreement between the PE results and data are observed. Range-dependent mechanisms are seen to have a strong azimuthal dependence in both TL and wave number results for this site. Shear losses are considered by exploiting equivalent fluid relationships. The effects of sediment dispersion through frequency dependent sound speed are also examined.

On degenerate quasilinear parabolic equations of higher order

In the paper existence results for degenerate quasilinear parabolic initial boundary value problems of higher order are proved. The weak solution is sought in a suitable weighted Sobolev space using the generalized degree theory.

On the behaviour near the boundary of the solutions of nonlinear degenerate parabolic equations of higher order

On the behaviour near the boundary of the solutions of nonlinear degenerate parabolic equations of higher order

Composition of stochastic processes governed by higher-order parabolic and hyperbolic equations

We consider compositions of stochastic processes that are governed by higherorder partial differential equations. The processes studied include compositions of Brownian motions, stable-like processes with Brownian time, Brownian motion whose time is an integrated telegraph process, and an iterated integrated telegraph process. The governing higher-order equations that are obtained are shown to be either of the usual parabolic type or, as in the last example, of hyperbolic type.

On Boundary Value Problems for Parabolic Equations of Higher Order in Time

On Boundary Value Problems for Parabolic Equations of Higher Order in Time

Performance analysis of the holographic array processing algorithm

This paper provides detailed analysis and performance evaluation of the holographic array processing (HAP) algorithm. The HAP is a source localization method that is based on medium calibration. Conventional array processing algorithms, such as matched-field processing (MFP), require precise knowledge of the medium between the source and the receiving array, but the HAP method relaxes this stiff requirement. It calibrates the integrated effect of a great portion of the medium, and geoacoustic parameter estimation is needed only for a small portion of the ocean between the unknown source and the virtual array. The virtual array is constructed by moving a reference source to incremental depths of the water column near the target. Theoretical analysis is provided using the WKB approximation for a range dependent ocean. The numerical simulation is performed using a high-order parabolic equation (PE) code for a range-dependent analytical sound-speed profile (SSP), and measured sound-speed data from the North Pacific ocean. The results of the analysis and simulation show the possibility of localizing a source at large distance with great accuracy.

Split-step/finite-difference and split-step/Lanczos algorithms for solving alternative higher-order parabolic equations

Recently Porter and Jensen [J. Acoust. Soc. Am. 94, 1510–1516 (1993)] reported on a seemingly benign propagation problem in which two wide-angle split-step parabolic equations (PEs) performed poorly compared to other PEs. That is, for a source and receiver located within a leaky surface duct, anomalously high transmission losses were predicted for ranges at which the leakage energy is refracted back into the duct. Moreover, both equations displayed greater sensitivity to the choice of reference wave number than the standard PE. In this paper, a new propagation operator is formulated that retains the computational efficiency of the split-step algorithm but is considerably more accurate. Two rapid numerical solution algorithms valid in both two and three dimensions are then introduced and applied to the leaky surface duct. To implement the simplest second-order method a tridiagonal finite-difference system of equations is solved at each range step in addition to the usual split-step Fourier computations. Higher-order procedures instead involve applying the Lanczos algorithm solely to the higher-order terms in the wide-angle expansion, thereby circumventing the convergence difficulties associated with the direct Lanczos evaluation of the Helmholtz propagator [Hermansson et al., IEEE J. Light. Technol. 10, 772–776 (1992)].

Qualitative properties of solutions of the Neumann problem for a higher-order quasilinear parabolic equation

The property of localization of perturbations is proved for a solution of an initial boundary-value Neumann problem in a regionD=Ωx, t>0, where Ω is a region in Rnwith a noncompact boundary.

On the H�lder property for functions from the classB q,s

A class of functionsBq,s containing generalized solutions of some higher-order quasilinear parabolic equations is defined. We prove thatBq,s is imbedded in the space of Holder functions.

