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

Wavelet Methods for the Solutions of Partial and Fractional Differential Equations Arising in Physical Problems

The subject of fractional calculus has gained considerable popularity and importance during the past three decades or so, mainly due to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. It deals with derivatives and integrals of arbitrary orders. The fractional derivative has been occurring in many physical problems, such as frequency-dependent damping behavior of materials, motion of a large thin plate in a Newtonian fluid, creep and relaxation functions for viscoelastic materials, the PI D controller for the control of dynamical systems etc. Phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry, control theory, neutron point kinetic model, anomalous diffusion, Brownian motion, signal and image processing, fluid dynamics and material science are well described by differential equations of fractional order. Generally, nonlinear partial differential equations of fractional order are difficult to solve. So for the last few decades, a great deal of attention has been directed towards the solution (both exact and numerical) of these problems. The aim of this dissertation is to present an extensive study of different wavelet methods for obtaining numerical solutions of mathematical problems occurring in disciplines of science and engineering. This present work also provides a comprehensive foundation of different wavelet methods comprising Haar wavelet method, Legendre wavelet method, Legendre multi-wavelet methods, Chebyshev wavelet method, Hermite wavelet method and Petrov-Galerkin method. The intension is to examine the accuracy of various wavelet methods and their efficiency for solving nonlinear fractional differential equations. With the widespread applications of wavelet methods for solving difficult problems in diverse fields of science and engineering such as wave propagation, data compression, image processing, pattern recognition, computer graphics and in medical technology, these methods have been implemented to develop accurate and fast algorithms for solving integral, differential and integro-differential equations, especially those whose solutions are highly localized in position and scale. The main feature of wavelets is its ability to convert the given differential and integral equations to a system of linear or nonlinear algebraic equations, which can be solved by numerical methods. Therefore, our main focus in the present work is to analyze the application of wavelet based transform methods for solving the problem of fractional order partial differential equations. The introductory concept of wavelet, wavelet transform and multi-resolution analysis (MRA) have been discussed in the preliminary chapter. The basic idea of various analytical and numerical methods viz. Variational Iteration Method (VIM), Homotopy Perturbation Method (HPM), Homotopy Analysis Method (HAM), First Integral Method (FIM), Optimal Homotopy Asymptotic Method (OHAM), Haar Wavelet Method, Legendre Wavelet Method, Chebyshev Wavelet Method and Hermite Wavelet Method have been presented in chapter 1. In chapter 2, we have considered both analytical and numerical approach for solving some particular nonlinear partial differential equations like Burgers’ equation, modified Burgers’ equation, Huxley equation, Burgers-Huxley equation and modified KdV equation, which have a wide variety of applications in physical models. Variational Iteration Method and Haar wavelet Method are applied to obtain the analytical and numerical approximate solution of Huxley and Burgers-Huxley equations. Comparisons between analytical solution and numerical solution have been cited in tables and also graphically. The Haar wavelet method has also been applied to solve Burgers’, modified Burgers’, and modified KdV equations numerically. The results thus obtained are compared with exact solutions as well as solutions available in open literature. Error of collocation method has been presented in this chapter. Methods like Homotopy Perturbation Method (HPM) and Optimal Homotopy Asymptotic Method (OHAM) are very powerful and efficient techniques for solving nonlinear PDEs. Using these methods, many functional equations such as ordinary, partial differential equations and integral equations have been solved. We have implemented HPM and OHAM in chapter 3, in order to obtain the analytical approximate solutions of system of nonlinear partial differential equation viz. the Boussinesq-Burgers’ equations. Also, the Haar wavelet method has been applied to obtain the numerical solution of BoussinesqBurgers’ equations. Also, the convergence of HPM and OHAM has been discussed in this chapter. The mathematical modeling and simulation of systems and processes, based on the description of their properties in terms of fractional derivatives, naturally leads to differential equations of fractional order and the necessity to solve such equations. The mathematical preliminaries of fractional calculus, definitions and theorems have been presented in chapter 4. Next, in this chapter, the Haar wavelet method has been analyzed for solving fractional differential equations. The time-fractional Burgers-Fisher, generalized Fisher type equations, nonlinear time- and space-fractional Fokker-Planck equations have been solved by using two-dimensional Haar wavelet method. The obtained results are compared with the Optimal Homotopy Asymptotic Method (OHAM), the exact solutions and the results available in open literature. Comparison of obtained results with OHAM, Adomian Decomposition Method (ADM), VIM and Operational Tau Method (OTM) has been demonstrated in order to justify the accuracy and efficiency of the proposed schemes. The convergence of two-dimensional Haar wavelet technique has been provided at the end of this chapter. In chapter 5, the fractional differential equations such as KdV-Burger-Kuramoto (KBK) equation, seventh order KdV (sKdV) equation and Kaup-Kupershmidt (KK) equation have been solved by using two-dimensional Legendre wavelet and Legendre multi-wavelet methods. The main focus of this chapter is the application of two-dimensional Legendre wavelet technique for solving nonlinear fractional differential equations like timefractional KBK equation, time-fractional sKdV equation in order to demonstrate the efficiency and accuracy of the proposed wavelet method. Similarly in chapter 6, twodimensional Chebyshev wavelet method has been implemented to obtain the numerical solutions of the time-fractional Sawada-Kotera equation, fractional order Camassa-Holm equation and Riesz space-fractional sine-Gordon equations. The convergence analysis has been done for these wavelet methods. In chapter 7, the solitary wave solution of fractional modified Fornberg-Whitham equation has been attained by using first integral method and also the approximate solutions obtained by optimal homotopy asymptotic method (OHAM) are compared with the exact solutions acquired by first integral method. Also, the Hermite wavelet method has been implemented to obtain approximate solutions of fractional modified Fornberg-Whitham equation. The Hermite wavelet method is implemented to system of nonlinear fractional differential equations viz. the fractional Jaulent-Miodek equations. Convergence of this wavelet methods has been discussed in this chapter. Chapter 8 emphasizes on the application of Petrov-Galerkin method for solving the fractional differential equations such as the fractional KdV-Burgers’ (KdVB) equation and the fractional Sharma-TassoOlver equation with a view to exhibit the capabilities of this method in handling nonlinear equation. The main objective of this chapter is to establish the efficiency and accuracy of Petrov-Galerkin method in solving fractional differential equtaions numerically by implementing a linear hat function as the trial function and a quintic B-spline function as the test function. Various wavelet methods have been successfully employed to numerous partial and fractional differential equations in order to demonstrate the validity and accuracy of these procedures. Analyzing the numerical results, it can be concluded that the wavelet methods provide worthy numerical solutions for both classical and fractional order partial differential equations. Finally, it is worthwhile to mention that the proposed wavelet methods are promising and powerful methods for solving fractional differential equations in mathematical physics. This work also aimed at, to make this subject popular and acceptable to engineering and science community to appreciate the universe of wonderful mathematics, which is in between classical integer order differentiation and integration, which till now is not much acknowledged, and is hidden from scientists and engineers. Therefore, our goal is to encourage the reader to appreciate the beauty as well as the usefulness of these numerical wavelet based techniques in the study of nonlinear physical system

