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

The study of comets, part 1

Papers are presented dealing with observations of comets. Topic discussed include: photometry, polarimetry, and astrometry of comets; detection of water and molecular transitions in comets; ion motions in comet tails; determination of comet brightness and luminosity; and evolution of cometary orbits. Emphasis is placed on analysis of observations of comet Kohoutek

The 1981 Goddard Space Flight Center Battery Workshop

Results of testing, analysis, and development of lithium, nickel-cadmium, and nickel-hydrogen batteries are reported. Focus is on the improvement of power systems in the areas of high capacity, high energy density, and long cycle and storage life. Applications of these batteries as spacecraft power supplies are discussed. Those spacecraft include deepspace probes, spacecraft in geostationary orbit, and large space systems in low-Earth orbit

Object Relations and Identity Disturbances in Bulimic Women

Problem. Although diagnostic criteria of bulimia center on weight- and food-related issues, eating disorders may be viewed as a response to deficits in self-regulatory functions. The purpose of this study was to investigate the relationship between the severity of bulimia, object relations, and identity. This study tested the hypotheses that women with a more severe eating pathology have high scores on object-relations disturbance as well as identity disturbance. It was postulated that women who have been assessed as having a more cohesive ego might respond to cognitive behavioral therapy while those who are assessed as having less intact ego resources may require more intensive psychodynamic approaches. Method. The study involved the administration of three tests by therapists who were treating women diagnosed with bulimia nervosa according to DSM-IV criteria. The test instruments included the following: Bulimia Test-Revised, Bell Object Relations Inventory, and the Erwin Identity Scale. An interview was conducted on a selected group of 12 subjects. Results. There were statistically significant correlations between the severity of bulimia and the severity of object relations and identity disturbance. Specifically, the Alienation subscale of the Bell Object Relations Inventory and the Confidence subscale of the Erwin Identity Scale had the strongest correlation with the BULIT-R. The qualitative results indicated that a number of themes were strongly identified by both High Bulimics and Low Bulimics. Conclusions. The quantitative analysis indicated there was a relationship between the severity of object relations, identity disturbance, and bulimia. However, the qualitative analysis identified many of the women, in the Low Bulimic group, had significant disturbances in their relationships as well as their opinion of their body. It was concluded that both groups exhibited significant object relations and identity disturbances. Therefore, it is suggested that a more psychodynamic approach is useful for understanding the adaptive functions of bulimia

The identification of microRNAs to predict glioma prognosis

Until now, personalised medicine for patients in oncology has been focused on the use of DNA-based techniques such as mutation detection and fluorescence in situ hybridisation, fluorescence-activated cell sorting and immuno-staining for classifying tumours. MicroRNAs are short non-coding RNAs that are involved in post-translational regulation of gene expression. Their expression levels are often altered in cancer. Due to their functional importance and stability in biological samples, they represent another tool that could be used to aid patient management. Glioblastoma is a disease that has had little improvement in survival over the past decade in comparison to other cancers. A number of new drugs have been explored but even successful trials have shown limited success. This thesis is focused on identification of microRNAs as signatures for prognosis prediction in glioblastoma. It is separated into four parts; the identification of a microRNA signature that can be used to predict prognosis in glioblastoma; the alignment of glioblastoma microRNA expression with the microRNA expression of oligodendrocyte precursors and its involvement in patient outcome; the use of the expression pattern of the most abundant and robust prognostic microRNA in glioma (miR-9) to delineate glioblastoma subtype and finally the identification of a microRNA signature to predict prognosis in patients treated with the anti-angiogenic drug bevacizumab. The research aims to create signatures suitable for clinical practice, with a small number of predictors, and where possible the function of the microRNAs has been predicted and reviewed to provide confirmation of their role in glioma biology. The key findings of this research are the formation of robust signatures using microRNAs in a disease where few markers are available and proof of a technique that can be used in future drug studies to improve performance at clinical trials

Collected Papers (on Physics, Artificial Intelligence, Health Issues, Decision Making, Economics, Statistics), Volume XI

