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Orientador: Edmundo Capelas de Oliveira Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica Resumo: Neste trabalho apresentamos um estudo sobre as funções de Mittag-Leffler de um, dois e três parâmetros. Apresentamos a função de Mittag-Leffler como uma generalização da função exponencial bem como a relação que esta possui com outras funções especiais, tais como as funções beta, gama, gama incompleta e erro. Abordamos, também, a integração fracionária que se faz necessária para introduzir o conceito de derivação fracionária. Duas formulações para a derivada fracionária são estudadas, as formulações proposta por Riemann-Liouville e por Caputo. Investigamos quais regras clássicas de derivação são estendidas para estas formulações. Por fim, como uma aplicação, utilizamos a metodologia da transformada de Laplace para resolver a equação diferencial fracionária associada ao problema do oscilador harmônico fracionário Abstract: This work presents a study about the one- two- and three-parameters Mittag-Leffler functions. We show that the Mittag-Leffler function is a generalization of the exponential function and present its relations to other special functions beta, gamma, incomplete gamma and error functions. We also approach fractional integration, which is necessary to introduce the concept of fractional derivatives. Two formulations for the fractional derivative are studied, the formulations proposed by Riemann-Liouville and by Caputo. We investigate which classical derivatives rules can be extended to these formulations. Finally, as an application, using the Laplace transform methodology, we discuss the fractional differential equation associated with the harmonic oscillator problem Mestrado Matemática Aplicada Mestra em Matemática Aplicada

Актуальность исследования обусловлена необходимостью снижения температуры воздуха в тупиковых горных выработках при ведении работ, связанных с выделением большого количества теплоты. Разработка способов нормализации термодинамических параметров рудничного воздуха в забоях на основе построения прогнозных математических моделей изменения теплового режима является необходимым условием обеспечения безопасности труда в горнодобывающей промышленности. Цель: получение зависимостей для определения скорости повышения температуры воздуха в тупиковой выработке при известных величинах интенсивности выделения теплоты и скорости ее отведения вентиляционной струей. Объекты: тупиковые выработки и окружающий породный массив. Методы: решение задачи нестационарного теплообмена рудничного воздуха и породного массива в сопряженной постановке с помощью преобразований Лапласа; оценка безопасности по газовому фактору интенсификации выноса теплоты из горной выработки путем увеличения подачи воздуха в забой за счет частично-повторного использования воздуха. Результаты. Проведен анализ проблемы отвода теплоты, выделяющейся при работе оборудования в замкнутом пространстве горных выработок. Показано, что процессов поглощения теплоты породным массивом и ее отведения за счет проветривания может быть недостаточно для обеспечения нормативных значений температуры воздуха. Разработана математическая модель сопряженного теплообмена воздуха и породного массива, позволяющая прогнозировать повышение температуры с течением времени в зависимости от длины выработки и суммарной интенсивности выделения теплоты в ней. Установлено, что теплообменные процессы продолжительностью в несколько часов могут моделироваться в приближении малых времен, что значительно упрощает расчетные зависимости. Произведен расчет динамики теплового режима при характерном для шахтных условий наборе физических параметров задачи, результаты которого явились обоснованием сделанного приближения и подтвердили опасность быстрого увеличения температуры воздуха при недостаточной скорости выноса теплоты из выработки вентиляционной струей. Доказано, что при необходимости дополнительного отведения теплоты можно применять источники тяги высокой производительности с частично-повторным использованием исходящего воздуха без риска увеличения содержания газов в тупиковых выработках. The relevance of the research is caused by the need to reduce the air temperature during mining operations in dead-end workings associated with the release of a large amount of heat. Development of ways to normalize the thermodynamic parameters of mine air in the faces based on the construction of predictive mathematical models of change of the thermal regime is a necessary condition for safety in the mining industry. The aim of the research is to obtain dependences for determining the rate of air temperature increase in a dead-end mine at known values of the intensity of heat release and its removal by a ventilation jet. Objects: dead-end workings and the surrounding rock mass. Methods: solution of the problem of non-stationary heat transfer of mine air and rock mass in the conjugate formulation using Laplace transformations; safety assessment on the gas factor of intensification of heat removal from the production by increasing the air supply to the bottom using recirculation. Results. The authors have analyzed the problem of heat removal, released during the operation of mining equipment in a confined space of mine workings. It is shown that heat absorption by the rock mass and its removal due to ventilation may not be enough to prevent excessive overheating of the air. A mathematical model of the conjugate heat exchange of air and rock is developed, which allows predicting the temperature increase over time depending on the length of production and the total intensity of heat release in it. It is established that heat exchange lasting several hours can be modeled in the approximation of small times, which greatly simplifies the calculated dependences. The calculation of the dynamics of the thermal regime under a set of physical parameters characteristic of mine conditions of the problem, the results of which were the basis of the approximation made and confirmed the danger of a rapid increase in air temperature with insufficient intensity of heat removal from the workings of the ventilation jet. It is proved that if additional heat removal is necessary, it is possible to use traction sources with a large flow rate and air recirculation, without fear of increasing gas content in the bottom hole.