New Challenges Arising in Engineering Problems with Fractional and Integer Order

Mathematical models have been frequently studied in recent decades, in order to obtain the deeper properties of real-world problems. In particular, if these problems, such as finance, soliton theory and health problems, as well as problems arising in applied science and so on, affect humans from all over the world, studying such problems is inevitable. In this sense, the first step in understanding such problems is the mathematical forms. This comes from modeling events observed in various fields of science, such as physics, chemistry, mechanics, electricity, biology, economy, mathematical applications, and control theory. Moreover, research done involving fractional ordinary or partial differential equations and other relevant topics relating to integer order have attracted the attention of experts from all over the world. Various methods have been presented and developed to solve such models numerically and analytically. Extracted results are generally in the form of numerical solutions, analytical solutions, approximate solutions and periodic properties. With the help of newly developed computational systems, experts have investigated and modeled such problems. Moreover, their graphical simulations have also been presented in the literature. Their graphical simulations, such as 2D, 3D and contour figures, have also been investigated to obtain more and deeper properties of the real world problem

A multi-domain implementation of the pseudo-spectral method and compact finite difference schemes for solving time-dependent differential equations

Abstract : In this dissertation, we introduce new numerical methods for solving time-dependant differential equations. These methods involve dividing the domain of the problem into multiple sub domains. The nonlinearity of the differential equations is dealt with by using a Gauss-Seidel like relaxation or quasilinearisation technique. To solve the linearized iteration schemes obtained we use either higher order compact finite difference schemes or spectral collocation methods and we call the resulting methods the multi-domain compact finite difference relaxation method (MD-CFDRM), multi-domain compact finite difference quasilinearisation method (MD-CFDQLM) and multi-domain bivariate spectral quasilinearisation method (MD-BSQLM) respectively. We test the applicability of these methods in a wide variety of differential equations. The accuracy is compared against other methods as well as other results from literature. The MD-CFDRM is used to solve famous chaotic systems and hyperchaotic systems. Chaotic and hyperchaotic systems are characterized by high sensitivity to small perturbation on initial data and rapidly changing solutions. Such rapid variations in the solution pose tremendous problems to a number of numerical approximations. We modify the CFDs to be able to deal with such systems of equations. We also used the MD-CFDQLM to solve the nonlinear evolution partial differential equations, namely, the Fisher’s equation, Burgers- Fisher equation, Burgers-Huxley equation and the coupled Burgers’ equations over a large time domain. The main advantage of this approach is that it offers better accuracy on coarser grids which significantly improves the computational speed of the method for large time domain. We also studied the generalized Kuramoto-Sivashinsky (GKS) equations. The KS equations exhibit chaotic behaviour under certain conditions. We used the multi-domain bivariate spectral quasilinearisation method (MD-BSQLM) to approximate the numerical solutions for the generalized KS equations.M.Sc. (Pure and Applied Mathematics

Wavelet Theory

The wavelet is a powerful mathematical tool that plays an important role in science and technology. This book looks at some of the most creative and popular applications of wavelets including biomedical signal processing, image processing, communication signal processing, Internet of Things (IoT), acoustical signal processing, financial market data analysis, energy and power management, and COVID-19 pandemic measurements and calculations. The editor’s personal interest is the application of wavelet transform to identify time domain changes on signals and corresponding frequency components and in improving power amplifier behavior

Mathematical and Numerical Aspects of Dynamical System Analysis

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”