Numerical computing approach for solving Hunter-Saxton equation arising in liquid crystal model through sinc collocation method

In this study, numerical treatment of liquid crystal model described through Hunter-Saxton equation (HSE) has been presented by sinc collocation technique through theta weighted scheme due to its enormous applications including, defects, phase diagrams, self-assembly, rheology, phase transitions, interfaces, and integrated biological applications in mesophase materials and processes. Sinc functions provide the procedure for function approximation over all types of domains containing singularities, semi-infinite or infinite domains. Sinc functions have been used to reduce HSE into an algebraic system of equations that makes the solution quite superficial. These algebraic equations have been interpreted as matrices. This projected that sinc collocation technique is considerably efficacious on computational ground for higher accuracy and convergence of numerical solutions. Stability analysis of the proposed technique has ensured the accuracy and reliability of the method, moreover, as the stability parameter satisfied the condition the proposed solution of the problem converges. The solution of the HSE is presented through graphical figures and tables for different cases that are constructed on various values of θ and collocation points. The accuracy and efficiency of the proposed technique is analyzed on the basis of absolute errors.This research has been partially supported by Ministerio de Ciencia, Innovación y Universidades grant number PGC2018-0971-B-100 and Fundación Séneca -Agencia de Ciencia y Tecnología de la Región de Murcia grant number 20783/PI/18. Also, It has been supported by the National Research Program for Universities (NRPU), Higher Education Commission, Pakistan, No. 8103/Punjab/NRPU/R and D/HEC/2017

Generalized Spectral Decomposition for Stochastic Non Linear Problems

International audienceWe present an extension of the Generalized Spectral Decomposition method for the resolution of non-linear stochastic problems. The method consists in the construction of a reduced basis approximation of the Galerkin solution and is independent of the stochastic discretization selected (polynomial chaos, stochastic multi-element or multiwavelets). Two algorithms are proposed for the sequential construction of the successive generalized spectral modes. They involve decoupled resolutions of a series of deterministic and low dimensional stochastic problems. Compared to the classical Galerkin method, the algorithms allow for significant computational savings and require minor adaptations of the deterministic codes. The methodology is detailed and tested on two model problems, the one-dimensional steady viscous Burgers equation and a two-dimensional non-linear diffusion problem. These examples demonstrate the effectiveness of the proposed algorithms which exhibit convergence rates with the number of modes essentially dependent on the spectrum of the stochastic solution but independent of the dimension of the stochastic approximation space

Time-Domain Simulation of Sound Propagation in Frequency-Dependent Materials

This dissertation investigates sound propagation in frequency-dependent materials. The study provides an improved understanding of how to numerically model the porous impedance materials more accurately under the conditions of complicated geometries. The finite difference time-domain (FDTD) method is implemented on the linearized Euler equation (LEE), along with the immersed boundary (IB) method and other numerical techniques to simulate the acoustic wave propagation in air, water, porous media and biological tissues. When material properties vary in the frequency domain, their time-domain counterpart may contain either convolution operation or fractional derivative operation. Both operations have been studied in this dissertation. Recursive algorithm methods, piece-wise constant recursive methods (PCRC) and piece-wise linear recursive methods (PLRC) are used to numerically solve for convolution operations, and fractional central difference (FCD) methods are used to solve for fractional Laplacians. Both methods show good results in comparison with analytical solutions. A variety of models have been implemented to simulate the acoustic wave propagation inside porous media. The techniques include: the Zwicker and Kosten (ZK) phenomenological model, the Delany and Bazley model, various porosity two-parameter models, the time-domain boundary condition (TDBC) models, and Wilson’s relaxation model (WRX). A new method is also proposed that utilizes the ANSI/ASA-S1.18 measurements to construct a new relaxation function. The new relaxation function can improve the prediction from the TDBC and WRX models significantly. The ZK and WRX models have also been used in predicting the noise reduction of a house. The noise due to transmission and vibration of the wall is modeled as a simple wave transmission through a porous material layer. A curve fitting method is used to match acoustic properties of the wall material. By assembling all the materials together, the over-all acoustic response of a house can be simulated. When acoustic wave propagating in biological tissues, wave propagation equations were previously solved either with convolutions, which consume a large amount of memory, or with pseudo-spectral methods, which cannot handle complicated geometries effectively. The approach described in this study employs FCD method, combined with the IB method for the FDTD simulation. It also works naturally with the IB method which enables a simple Cartesian-type grid mesh to be used to solve problems with complicated geometries. This work also studies acoustic scattering effects caused by 2D or 3D vortices. The LEE is used to investigate sound wave propagation over subsonic vortices. Instead of traditional direct numerical simulation (DNS) methods, the new approach treats vortex flow field as a scattering background flow and solves the acoustic field with the LEE solver. The numerical method uses a high-order WENO scheme to accommodate the highly convective background flow at high Mach numbers. The study focuses on the acoustic field scaling laws scattered by the 2D and 3D vortices

Нелінійна динаміка — 2013

The book of Proceedings includes extended abstracts of presentations on the Fourth International conference on nonlinear dynamics

New Trends in Differential and Difference Equations and Applications

This is a reprint of articles from the Special Issue published online in the open-access journal Axioms (ISSN 2075-1680) from 2018 to 2019 (available at https://www.mdpi.com/journal/axioms/special issues/differential difference equations)