34 research outputs found

    Solution of the Nonlinear Mixed Volterra-Fredholm Integral Equations by Hybrid of Block-Pulse Functions and Bernoulli Polynomials

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    A new numerical method for solving the nonlinear mixed Volterra-Fredholm integral equations is presented. This method is based upon hybrid functions approximation. The properties of hybrid functions consisting of block-pulse functions and Bernoulli polynomials are presented. The operational matrices of integration and product are given. These matrices are then utilized to reduce the nonlinear mixed Volterra-Fredholm integral equations to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique

    Fibonacci Collocation Method for Solving High-Order Linear Fredholm Integro-Differential-Difference Equations

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    A new collocation method based on the Fibonacci polynomials is introduced for the approximate solution of high order-linear Fredholm integro-differential-difference equations with the mixed conditions. The proposed method is analyzed to show the convergence of the method. Some further numerical experiments are carried out to demonstrate the method

    Optimal control of systems with memory

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    The “Optimal Control of Systems with memory” is a PhD project that is borne from the collaboration between the Department of Mechanical and Aerospace Engineering of Sapienza University of Rome and CNR-INM the Institute for Marine Engineering of the National Research Council of Italy (ex INSEAN). This project is part of a larger EDA (European Defence Agency) project called ETLAT: Evaluation of State of the Art Thin Line Array Technology. ETLAT is aimed at improving the scientific and technical knowledge of potential performance of current Thin Line Towed Array (TLA) technologies (element sensors and arrays) in view of Underwater Surveillance applications. A towed sonar array has been widely employed as an important tool for naval defence, ocean exploitation and ocean research. Two main operative limitations costrain the TLA design such as: a fixed immersion depth and the stabilization of its horizontal trim. The system is composed by a towed vehicle and a towed line sonar array (TLA). The two subsystems are towed by a towing cable attached to the moving boat. The role of the vehicle is to guarantee a TLA’s constant depth of navigation and the reduction of the entire system oscillations. The vehicle is also called "depressor" and its motion generates memory effects that influence the proper operation of the TLA. The dynamic of underwater towed system is affected by memory effects induced by the fluid-structure interaction, namely: vortex shedding and added damping due to the presence of a free surface in the fluid. In time domain, memory effects are represented by convolution integral between special kernel functions and the state of the system. The mathematical formulation of the underwater system, implies the use of integral-differential equations in the time domain, that requires a nonstandard optimal control strategy. The goal of this PhD work is to developed a new optimal control strategy for mechanical systems affected by memory effects and described by integral-differential equations. The innovative control method presented in this thesis, is an extension of the Pontryagin optimal solution which is normally applied to differential equations. The control is based on the variational control theory implying a feedback formulation, via model predictive control. This work introduces a novel formulation for the control of the vehicle and cable oscillations that can include in the optimal control integral terms besides the more conventional differential ones. The innovative method produces very interesting results, that show how even widely applied control methods (LQR) fail, while the present formulation exhibits the advantage of the optimal control theory based on integral-differential equations of motion

    Numerical Approximate Methods for Solving Linear and Nonlinear Integral Equations

