Superconvergent Nyström and Degenerate Kernel Methods for Integro-Differential Equations

This research received no external funding and APC was funded by University of Granada.The aim of this paper is to carry out an improved analysis of the convergence of the Nystrom and degenerate kernel methods and their superconvergent versions for the numerical solution of a class of linear Fredholm integro-differential equations of the second kind. By using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomial functions of degree <= r - 1, we obtain convergence order 2r for degenerate kernel and Nystrom methods, while, for the superconvergent and the iterated versions of theses methods, the obtained convergence orders are 3r + 1 and 4r, respectively. Moreover, we show that the optimal convergence order 4r is restored at the partition knots for the approximate solutions. The obtained theoretical results are illustrated by some numerical examples.University of Granad

On the Solution of Volterra Integro-differential Equations using a Modified Adomian Decomposition Method

The Adomian decomposition method’s effectiveness has been demonstrated in recent research, the process requires several iterations and can be time-consuming. By breaking down the source term function into series, the current work introduced a new decomposition approach to the Adomian decomposition method. As compared to the conventional Adomian decomposition approach, the newly devised method hastens the convergence of the solution. Numerical experiments were provided to show the superiority qualities

Multiscale wavelet analysis for integral and differential problems

2009 - 2010The object of the present research is wavelet analysis of integral and differential problems by means of harmonic and circular wavelets. It is shown that circular wavelets constitute a complete basis for L2[0; 1] functions, and form multiresolution analysis. Multiresolution analysis can be briefly considered as a decomposition of L2[0; 1] into a complete set of scale depending subspaces of wavelets. Thus, integral operators, differential operators, and L2(R) functions were investigated as scale depending functions through their projection onto these subspaces of wavelets. In particular: - conditions when a certain wavelet can be applied for solution of integral or differential problem are given; - it is shown that the accuracy of this approach exponentially grows when increasing the number of vanishing moments and scaling parameter; - wavelet solutions of low-dimensional nonlinear partial differential equations are compared with other methods; - wavelet-based approach is applied to low-dimensional Fredholm integral equations and the Galerkin method for two-dimensional Fredholm integral equations.[edited by author]. Oggetto della seguente ricerca `e l’analisi di problemi differenziali e integrali, utilizzando wavelet armoniche e wavelet armoniche periodiche. Si dimostra che le wavelet periodiche costituiscono una base completa per le funzioni L2[0; 1] e formano un’analisi multiscala. L’analisi multirisoluzione pu`o essere brevemente considerata come la decomposizione di L2[0; 1] in un insieme completo di sottospazi di wavelet dipendenti da un fattore di scala. Pertanto gli operatori integrali e differenziali e le funzioni L2(R) vengono studiati come funzioni di scala mediante le corrispondenti proiezioni in questi sottospazi di wavelet. In particolare, vengono sviluppati quattro principali argomenti: - sono state individuate le condizioni per applicare una data famiglia di wavelets alla soluzione di un data problema differenziale o integrale; - si `e dimostrato che la precisione di questo approccio cresce esponenzialmente quando decresce il numero dei momenti nulli e del parametro di scala; - soluzioni wavelet di equazioni differenziali a derivate parziali nonlineari di dimensione bassa sono state confrontate con altri metodi di soluzioni; - l’approccio basato sull’uso delle wavelet `e stato applicato anche per ricerca di soluzioni di alcune equazioni integrali di Fredholm e insieme al metodo di Galerkin per risolvere equazioni integrali Fredholm di dimensioni due.[a cura dell'autore]IX n.s

Effective numerical methods for nonlinear singular two-point boundary value Fredholm integro-differential equations

We deal with some effective numerical methods for solving a class of nonlinear singular two-point boundary value Fredholm integro-differential equations. Using an appropriate interpolation and a q-order quadrature rule of integration, the original problem will be approximated by the non-linear finite difference equations and so reduced to a nonlinear algebraic system that can be simply implemented. The convergence properties of the proposed method are discussed, and it is proved that its convergence order will be of O(hmin{ 72 ,q− 12 }). Ample numerical results are addressed to con-firm the expected convergence order as well as the accuracy and efficiency of the proposed method

Theory and applications of the multiwavelets for compression of boundary integral operators

In general the numerical solution of boundary integral equations leads to full coefficientmatrices. The discrete system can be solved in O(N2) operations by iterative solvers ofthe Conjugate Gradient type. Therefore, we are interested in fast methods such as fastmultipole and wavelets, that reduce the computational cost to O(N lnp N).In this thesis we are concerned with wavelet methods. They have proved to be veryefficient and effective basis functions due to the fact that the coefficients of a wavelet expansiondecay rapidly for a large class of functions. Due to the multiresolution propertyof wavelets they provide accurate local descriptions of functions efficiently. For examplein the presence of corners and edges, the functions can still be approximated with a linearcombination of just a few basis functions. Wavelets are attractive for the numericalsolution of integral equations because their vanishing moments property leads to operatorcompression. However, to obtain wavelets with compact support and high order of vanishingmoments, the length of the support increases as the order of the vanishingmomentsincreases. This causes difficulties with the practical use of wavelets particularly at edgesand corners. However, with multiwavelets, an increase in the order of vanishing momentsis obtained not by increasing the support but by increasing the number of mother wavelets.In chapter 2 we review the methods and techniques required for these reformulations,we also discuss how these boundary integral equations may be discretised by a boundaryelement method. In chapter 3, we discuss wavelet and multiwavelet bases. In chapter4, we consider two boundary element methods, namely, the standard and non-standardGalerkin methods with multiwavelet basis functions. For both methods compressionstrategies are developed which only require the computation of the significant matrix elements.We show that they are O(N logp N) such significant elements. In chapters 5 and6 we apply the standard and non-standard Galerkin methods to several test problems

Numerical homogenization of time-dependent Maxwell\u27s equations with dispersion effects

This thesis studies the propagation of electromagnetic waves in heterogeneous structures such as metamaterials. The governing equations for this problem are Maxwell\u27s equations with highly oscillatory parameters. We use an analytic homogenization result which yields an effective Maxwell system that involves a convolution integral. This convolution represents dispersive effects that result from the interaction of the wave with the (locally) periodic microscopic structure. We discretize in space using the Finite Element Heterogeneous Multiscale Method (FE-HMM) and provide a semi-discrete error estimate. The rigorous error analysis in space is supplemented by a rather standard time discretization at the end of which an efficient, fully discrete method is proposed. This method uses a recursive approximation of the convolution that relies on the assumption that the convolution kernel is an exponential function. Eventually, we present numerical experiments both for the microscopic and the macroscopic scale

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