Singularity Preserving Numerical Methods for Boundary Integral Equations

In the past twelve months (May 8, 1995 - May 8, 1996), under the cooperative agreement with Division of Multidisciplinary Optimization at NASA Langley, we have accomplished the following five projects: a note on the finite element method with singular basis functions; numerical quadrature for weakly singular integrals; superconvergence of degenerate kernel method; superconvergence of the iterated collocation method for Hammersteion equations; and singularity preserving Galerkin method for Hammerstein equations with logarithmic kernel. This final report consists of five papers describing these projects. Each project is preceeded by a brief abstract

Non-uniform UE-spline quasi-interpolants and their application to the numerical solution of integral equations

A construction of Marsden’s identity for UE-splines is developed and a complete proof is given. With the help of this identity, a new non-uniform quasi-interpolant that repro-duces the spaces of polynomial, trigonometric and hyperbolic functions are defined. Effi-cient quadrature rules based on integrating these quasi-interpolation schemes are derived and analyzed. Then, a quadrature formula associated with non-uniform quasi-interpolation along with Nyström’s method is used to numericallysolve Hammerstein and Fredholm integral equations. Numerical results that illustrate the effectiveness of these rules are pre-sented.Universidad de Granada / CBU

Superconvergence in Iterated Solutions of Integral Equations

In this thesis, we investigate the superconvergence phenomenon of the iterated numerical solutions for the Fredholm integral equations of the second kind as well as a class of nonliner Hammerstein equations. The term superconvergence was first described in the early 70s in connection with the solution of two-point boundary value problems and other related partial differential equations. Superconvergence in this context was understood to mean that the order of convergence of the numerical solutions arising from the Galerkin as well as the collocation method is higher at the knots than we might expect from the numerical solutions that are obtained by applying a class of piecewise polynomials as approximating functions. The type of superconvergence that we investigate in this thesis is different. We are interested in finding out whether or not we obtain an enhancement in the global rate of convergence when the numerical solutions are iterated through integral operators. A general operator approximation scheme for the second kind linear equation is described that can be used to explain some of the existing superconvergence results. Moreover, a corollary to the general approximation scheme will be given which can be used to establish the superconvergence of the iterated degenerate kernel method for the Fredholm equations of the second kind. We review the iterated Galerkin method for Hammerstein equations and discuss the iterated degenerate kernel method for the Fredholm equations of the second kind. We review the iterated Galerkin method for Hammerstein equations and discuss the iterated degenerate kernel method for Hammerstein and weakly singular Hammerstein equations and its corresponding superconvergence phenomena for the iterated solutions. The type of regularities that the solution of weakly singular Hammerstein equations possess is investigated. Subsequently, we establish the singularity preserving Galerkin method for Hammerstein equations. Finally, the superconvergence results for the iterated solutions corresponding to this method will be described

Numerical Approximate Methods for Solving Linear and Nonlinear Integral Equations

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

Chebyshev cardinal functions for solving volterra-fredholm integro- differential equations using operational matrices

Abstract In this paper, an effective direct method to determine the numerical solution of linear and nonlinear Fredholm and Volterra integral and integro-differential equations is proposed. The method is based on expanding the required approximate solution as the elements of Chebyshev cardinal functions. The operational matrices for the integration and product of the Chebyshev cardinal functions are described in detail. These matrices play the important role of reducing an integral equation to a system of algebraic equations. Illustrative examples are shown, which confirms the validity and applicability of the presented technique

Applied Mathematics and Fractional Calculus

In the last three decades, fractional calculus has broken into the field of mathematical analysis, both at the theoretical level and at the level of its applications. In essence, the fractional calculus theory is a mathematical analysis tool applied to the study of integrals and derivatives of arbitrary order, which unifies and generalizes the classical notions of differentiation and integration. These fractional and derivative integrals, which until not many years ago had been used in purely mathematical contexts, have been revealed as instruments with great potential to model problems in various scientific fields, such as: fluid mechanics, viscoelasticity, physics, biology, chemistry, dynamical systems, signal processing or entropy theory. Since the differential and integral operators of fractional order are nonlinear operators, fractional calculus theory provides a tool for modeling physical processes, which in many cases is more useful than classical formulations. This is why the application of fractional calculus theory has become a focus of international academic research. This Special Issue "Applied Mathematics and Fractional Calculus" has published excellent research studies in the field of applied mathematics and fractional calculus, authored by many well-known mathematicians and scientists from diverse countries worldwide such as China, USA, Canada, Germany, Mexico, Spain, Poland, Portugal, Iran, Tunisia, South Africa, Albania, Thailand, Iraq, Egypt, Italy, India, Russia, Pakistan, Taiwan, Korea, Turkey, and Saudi Arabia