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    Integral equation has been one of the essential tools for various area of applied mathematics. In this work, we employed different numerical methods for solving both linear and nonlinear Fredholm integral equations. A goal is to categorize the selected methods and assess their accuracy and efficiency. We discuss challenges faced by researchers in this field, and we emphasize the importance of interdisciplinary effort for advancing the study on numerical methods for solving integral equations. Integral equations can be viewed as equations which are results of transformation of points in a given vector spaces of integrable functions by the use of certain specific integral operators to points in the same space. If, in particular, one is concerned with function spaces spanned by polynomials for which the kernel of the corresponding transforming integral operator is separable being comprised of polynomial functions only, then several approximate methods of solution of integral equations can be developed. This work, specially, deals with the development of different wavelet methods for solving integral and intgro-differential equations. Wavelets theory is a relatively new and emerging area in mathematical research. It has been applied in a wide range of engineering disciplines; particularly, wavelets are very successfully used in signal analysis for waveform representations and segmentations, time frequency analysis, and fast algorithms for easy implementation. Wavelets permit the accurate representation of a variety of functions and operators. Moreover, wavelets establish a connection with fast numerical algorithms. Wavelets can be separated into two distinct types, orthogonal and semi-orthogonal. The preliminary concept of integral equations and wavelets are first presented in Chapter 1. Classification of integral equations, construction of wavelets and multi-resolution analysis (MRA) have been briefly discussed and provided in this chapter. In Chapter 2, different wavelet methods are constructed and function approximation by these methods with convergence analysis have been presented. In Chapter 3, linear semi-orthogonal compactly supported B-spline wavelets together with their dual wavelets have been applied to approximate the solutions of Fredholm integral equations (both linear and nonlinear) of the second kind and their systems. Properties of these wavelets are first presented; these properties are then utilized to reduce the computation of integral equations to some algebraic equations. Convergence analysis of B-spline method has been discussed in this chapter. Again, in Chapter 4, system of nonlinear Fredholm integral equations have been solved by using hybrid Legendre Block-Pulse functions and xiii Bernstein collocation method. In Chapter 5, two practical problems arising from chemical phenomenon, have been modeled as Fredholm- Hammerstein integral equations and solved numerically by different numerical techniques. First, COSMO-RS model has been solved by Bernstein collocation method, Haar wavelet method and Sinc collocation method. Second, Hammerstein integral equation arising from chemical reactor theory has been solved by B-spline wavelet method. Comparison of results have been demonstrated through illustrative examples. In Chapter 6, Legendre wavelet method and Bernoulli wavelet method have been developed to solve system of integro-differential equations. Legendre wavelets along with their operational matrices are developed to approximate the solutions of system of nonlinear Volterra integro-differential equations. Also, nonlinear Volterra weakly singular integro-differential equations system has been solved by Bernoulli wavelet method. The properties of these wavelets are used to reduce the system of integral equations to a system of algebraic equations which can be solved numerically by Newton's method. Rigorous convergence analysis has been done for these wavelet methods. Illustrative examples have been included to demonstrate the validity and applicability of the proposed techniques. In Chapter 7, we have solved the second order Lane-Emden type singular differential equation. First, the second order differential equation is transformed into integro-differential equation and then solved by Legendre multi-wavelet method and Chebyshev wavelet method. Convergence of these wavelet methods have been discussed in this chapter. In Chapter 8, we have developed a efficient collocation technique called Legendre spectral collocation method to solve the Fredholm integro-differential-difference equations with variable coefficients and system of two nonlinear integro-differential equations which arise in biological model. The proposed method is based on the Gauss-Legendre points with the basis functions of Lagrange polynomials. The present method reduces this model to a system of nonlinear algebraic equations and again this algebraic system has been solved numerically by Newton's method. The study of fuzzy integral equations and fuzzy differential equations is an emerging area of research for many authors. In Chapter 9, we have proposed some numerical techniques for solving fuzzy integral equations and fuzzy integro-differential equations. Fundamentals of fuzzy calculus have been discussed in this chapter. Nonlinear fuzzy Hammerstein integral equation has been solved by Bernstein polynomials and Legendre wavelets, and then compared with homotopy analysis method. We have solved nonlinear fuzzy Hammerstein Volterra integral equations with constant delay by Bernoulli wavelet method and then compared with B-spline wavelet method. Finally, fuzzy integro-differential equation has been solved by Legendre wavelet method and compared with homotopy analysis method. In fuzzy case, we have applied two-dimensional numerical methods which are discussed in chapter 2. Convergence analysis and error estimate have been also provided for Bernoulli wavelet method. xiv The study of fractional calculus, fractional differential equations and fractional integral equations has a great importance in the field of science and engineering. Most of the physical phenomenon can be best modeled by using fractional calculus. Applications of fractional differential equations and fractional integral equations create a wide area of research for many researchers. This motivates to work on fractional integral equations, which results in the form of Chapter 10. First, the preliminary definitions and theorems of fractional calculus have been presented in this chapter. The nonlinear fractional mixed Volterra-Fredholm integro-differential equations along with mixed boundary conditions have been solved by Legendre wavelet method. A numerical scheme has been developed by using Petrov-Galerkin method where the trial and test functions are Legendre wavelets basis functions. Also, this method has been applied to solve fractional Volterra integro-differential equations. Uniqueness and existence of the problem have been discussed and the error estimate of the proposed method has been presented in this work. Sinc Galerkin method is developed to approximate the solution of fractional Volterra-Fredholm integro-differential equations with weakly singular kernels. The proposed method is based on the Sinc function approximation. Uniqueness and existence of the problem have been discussed and the error analysis of the proposed method have been presented in this chapte

