328,386 research outputs found
Boundary knot method: A meshless, exponential convergence, integration-free, and boundary-only RBF technique
Based on the radial basis function (RBF), non-singular general solution and
dual reciprocity principle (DRM), this paper presents an inheretnly meshless,
exponential convergence, integration-free, boundary-only collocation techniques
for numerical solution of general partial differential equation systems. The
basic ideas behind this methodology are very mathematically simple and
generally effective. The RBFs are used in this study to approximate the
inhomogeneous terms of system equations in terms of the DRM, while non-singular
general solution leads to a boundary-only RBF formulation. The present method
is named as the boundary knot method (BKM) to differentiate it from the other
numerical techniques. In particular, due to the use of non-singular general
solutions rather than singular fundamental solutions, the BKM is different from
the method of fundamental solution in that the former does no need to introduce
the artificial boundary and results in the symmetric system equations under
certain conditions. It is also found that the BKM can solve nonlinear partial
differential equations one-step without iteration if only boundary knots are
used. The efficiency and utility of this new technique are validated through
some typical numerical examples. Some promising developments of the BKM are
also discussed.Comment: 36 pages, 2 figures, Welcome to contact me on this paper: Email:
[email protected] or [email protected]
A note on fractional derivative modeling of broadband frequency-dependent absorption: Model III
By far, the fractional derivative model is mainly related to the modelling of
complicated solid viscoelastic material. In this study, we try to build the
fractional derivative PDE model for broadband ultrasound propagation through
human tissues
A new definition of the fractional Laplacian
It is noted that the standard definition of the fractional Laplacian leads to
a hyper-singular convolution integral and is also obscure about how to
implement the boundary conditions. This purpose of this note is to introduce a
new definition of the fractional Laplacian to overcome these major drawbacks.Comment: This study is carred out with the ongoing project of "mathematical
and numerical modelling of medical ultasound wave propagation" sponsored by
the Simula Research Laborator
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