15,489 research outputs found
Adaptive Multi-Rate Wavelet Method for Circuit Simulation
In this paper a new adaptive algorithm for multi-rate circuit simulation encountered in the design of RF circuits based on spline wavelets is presented. The ordinary circuit differential equations are first rewritten by a system of (multi-rate) partial differential equations (MPDEs) in order to decouple the different time scales. Second, a semi-discretization by Rothe's method of the MPDEs results in a system of differential algebraic equations DAEs with periodic boundary conditions. These boundary value problems are solved by a Galerkin discretization using spline functions. An adaptive spline grid is generated, using spline wavelets for non-uniform grids. Moreover the instantaneous frequency is chosen adaptively to guarantee a smooth envelope resulting in large time steps and therefore high run time efficiency. Numerical tests on circuits exhibiting multi-rate behavior including mixers and PLL conclude the paper
Wavelets operational methods for fractional differential equations and systems of fractional differential equations
In this thesis, new and effective operational methods based on polynomials and
wavelets for the solutions of FDEs and systems of FDEs are developed. In particular
we study one of the important polynomial that belongs to the Appell family of
polynomials, namely, Genocchi polynomial. This polynomial has certain great
advantages based on which an effective and simple operational matrix of derivative
was first derived and applied together with collocation method to solve some singular
second order differential equations of Emden-Fowler type, a class of generalized
Pantograph equations and Delay differential systems. A new operational matrix of
fractional order derivative and integration based on this polynomial was also
developed and used together with collocation method to solve FDEs, systems of
FDEs and fractional order delay differential equations. Error bound for some of the
considered problems is also shown and proved. Further, a wavelet bases based on
Genocchi polynomials is also constructed, its operational matrix of fractional order
derivative is derived and used for the solutions of FDEs and systems of FDEs. A
novel approach for obtaining operational matrices of fractional derivative based on
Legendre and Chebyshev wavelets is developed, where, the wavelets are first
transformed into corresponding shifted polynomials and the transformation matrices
are formed and used together with the polynomials operational matrices of fractional
derivatives to obtain the wavelets operational matrix. These new operational matrices
are used together with spectral Tau and collocation methods to solve FDEs and
systems of FDEs
Compactly Supported Wavelets Derived From Legendre Polynomials: Spherical Harmonic Wavelets
A new family of wavelets is introduced, which is associated with Legendre
polynomials. These wavelets, termed spherical harmonic or Legendre wavelets,
possess compact support. The method for the wavelet construction is derived
from the association of ordinary second order differential equations with
multiresolution filters. The low-pass filter associated with Legendre
multiresolution analysis is a linear phase finite impulse response filter
(FIR).Comment: 6 pages, 6 figures, 1 table In: Computational Methods in Circuits and
Systems Applications, WSEAS press, pp.211-215, 2003. ISBN: 960-8052-88-
Wavelets for Differential Equations and Numerical Operator Calculus
Differential equations are commonplace in engineering, and lots of research have been carried out in developing methods, both efficient and precise, for their numerical solution. Nowadays the numerical practitioner can rely on a wide range of tools for solving differential equations: finite difference methods, finite element methods, meshless, and so on. Wavelets, since their appearance in the early 1990s, have attracted attention for their multiresolution nature that allows them to act as a “mathematical zoom,” a characteristic that promises to describe efficiently the functions involved in the differential equation, especially in the presence of singularities. The objective of this chapter is to introduce the main concepts of wavelets and differential equation, allowing the reader to apply wavelets to the solution of differential equations and in numerical operator calculus
Homotopy Analysis And Legendre Multi-Wavelets Methods For Solving Integral Equations
Due to the ability of function representation, hybrid functions and wavelets have a
special position in research. In this thesis, we state elementary definitions, then we
introduce hybrid functions and some wavelets such as Haar, Daubechies, Cheby-
shev, sine-cosine and linear Legendre multi wavelets. The construction of most
wavelets are based on stepwise functions and the comparison between two categories of wavelets will become easier if we have a common construction of them.
The properties of the Floor function are used to and a function which is one on the
interval [0; 1) and zero elsewhere. The suitable dilation and translation parameters
lead us to get similar function corresponding to the interval [a; b). These functions
and their combinations enable us to represent the stepwise functions as a function of
floor function. We have applied this method on Haar wavelet, Sine-Cosine wavelet,
Block - Pulse functions and Hybrid Fourier Block-Pulse functions to get the new
representations of these functions.
The main advantage of the wavelet technique for solving a problem is its ability
to transform complex problems into a system of algebraic equations. We use the Legendre multi-wavelets on the interval [0; 1) to solve the linear integro-differential
and Fredholm integral equations of the second kind. We also use collocation points
and linear legendre multi wavelets to solve an integro-differential equation which describes the charged particle motion for certain configurations of oscillating magnetic
fields. Illustrative examples are included to reveal the sufficiency of the technique.
In linear integro-differential equations and Fredholm integral equations of the second
kind cases, comparisons are done with CAS wavelets and differential transformation
methods and it shows that the accuracy of these results are higher than them.
Homotopy Analysis Method (HAM) is an analytic technique to solve the linear
and nonlinear equations which can be used to obtain the numerical solution too.
We extend the application of homotopy analysis method for solving Linear integro-
differential equations and Fredholm and Volterra integral equations. We provide
some numerical examples to demonstrate the validity and applicability of the technique. Numerical results showed the advantage of the HAM over the HPM, SCW,
LLMW and CAS wavelets methods. For future studies, some problems are proposed
at the end of this thesis
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