The split-step Padé solution

The parabolic equation (PE) method has been the most efficient range-dependent propagation model in ocean acoustics ever since the split-step Fourier solution was developed [F. D. Tappert, ‘‘Numerical solutions of a canonical nonlinear dispersive wave equation,’’ SIAM Rev. 16, 140 (1974)]. Since the asymptotic accuracy of the split-step Fourier solution is restricted to leading order (a serious limitation for some problems), the PE method was extended to higher-order asymptotic accuracy using Padé approximations. Until recently, the higher-order PE has been solved using relatively inefficient finite-difference solutions, and the split-step Fourier solution has remained in widespread use despite the accuracy improvements. To improve efficiency without sacrificing accuracy, we apply Padé approximations to achieve higher-order accuracy in both the numerics and the asymptotics. The split-step Padé solution, which has been implemented for both fluid and elastic media, achieves both higher-order asymptotic accuracy and the efficiency of the split-step Fourier solution. This finite-difference solution is one to three orders of magnitude faster than other finite-difference solutions.

Estimates of rate of propagation of perturbations in quasilinear divergent degenerate parabolic equations of high order

A method is given for obtaining integral growth lemmas for solutions of boundary problems for a large class of quasilinear evolution equations. As a possible application a sharp estimate is obtained of the dependence of the support of a solution of a mixed problem and a Cauchy problem for a quasilinear divergent parabolic equation on time.

Uniqueness of solutions of mixed problems and a Cauchy problem for parabolic equations of high order with unbounded coefficients

New uniqueness classes of generalized solutions (generalized Tacklind classes) of initial-boundary-value problems are found for linear and quasilinear divergent parabolic equations of high order with coefficients increasing at infinity.

Stable and accurate interface conditions for wide-angle parabolic wave equations

The proper formulation of mathematical and numerical conditions at a fluid/elastic interface is studied for the standard and higher-order parabolic wave equations. For wide-range integrations in shallow oceans, the amplitude and phase error of the bottom reflection coefficient and the error growth of the scheme must be as small as possible. Schemes with minimal local error and error growth are given, both for the usual parabolic wave equation, the higher-order variants based on Padé approximations, and the shallow-water variants suggested by Collins [J. Acoust. Soc. Am. 86, 1097–1102 (1989)].

Benchmark calculations for higher-order parabolic equations

Benchmark solutions generated with parabolic equation (PE) models are presented for range-dependent underwater acoustic propagation problems involving both penetrable and perfectly reflecting ocean bottoms. The solution of the wide-angle PE of Claerbout [J. F. Claerbout, Fundamentals of Geophysical Data Processing (McGraw-Hill, New York, 1976), pp. 206–207] agrees with the outgoing coupled-mode solution for the problems involving penetrable bottoms. The solution of the higher-order PE of Bamberger et al. [Bamberger et al., ‘‘Higher Order Paraxial Wave Equation Approximations in Heterogeneous Media,’’ SIAM J. Appl. Math. 48, 129–154 (1988)], which is a generalization of Claerbout’s PE, agrees with the outgoing coupled-mode solution for problems involving large variations in sound speed and propagation nearly orthogonal to the preferred direction. The computer code FEPE was used to generate the benchmark solutions and was found to run several times faster than the IFDPE computer code due to a tridiagonal system solver in FEPE that is optimized for range-dependent problems.

A note on strongly nonlinear parabolic equations of higher order

A note on strongly nonlinear parabolic equations of higher order

A higher-order parabolic equation for wave propagation in an ocean overlying an elastic bottom

A higher-order elastic parabolic equation (HEPE) is derived for wave propagation in depth-dependent and weakly range-dependent fluid/solid media. Galerkin’s method is used to discretize the depth operators in the HEPE within layers in which depth variations in the Lamé constants and density are continuous. Discontinuities in material properties are handled with centered differences for the interface conditions between layers. The numerical solution of the HEPE also involves the method of alternating directions and Crank–Nicolson integration. The HEPE is applied to underwater wave propagation problems involving a water column over an elastic bottom including a weakly range-dependent problem. The accuracy of the HEPE is demonstrated with benchmark calculations.

Applications and time-domain solution of higher-order parabolic equations in underwater acoustics

A higher-order parabolic equation and the corresponding higher-order time-domain parabolic equation are derived from a Padé series and solved numerically. These models provide accurate solutions for problems involving very-wide-angle propagation (e.g., propagation in the nearfield or over a hard ocean bottom); propagation in domains in which sound-speed variations are large (e.g., propagation in deep water, deep within the ocean bottom, in high-speed ocean bottoms, and possibly of different wave types); and propagation out to very long ranges. The possibility of modeling elastic wave propagation with a similar approach is considered.

On fundamental solutions of linear parabolic equations of higher order in time and associated Volterra equations

On fundamental solutions of linear parabolic equations of higher order in time and associated Volterra equations