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

Control And Inverse Problems For One Dimensional Systems

Thesis (Ph.D.) University of Alaska Fairbanks, 2009The thesis is devoted to control and inverse problems (dynamical and spectral) for systems on graphs and on the half line. In the first part we study the boundary control problems for the wave, heat, and Schrodinger equations on a finite graph. We suppose that the graph is a tree (i.e., it does not contain cycles), and on each edge an equation is defined. The control is acting through the Dirichlet condition applied to all or all but one boundary vertices. The exact controllability in L2-classes of controls is proved and sharp estimates of the time of controllability are obtained for the wave equation. The null controllability for the heat equation and exact controllability for the Schrodinger equation in arbitrary time interval are obtained. In the second part we consider the in-plane motion of elastic strings on a tree-like network, observed from the 'leaves.' We investigate the inverse problem of recovering not only the physical properties, i.e. the 'optical lengths' of each string, but also the topology of the tree which is represented by the edge degrees and the angles between branching edges. It is shown that under generic assumptions the inverse problem can be solved by applying measurements at all leaves, the root of the tree being fixed. In the third part of the thesis we consider Inverse dynamical and spectral problems for the Schrodinger operator on the half line. Using the connection between dynamical (Boundary Control method) and spectral approaches (due to Krein, Gelfand-Levitan, Simon and Remling), we improved the result on the representation of so-called A---amplitude and derive the "local" version of the classical Gelfand-Levitan equations

Adaptive meshless point collocation methods: investigation and application to geometrically non-linear solid mechanics

Conventional mesh-based methods for solid mechanics problems suffer from issues resulting from the use of a mesh, therefore, various meshless methods that can be grouped into those based on weak or strong forms of the underlying problem have been proposed to address these problems by using only points for discretisation. Compared to weak form meshless methods, strong form meshless methods have some attractive features because of the absence of any background mesh and avoidance of the need for numerical integration, making the implementation straightforward. The objective of this thesis is to develop a novel numerical method based on strong form point collocation methods for solving problems with geometric non-linearity including membrane problems. To address some issues in existing strong form meshless methods, the local maximum entropy point collocation method is developed, where the basis functions possess some advantages such as the weak Kronecker-Delta property on boundaries. r- and h-adaptive strategies are investigated in the proposed method and are further combined into a novel rh-adaptive approach, achieving the prescribed accuracy with the optimised locations and limited number of points. The proposed meshless method with h-adaptivity is then extended to solve geometrically non-linear problems described in a Total Lagrangian formulation, where h-adaptivity is again employed after the initial calculation to improve the accuracy of the solution effciently. This geometrically non-linear method is finally developed to analyse membrane problems, in which the out-of-plane deformation for membranes complicates the governing PDEs and the use of hyperelastic materials makes the computational modelling of membrane problems challenging. The Newton-Raphson arc-length method is adopted here to capture the snap-through behaviour in hyperelastic membrane problems. Several numerical examples are presented for each proposed algorithm to validate the proposed methodology and suggestions are made for future work leading on from the findings of this thesis

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described