NEW TECHINQE FOR SOLVIND FINITE LEVEL FUZZY NON-LINEAR INTEGRAL EQUATION

In this paper, non linear  finite fuzzy Volterra integral equation of the second kind is considered. The successive approximate method  will be used t o solve it, and comparing with the exact solution and calculate the absolute error between exact and approximate method .  Some numerical examples are prepared to show the efficiency and simplicity of the method

International Conference on Mathematical Analysis and Applications in Science and Engineering – Book of Extended Abstracts

The present volume on Mathematical Analysis and Applications in Science and Engineering - Book of Extended Abstracts of the ICMASC’2022 collects the extended abstracts of the talks presented at the International Conference on Mathematical Analysis and Applications in Science and Engineering – ICMA2SC'22 that took place at the beautiful city of Porto, Portugal, in June 27th-June 29th 2022 (3 days). Its aim was to bring together researchers in every discipline of applied mathematics, science, engineering, industry, and technology, to discuss the development of new mathematical models, theories, and applications that contribute to the advancement of scientific knowledge and practice. Authors proposed research in topics including partial and ordinary differential equations, integer and fractional order equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, probability, computational mathematics, amongst others. The conference was designed to maximize the involvement of all participants and will present the state-of- the-art research and the latest achievements.info:eu-repo/semantics/publishedVersio

SINGULAR INTEGRATION BY INTERPOLATION FOR INTEGRAL EQUATIONS

Maxwell’s equations and the laws of Electromagnetics (EM) govern a plethora of electrical, optical phenomena with applications on wireless, cellular, communications, medical and computer hardware technologies to name a few. A major contributor to the technological progress in these areas has been due to the development of simulation and design tools that enable engineers and scientists to model, analyze and predict the EM interactions in their systems of interest. At the core of such tools is the field of Computational Electromagnetics (CEM), which studies the solution of Maxwell’s equations with the aid of computers. The advances in these applications technologies, in return, demand increasingly more efficient and accurate CEM methods. Among the many CEM methodologies that are currently in broad use, the Boundary Element Method (BEM) or surface Method of Moments (MoM), is perhaps the most popular in solving electrically large or electrically small multi-layered structures. In BEM, the surfaces of conductors and dielectrics are discretized to triangular or quadrilateral elements and the equivalent currents on them are convolved with the appropriate Green’s function at all observations on the mesh to produce a fully populated impedance matrix to be solved with an appropriate excitation. The reliability, accuracy and speed of BEM, among others, critically depends on the method used to perform the singular four-dimensional convolution integrals between source and observation surface currents through a Green’s function, that exhibits a singularity when observation and source elements touch or overlap. Large literature has been devoted in addressing this important issue, and methods involving using singularity subtraction, cancellation or even full 4D integral evaluations. Each of these approaches offer certain advantages, but they tend to require thousands of (often complicated) function evaluations for a single impedance matrix singular integration, it is noted that a typical problem may involve tens or hundreds of millions of such singular integrations. In this dissertation, an unconventional approach of calculating all weakly singular and near weakly singular integrals, encountered in the BEM solution of the Electric Field Integral Equation (EFIE), as well as near singular integrals encountered in the BEM solution of the Magnetic Field Integral Equation (MFIE) in flat triangular meshes, is presented. Instead of specialized integration rules such as singularity subtraction or cancellation, universal look-up-tables and multi-dimensional interpolation are used. Firstly, frequency independent integral expressions, equivalent to the original EFIE-BEM, MFIE-BEM element matrix expressions are derived, in order to facilitate the construction of said universal look-up-tables of integrals. The domain of these functions is discretized by hp refinement, i.e., the size, h and approximation order, p, of the interpolation elements of the entire interpolation domain can be varied independently. Because of the high-dimensional nature of the interpolation domain, from three dimensional to six dimensional, the interpolation over each element is performed with either sparse grids or low-rank tensor train approximations. The integrals are pre-computed into the tables using a state-of-the-art singularity subtraction method at maximum accuracy. Consequently, during run-time, these tables are loaded and any arbitrary singular integral is recovered by multi-dimensional interpolation. The method is compared to a state-of-the-art singularity subtraction technique for the lowest order Rao-Wilton-Glisson (RWG) basis functions in various PEC flat triangular meshes. For EFIE common triangle, weakly singular, in accuracy, while offering over 150× speed-ups. Similarly for EFIE common edge, near weakly singular, interactions it shows about 50× speed-ups but at a somewhat lower, yet acceptable, accuracy. The tensor decomposition approach improves the accuracy to the level of the state-of-the-art and offers about 20× speed-ups, while it also has a controllable accuracy and speed. Lastly, for MFIE common edge, near hyper singular, interactions accuracy is improved by 1 − 2 decimal digits, while offering 20× speed-ups. For a typical BEM run using the single level fast multiple method (FMM) accelerator, the end-to-end set-up time speed improvement with the proposed approach is 15 − 20%

The 2nd International Conference on Mathematical Modelling in Applied Sciences, ICMMAS’19, Belgorod, Russia, August 20-24, 2019 : book of abstracts

The proposed Scientific Program of the conference is including plenary lectures, contributed oral talks, poster sessions and listeners. Five suggested special sessions / mini-symposium are also considered by the scientific committe

