A Wavelet-Galerkin Method for the Nonlinear Schrödinger Equation

A general implementation is presented for constructing a wavelet method for solving the nonlinear equation of Schr¨odinger. An explicit formula is derived which yields a stability in of the numerical solution. A simulation is elaborated to show the general behavior of the distribution function. Numerical results and comparison with classical algorithms are provided. This approach prove an attractive scheme for solving such equation.

An Efficient Modified Taylor Wavelet Galerkin Method for One Dimensional Partial Differential Equations

Background: In this paper, a modified Taylor wavelet Galerkin method (MTWGM) based on approximation scheme is used to solve partial differential equations (PDEs), which is play an important role in electrical circuit models. Objective: The objective of this work is to give fine and accurate implementation of proposed method for the solution of PDEs, which is the best tool for the analysis of electric circuit problems. Methods: In this work, we used an effective, modified Taylor wavelet Galerkin method with its residual technique and we obtained more accurate numerical solution of the one dimensional PDEs. The Introduced wavelet method is more efficiently applicable in the comparison of some existing numerical methods such as, finite difference method, finite element method, finite volume method, spectral method etc. This method is the best tool for solving PDEs. Therefore, it has significance in the field of electrical engineering and others. Results: The experimentally four numerical problems are given which are showing the numerical results extractive by introduced method and those results compared with exact solution and other available numerical methods i.e., Hermite wavelet Galarkin method (HWGM), Finite difference method (FDM) and spectral procedures which shows that proposed method is more effective. Conclusion: This work is significantly helpful for the electrical circuits in which the governing models are available in the form of PDEs.

An effective computational approach based on Gegenbauer wavelets for solving the time-fractional Kdv-Burgers-Kuramoto equation

In this paper, our purpose is to present a wavelet Galerkin method for solving the time-fractional KdV-Burgers-Kuramoto (KBK) equation, which describes nonlinear physical phenomena and involves instability, dissipation, and dispersion parameters. The presented computational method in this paper is based on Gegenbauer wavelets. Gegenbauer wavelets have useful properties. Gegenbauer wavelets and the operational matrix of integration, together with the Galerkin method, were used to transform the time-fractional KBK equation into the corresponding nonlinear system of algebraic equations, which can be solved numerically with Newton’s method. Our aim is to show that the Gegenbauer wavelets-based method is efficient and powerful tool for solving the KBK equation with time-fractional derivative. In order to compare the obtained numerical results of the wavelet Galerkin method with exact solutions, two test problems were chosen. The obtained results prove the performance and efficiency of the presented method.

3-D marine controlled-source electromagnetic modelling in an anisotropic medium using a Wavelet–Galerkin method with a secondary potential formulation

SUMMARY This paper presents a new algorithm for solving 3-D MCSEM modelling problems in an anisotropic medium using a Wavelet–Galerkin method (WGM) based on compactly supported Daubechies wavelets which are differentiable according to the requirement. In order to avoid the source singularity, we adopted a secondary potential formulation for the quasi-static Maxwell's equation. The primary field on the modelling domain is calculated using fast Hankel transform. The domain can be discretized by locally intensive nodes to deal with the model's complexity, which is observed to improve the accuracy of the solution. The sparse system of the WGM equations is solved using the direct solver MUMPS. This study's algorithm is then applied to calculate the response of the MCSEM in isotropic and anisotropic mediums. The results generated are verified against with solution obtained by FE method and confirmed the performance of the algorithm presented in this study.