Gravity Field Refinement by Radial Basis Functions from In-situ Satellite Data

In this thesis, an integrated approach is developed for the regional refinement of global gravity field solutions. The analysis concepts are tailored to the in-situ type character of the observations provided by the new satellite missions CHAMP, GRACE, and GOCE. They are able to evaluate data derived from short arcs of the satellite's orbit and, therefore, offer the opportunity to use regional satellite data for the calculation of regional gravity field solutions. The regional character of the approach will be realized at various stages of the analysis procedure. The first step is the design of specifically tailored space localizing basis functions. In order to adapt the basis functions to the signal content to be expected in the gravity field solution, they will be derived from the covariance function of the gravitational potential. To use the basis functions in gravity field modeling, they have to be located at the nodal points of a spherical grid; therefore investigations will be performed regarding a suitable choice of such a nodal point distribution. Another important aspect in the regional gravity field analysis approach is the downward continuation process. In this context, a regionally adapted regularization will be introduced which assigns different regularization matrices to geographical areas with varying signal content. Regularization parameters individually determined for each region take into account the varying frequency behavior, allowing to extract additional information out of a given data set. To conclude the analysis chain, an approach will be described that combines regional solutions with global coverage to obtain a global solution and to derive the corresponding spherical harmonic coefficients by means of the Gauss-Legendre quadrature method. The capability of the method will be demonstrated by its successful application to real data provided by CHAMP and GRACE and to a simulation scenario based on a combination of GRACE and GOCE observations.Verfeinerungen des Gravitationsfeldes mit radialen Basisfunktionen aus in-situ Satellitendaten In der vorliegenden Arbeit wird ein ganzheitliches Konzept für die regionale Verfeinerung globaler Gravitationsfeldmodelle entwickelt. Die dazu verwendeten Analyseverfahren sind dem in-situ Charakter der Beobachtungen der neuen Satellitenmissionen CHAMP, GRACE und GOCE angepasst. Sie beruhen auf kurzen Bahnbögen und ermöglichen somit die Berechnung regionaler Gravitationsfeldmodelle aus regional begrenzten Satellitendaten. Der regionale Charakter des Ansatzes wird dabei auf verschiedenen Ebenen des Analyseprozesses realisiert. Der erste Schritt ist die Entwicklung angepasster orts-lokalisierender Basisfunktionen. Diese sollen das Frequenzverhalten des zu bestimmenden Gravitationsfeldes widerspiegeln; sie werden daher aus der Kovarianzfunktion des Gravitationspotentials abgeleitet. Um die Basisfunktionen für die Schwerefeldmodellierung zu verwenden, müssen sie an den Knotenpunkten eines sphärischen Gitters angeordnet werden. Daher werden Untersuchungen durchgeführt, welche Punktverteilung für diese Aufgabe besonders geeignet ist. Einen wichtigen Aspekt bei der regionalen Gravi-tationsfeldanalyse stellt der Fortsetzungsprozess nach unten dar. In diesem Zusammenhang wird ein regional angepasstes Regularisierungsverfahren entwickelt, das verschiedene Regularisierungsmatrizen für regionale Gebiete mit unterschiedlichem Schwerefeldsignal ermöglicht. Individuell angepasste Regularisierungsparameter berücksichtigen den variierenden Signalinhalt, wodurch erreicht wird, dass zusätzliche Informationen aus einem gegebenen Datensatz extrahiert werden können. Schließlich wird ein Ansatz vorgestellt, der regionale Lösungen mit globaler Überdeckung zu einer globalen Lösung zusammenfügt und die zugehörigen sphärischen harmonischen Koeffizienten mit Hilfe der Gauss-Legendre-Quadratur berechnet. Die Leistungsfähigkeit des beschriebenen Ansatzes wird durch eine erfolgreiche Anwendung auf die Echtdatenanalyse aus Daten der Satellitenmissionen CHAMP und GRACE und auf ein Simulationsszenario aus einer Kombination simulierter GRACE- und GOCE-Beobachtungen verdeutlicht