A Review of Computational Stochastic Elastoplasticity

Heterogeneous materials at the micro-structural level are usually subjected to several uncertainties. These materials behave according to an elastoplastic model, but with uncertain parameters. The present review discusses recent developments in numerical approaches to these kinds of uncertainties, which are modelled as random elds like Young's modulus, yield stress etc. To give full description of random phenomena of elastoplastic materials one needs adequate mathematical framework. The probability theory and theory of random elds fully cover that need. Therefore, they are together with the theory of stochastic nite element approach a subject of this review. The whole group of di erent numerical stochastic methods for the elastoplastic problem has roots in the classical theory of these materials. Therefore, we give here the classical formulation of plasticity in very concise form as well as some of often used methods for solving this kind of problems. The main issues of stochastic elastoplasticity as well as stochastic problems in general are stochastic partial di erential equations. In order to solve them we must discretise them. Methods of solving and discretisation are called stochastic methods. These methods like Monte Carlo, Perturbation method, Neumann series method, stochastic Galerkin method as well as some other very known methods are reviewed and discussed here

Mathematical Methods, Modelling and Applications

This volume deals with novel high-quality research results of a wide class of mathematical models with applications in engineering, nature, and social sciences. Analytical and numeric, deterministic and uncertain dimensions are treated. Complex and multidisciplinary models are treated, including novel techniques of obtaining observation data and pattern recognition. Among the examples of treated problems, we encounter problems in engineering, social sciences, physics, biology, and health sciences. The novelty arises with respect to the mathematical treatment of the problem. Mathematical models are built, some of them under a deterministic approach, and other ones taking into account the uncertainty of the data, deriving random models. Several resulting mathematical representations of the models are shown as equations and systems of equations of different types: difference equations, ordinary differential equations, partial differential equations, integral equations, and algebraic equations. Across the chapters of the book, a wide class of approaches can be found to solve the displayed mathematical models, from analytical to numeric techniques, such as finite difference schemes, finite volume methods, iteration schemes, and numerical integration methods

Foundations of realistic rendering : a mathematical approach

Die vorliegende Dissertation ist keine gewöhnliche Abhandlung, sondern sie ist als Lehrbuch zum realistischen Rendering für Studenten im zweiten Studienabschnitt, sowie Forscher und am Thema Interessierte konzipiert. Aus mathematischer Sicht versteht man unter realistischem Rendering das Lösen der stationären Lichttransportgleichung, einer komplizierten Fredholm Integralgleichung der 2tenArt, deren exakte Lösung, wenn überhaupt berechenbar, nur in einem unendlich- dimensionalen Funktionenraum existiert. Während in den existierenden Büchern, die sich mit globaler Beleuchtungstheorie beschäftigen, vorwiegend die praktische Implementierung von Lösungsansätzen im Vordergrund steht, sind wir eher daran interessiert, den Leser mit den mathematischen Hilfsmitteln vertraut zu machen, mit welchen das globale Beleuchtungsproblem streng mathematisch formuliert und letzendlich auch gelöst werden kann. Neue, effzientere und elegantere Algorithmen zur Berechnung zumindest approxima- tiver Lösungen der Lichttransportgleichung und ihrer unterschiedlichen Varianten können nur im Kontext mit einem vertieften Verständnis der Lichttransportgleichung entwickelt werden. Da die Probleme des realistischen Renderings tief in verschiedenen mathematis- chen Disziplinen verwurzelt sind, setzt das vollständige Verständnis des globalen Beleuch- tungsproblems Kenntnisse aus verschiedenen Bereichen der Mathematik voraus. Als zen- trale Konzepte kristallisieren sich dabei Prinzipien der Funktionalanalysis, der Theorie der Integralgleichungen, der Maß- und Integrationstheorie sowie der Wahrscheinlichkeitstheo- rie heraus. Wir haben uns zum Ziel gesetzt, dieses Knäuel an mathematischen Konzepten zu entflechten, sie für Studenten verständlich darzustellen und ihnen bei Bedarf und je nach speziellem Interesse erschöpfend Auskunft zu geben.The available doctoral thesis is not a usual paper but it is conceived as a text book for realistic rendering, made for students in upper courses, as well as for researchers and interested people. From mathematical point of view, realistic rendering means solving the stationary light transport equation, a complicated Fredholm Integral equation of 2nd kind. Its exact solution exists|if possible at all|in an infinite dimensional functional space. Whereas practical implementation of approaches for solving problems are in the center of attentionin the existing textbooks that treat global illumination theory, we are more interested in familiarizing our reader with the mathematical tools which permit them to formulate the global illumination problem in accordance with strong mathematical principles and last but not least to solve it. New, more eficient and more elegant algorithms to calculate approximate solutions for the light transport equation and their different variants must be developed in the context of deep and complete understanding of the light transport equation. As the problems of realistic rendering are deeply rooted in different mathematical disciplines, there must precede the complete comprehension of all those areas. There are evolving principles of functional analysis, theory of integral equations, measure and integration theory as well as probability theory. We have set ourselves the target to remerge this bundle of fluff of mathematical concepts and principles, to represent them to the students in an understandable manner, and to give them, if required, exhaustive information