Nonlinear equations and wavelets

Wavelet-inspired approaches for localized solutions of nonlinear partial differential equations (NDPE) were presented. A multi-scale similarity analysis was used which gives specific relations between the velocity, amplitude and width of localized solutions of nonlinear differential equations. This approach does not need the explicit form of the exact solutions

Signorini Cylindrical Waves and Shannon Wavelets

Hyperelastic materials based on Signorini's strain energy density are studied by using Shannon wavelets. Cylindrical waves propagating in a nonlinear elastic material from the circular cylindrical cavity along the radius are analyzed in the following by focusing both on the main nonlinear effects and on the method of solution for the corresponding nonlinear differential equation. Cylindrical waves' solution of the resulting equations can be easily represented in terms of this family of wavelets. It will be shown that Hankel functions can be linked with Shannon wavelets, so that wavelets can have some physical meaning being a good approximation of cylindrical waves. The nonlinearity is introduced by Signorini elastic energy density and corresponds to the quadratic nonlinearity relative to displacements. The configuration state of elastic medium is defined through cylindrical coordinates but the deformation is considered as functionally depending only on the radial coordinate. The physical and geometrical nonlinearities arising from the wave propagation are discussed from the point of view of wavelet analysis

The Legendre wavelet method for solving initial value problems of Bratu-type

AbstractThe aim of this work is to study the Legendre wavelets for the solution of initial value problems of Bratu-type, which is widely applicable in fuel ignition of the combustion theory and heat transfer. The properties of Legendre wavelets together with the Gaussian integration method are used to reduce the problem to the solution of nonlinear algebraic equations. Also a reliable approach for convergence of the Legendre wavelet method when applied to a class of nonlinear Volterra equations is discussed and an error estimation for the proposed method is also introduced. Illustrative examples have been discussed to demonstrate the validity and applicability of the technique and the results have been compared with the exact solution. We finally show the high accuracy and efficiency of the proposed method

Variational-Wavelet Approach to RMS Envelope Equations

We present applications of variational-wavelet approach to nonlinear (rational) rms envelope equations. We have the solution as a multiresolution (multiscales) expansion in the base of compactly supported wavelet basis. We give extension of our results to the cases of periodic beam motion and arbitrary variable coefficients. Also we consider more flexible variational method which is based on biorthogonal wavelet approach.Comment: 21 pages, 8 figures, LaTeX2e, presented at Second ICFA Advanced Accelerator Workshop, UCLA, November, 199

Laguerre wavelet solution of Bratu and Duffing equations

The aim of this study is to solve the Bratu and Duffing equations by using the Laguerre wavelet method. The solution of these nonlinear equations is approximated by Laguerre wavelets which are defined by well known Laguerre polynomials. One of the advantages of the proposed method is that it does not require the approximation of the nonlinear term like other numerical methods. The application of the method converts the nonlinear differential equation to a system of algebraic equations. The method is tested on four examples and the solutions are compared with the analytical and other numerical solutions and it is observed that the proposed method has a better accuracy.Publisher's Versio

Variational Approach in Wavelet Framework to Polynomial Approximations of Nonlinear Accelerator Problems

In this paper we present applications of methods from wavelet analysis to polynomial approximations for a number of accelerator physics problems. According to variational approach in the general case we have the solution as a multiresolution (multiscales) expansion in the base of compactly supported wavelet basis. We give extension of our results to the cases of periodic orbital particle motion and arbitrary variable coefficients. Then we consider more flexible variational method which is based on biorthogonal wavelet approach. Also we consider different variational approach, which is applied to each scale.Comment: LaTeX2e, aipproc.sty, 21 Page