This eleventh volume of Collected Papers includes 90 papers comprising 988 pages on Physics, Artificial Intelligence, Health Issues, Decision Making, Economics, Statistics, written between 2001-2022 by the author alone or in collaboration with the following 84 co-authors (alphabetically ordered) from 19 countries: Abhijit Saha, Abu Sufian, Jack Allen, Shahbaz Ali, Ali Safaa Sadiq, Aliya Fahmi, Atiqa Fakhar, Atiqa Firdous, Sukanto Bhattacharya, Robert N. Boyd, Victor Chang, Victor Christianto, V. Christy, Dao The Son, Debjit Dutta, Azeddine Elhassouny, Fazal Ghani, Fazli Amin, Anirudha Ghosha, Nasruddin Hassan, Hoang Viet Long, Jhulaneswar Baidya, Jin Kim, Jun Ye, Darjan Karabašević, Vasilios N. Katsikis, Ieva Meidutė-Kavaliauskienė, F. Kaymarm, Nour Eldeen M. Khalifa, Madad Khan, Qaisar Khan, M. Khoshnevisan, Kifayat Ullah,, Volodymyr Krasnoholovets, Mukesh Kumar, Le Hoang Son, Luong Thi Hong Lan, Tahir Mahmood, Mahmoud Ismail, Mohamed Abdel-Basset, Siti Nurul Fitriah Mohamad, Mohamed Loey, Mai Mohamed, K. Mohana, Kalyan Mondal, Muhammad Gulfam, Muhammad Khalid Mahmood, Muhammad Jamil, Muhammad Yaqub Khan, Muhammad Riaz, Nguyen Dinh Hoa, Cu Nguyen Giap, Nguyen Tho Thong, Peide Liu, Pham Huy Thong, Gabrijela Popović‬‬‬‬‬‬‬‬‬‬, Surapati Pramanik, Dmitri Rabounski, Roslan Hasni, Rumi Roy, Tapan Kumar Roy, Said Broumi, Saleem Abdullah, Muzafer Saračević, Ganeshsree Selvachandran, Shariful Alam, Shyamal Dalapati, Housila P. Singh, R. Singh, Rajesh Singh, Predrag S. Stanimirović, Kasan Susilo, Dragiša Stanujkić, Alexandra Şandru, Ovidiu Ilie Şandru, Zenonas Turskis, Yunita Umniyati, Alptekin Ulutaș, Maikel Yelandi Leyva Vázquez, Binyamin Yusoff, Edmundas Kazimieras Zavadskas, Zhao Loon Wang.‬‬‬

Solar Irradiance Variations on Active Region Time Scales

The variations of the total solar irradiance is an important tool for studying the Sun, thanks to the development of very precise sensors such as the ACRIM instrument on board the Solar Maximum Mission. The largest variations of the total irradiance occur on time scales of a few days are caused by solar active regions, especially sunspots. Efforts were made to describe the active region effects on total and spectral irradiance

Protein-protein recognition: The neonatal Fe receptor and immunoglobulin G

The neonatal Fe receptor (FeRn) binds the Fe portion of immunoglobulin G (IgG) at the acidic pH of endosomes or the gut and releases IgG at the alkaline pH of blood. FeRn is responsible for the maternofetal transfer of IgG and for rescuing endocytosed IgG from a default degradative pathway. We investigated how FeRn interacts with IgG by constructing a heterodimeric form of the Fe (hdFc) that contains one FeRn binding site. This molecule was used to characterize the interaction between one FeRn molecule and one Fe and to determine under what conditions FeRn forms a dimer. The hdFc binds one FeRn molecule at pH 6.0 with a K_d of 80 nM. In solution and with FeRn anchored to solid supports, the heterodimeric Fe does not induce a dimer of FeRn molecules. FcRnhdFc complex crystals were obtained and the complex structure was solved to 2.8 Å resolution. Analysis of this structure refined the understanding of the mechanism of the pH-dependent binding, shed light on the role played by carbohydrates in the Fe binding, and provided insights on how to design therapeutic IgG antibodies with longer serum half-lives. The FcRn-hdFc complex in the crystal did not contain the FeRn dimer. To characterize the tendency of FeRn to form a dimer in a membrane we analyzed the tendency of the hdFc to induce cross-phosphorylation of FeRn-tyrosine kinase chimeras. We also constructed FeRn-cyan and FeRn-yellow fluorescent proteins and have analyzed the tendency of these molecules to exhibit fluorescence resonance energy transfer. As of now, neither of these analyses have lead to conclusive results. In the process of acquiring the context to appreciate the structure of the FcRn-hdFc interface, we developed a study of 171 other nonobligate protein-protein interfaces that includes an original principal component analysis of the quantifiable aspects of these interfaces