415 research outputs found
A fractional B-spline collocation method for the numerical solution of fractional predator-prey models
We present a collocation method based on fractional B-splines for the solution of fractional differential problems. The key-idea is to use the space generated by the fractional B-splines, i.e., piecewise polynomials of noninteger degree, as approximating space. Then, in the collocation step the fractional derivative of the approximating function is approximated accurately and efficiently by an exact differentiation rule that involves the generalized finite difference operator. To show the effectiveness of the method for the solution of nonlinear dynamical systems of fractional order, we solved the fractional Lotka-Volterra model and a fractional predator-pray model with variable coefficients. The numerical tests show that the method we proposed is accurate while keeping a low computational cost
Spectral collocation method for compact integral operators
We propose and analyze a spectral collocation method for integral
equations with compact kernels, e.g. piecewise smooth kernels and
weakly singular kernels of the form We prove that 1) for integral equations, the convergence
rate depends on the smoothness of true solutions . If
satisfies condition (R): }, we obtain a geometric rate of convergence; if
satisfies condition (M): ,
we obtain supergeometric rate of convergence for both Volterra
equations and Fredholm equations and related integro differential
equations; 2) for eigenvalue problems, the convergence rate depends
on the smoothness of eigenfunctions. The same convergence rate for
the largest modulus eigenvalue approximation can be obtained.
Moreover, the convergence rate doubles for positive compact
operators. Our numerical experiments confirm our theoretical
results
Tensor Computation: A New Framework for High-Dimensional Problems in EDA
Many critical EDA problems suffer from the curse of dimensionality, i.e. the
very fast-scaling computational burden produced by large number of parameters
and/or unknown variables. This phenomenon may be caused by multiple spatial or
temporal factors (e.g. 3-D field solvers discretizations and multi-rate circuit
simulation), nonlinearity of devices and circuits, large number of design or
optimization parameters (e.g. full-chip routing/placement and circuit sizing),
or extensive process variations (e.g. variability/reliability analysis and
design for manufacturability). The computational challenges generated by such
high dimensional problems are generally hard to handle efficiently with
traditional EDA core algorithms that are based on matrix and vector
computation. This paper presents "tensor computation" as an alternative general
framework for the development of efficient EDA algorithms and tools. A tensor
is a high-dimensional generalization of a matrix and a vector, and is a natural
choice for both storing and solving efficiently high-dimensional EDA problems.
This paper gives a basic tutorial on tensors, demonstrates some recent examples
of EDA applications (e.g., nonlinear circuit modeling and high-dimensional
uncertainty quantification), and suggests further open EDA problems where the
use of tensor computation could be of advantage.Comment: 14 figures. Accepted by IEEE Trans. CAD of Integrated Circuits and
System
An improvement of the product integration method for a weakly singular Hammerstein equation
We present a new method to solve nonlinear Hammerstein equations with weakly
singular kernels. The process to approximate the solution, followed usually,
consists in adapting the discretization scheme from the linear case in order to
obtain a nonlinear system in a finite dimensional space and solve it by any
linearization method. In this paper, we propose to first linearize, via Newton
method, the nonlinear operator equation and only then to discretize the
obtained linear equations by the product integration method. We prove that the
iterates, issued from our method, tends to the exact solution of the nonlinear
Hammerstein equation when the number of Newton iterations tends to infinity,
whatever the discretization parameter can be. This is not the case when the
discretization is done first: in this case, the accuracy of the approximation
is limited by the mesh size discretization. A Numerical example is given to
confirm the theorical result
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