    Abstract book

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    Welcome at the International Conference on Differential and Difference Equations & Applications 2015. The main aim of this conference is to promote, encourage, cooperate, and bring together researchers in the fields of differential and difference equations. All areas of differential & difference equations will be represented with special emphasis on applications. It will be mathematically enriching and socially exciting event. List of registered participants consists of 169 persons from 45 countries. The five-day scientific program runs from May 18 (Monday) till May 22, 2015 (Friday). It consists of invited lectures (plenary lectures and invited lectures in sections) and contributed talks in the following areas: Ordinary differential equations, Partial differential equations, Numerical methods and applications, other topics

    Numerical approximations of fractional differential equations: a Chebyshev pseudo-spectral approach.

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    Doctoral Degree. University of KwaZulu-Natal, Pietermaritzburg.This study lies at the interface of fractional calculus and numerical methods. Recent studies suggest that fractional differential and integral operators are well suited to model physical phenomena with intrinsic memory retention and anomalous behaviour. The global property of fractional operators presents difficulties in fnding either closed-form solutions or accurate numerical solutions to fractional differential equations. In rare cases, when analytical solutions are available, they often exist only in terms of complex integrals and special functions, or as infinite series. Similarly, obtaining an accurate numerical solution to arbitrary order differential equation is often computationally demanding. Fractional operators are non-local, and so it is practicable that when approximating fractional operators, non-local methods should be preferred. One such non-local method is the spectral method. In this thesis, we solve problems that arise in the ow of non-Newtonian fluids modelled with fractional differential operators. The recurrent theme in this thesis is the development, testing and presentation of tractable, accurate and computationally efficient numerical schemes for various classes of fractional differential equations. The numerical schemes are built around the pseudo{spectral collocation method and shifted Chebyshev polynomials of the first kind. The literature shows that pseudo-spectral methods converge geometrically, are accurate and computationally efficient. The objective of this thesis is to show, among other results, that these features are true when the method is applied to a variety of fractional differential equations. A survey of the literature shows that many studies in which pseudo-spectral methods are used to numerically approximate the solutions of fractional differential equations often to do this by expanding the solution in terms of certain orthogonal polynomials and then simultaneously solving for the coefficients of expansion. In this study, however, the orthogonality condition of the Chebyshev polynomials of the first kind and the Chebyshev-Gauss-Lobatto quadrature are used to numerically find the coefficients of the series expansions. This approach is then applied to solve various fractional differential equations, which include, but are not limited to time{space fractional differential equations, two{sided fractional differential equations and distributed order differential equations. A theoretical framework is provided for the convergence of the numerical schemes of each of the aforementioned classes of fractional differential equations. The overall results, which include theoretical analysis and numerical simulations, demonstrate that the numerical method performs well in comparison to existing studies and is appropriate for any class of arbitrary order differential equations. The schemes are easy to implement and computationally efficient

    New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus

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    This reprint focuses on exploring new developments in both pure and applied mathematics as a result of fractional behaviour. It covers the range of ongoing activities in the context of fractional calculus by offering alternate viewpoints, workable solutions, new derivatives, and methods to solve real-world problems. It is impossible to deny that fractional behaviour exists in nature. Any phenomenon that has a pulse, rhythm, or pattern appears to be a fractal. The 17 papers that were published and are part of this volume provide credence to that claim. A variety of topics illustrate the use of fractional calculus in a range of disciplines and offer sufficient coverage to pique every reader's attention

    Nonlinear Systems

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    The editors of this book have incorporated contributions from a diverse group of leading researchers in the field of nonlinear systems. To enrich the scope of the content, this book contains a valuable selection of works on fractional differential equations.The book aims to provide an overview of the current knowledge on nonlinear systems and some aspects of fractional calculus. The main subject areas are divided into two theoretical and applied sections. Nonlinear systems are useful for researchers in mathematics, applied mathematics, and physics, as well as graduate students who are studying these systems with reference to their theory and application. This book is also an ideal complement to the specific literature on engineering, biology, health science, and other applied science areas. The opportunity given by IntechOpen to offer this book under the open access system contributes to disseminating the field of nonlinear systems to a wide range of researchers

    Fractional Calculus and the Future of Science

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    Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding
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