El texto "Ecuaciones diferenciales", trata esencialmente del cambio, el flujo, el movimiento y, en particular, de la rapidez a la que estos cambios se dan en los diferentes fenómenos de la naturaleza. Cada ser viviente experimenta cambios. Representarlos debe ser el marco de trabajo de cualquier documento de ecuaciones diferenciales. Es por ello que el presente texto trabaja en particular, la dinámica del estudio de los sistemas que evolucionan con el paso del tiempo. Universidad EAN The text "Differential equations" deals essentially with change, flow, movement and, in particular, with the speed at which these changes occur in the different phenomena of nature. Every living being experiences changes. Representing them should be the framework of any differential equations document. That is why this text works in particular, the dynamics of the study of systems that evolve over time.

U ovom radu bavimo se integralnim jednadžbama i njihovim primjenama. Integralne jednadžbe su jednadžbe kod kojih se nepoznata funkcija, koju trebamo odrediti, pojavljuje pod znakom integrala. Integralne jednadžbe su jedan od korisnijih matematičkih alata u teoretskoj i primijenjenoj matematici te imaju značajnu primjenu u mnogim fizikalnim problemima. Rad se sastoji od četiri poglavlja. U Poglavlju 1 bavimo se konceptom integralnih jednadžbi, njihovom povijesnom pozadinom te donosimo njihovu klasifikaciju. U Poglavlju 2 obratili smo posebnu pozornost na Volterrine integralne jednadžbe i različite metode njihovog rješavanja. U Poglavlju 3 fokusirali smo se na Fredholmove integralne jednadžbe i pripadajuće tehnike rješavanja. U posljednjem, Poglavlju 4 kao dodatak obradili smo Laplaceove transformacije i osnovna svojstva koja smo primjenjivali u glavnom dijelu rada. In this work we deal with the integral equations and their applications. Integral equation is an equation in which the unknown function to be determined appear under the integral sign. Integral equations are one of the most useful mathematical tools in pure and applied mathematics, and are widely used in many physical problems. This work contains four chapters. In Chapter 1 we deal with the concept of integral equations, its historical background and classification. In Chapter 2 we have paid special attention to Volterra integral equations and the corresponding solving techniques. In chapter 3 we focused on the Fredholm integral equation and different solution methods. In the last, fourth, chapter written as a supplement we introduced Laplace transformations, and its basic properties that we applied in the main part of the work.

For operators of many different kinds it has been proved that (generalized) Darboux transformations can be built using so called Wronskian formulae. Such Darboux transformations are not invertible in the sense that the corresponding mappings of the opera- tor kernels are not invertible. The only known invertible ones were Laplace transformations (and their compositions), which are special cases of Darboux transformations for hyperbolic bivariate operators of order 2. In the present paper we find a criteria for a bivariate linear partial differential operator of an arbitrary order d to have an invertible Darboux trans- formation. We show that Wronkian formulae may fail in some cases, and find sufficient conditions for such formulae to work.

This paper presents a semi-analytical method for the modeling of productivity testing of vertical, horizontal or multilateral wells. The method, which is applicable to both oil and gas reservoirs, automatically accounts for well interference. The use of analytical expressions ensures proper handling of transient short time behavior and semi-steady-state longtime behavior, both close to the well and further into the reservoir. Calculation times are still very limited, in the order of a few minutes down to a few seconds when there are vertical wells only. This makes the tool suitable for well testing evaluation. The approach is based on an earlier derived productivity prediction tool, in which the steady-state equations were solved. It has now been extended to solve the time-dependent diffusion equation and it is thus more rigorous than the extension to time-dependent behavior using solutions to the Laplace equation and moving pressure boundaries, which was presented recently. In our current method, the equations have first been transformed using the Laplace transformation. The expressions for the producing wells are combined with auxiliary sources outside the reservoir. The core of the semi-analytic method involves an adjustment of the positions and strengths of these sources in order to approximate the boundary conditions at the reservoir boundaries. The solution that is obtained is transformed back into the time domain using a Stehfest algorithm. The new approach has been validated with numerical tools, including both reservoir simulators and well-test interpretation software. Validations were performed with artificial cases using both single-well and multiple-well production tests. The results of these tests were excellent.

This paper presents a semi-analytical method for the modeling of productivity testing of vertical, horizontal or multilateral wells. The method, which is applicable to both oil and gas reservoirs, automatically accounts for well interference. The use of analytical expressions ensures proper handling of transient short time behavior and semi-steady-state longtime behavior, both close to the well and further into the reservoir. Calculation times are still very limited, in the order of a few minutes down to a few seconds when there are vertical wells only. This makes the tool suitable for well testing evaluation. The approach is based on an earlier derived productivity prediction tool, in which the steady-state equations were solved. It has now been extended to solve the time-dependent diffusion equation and it is thus more rigorous than the extension to time-dependent behavior using solutions to the Laplace equation and moving pressure boundaries, which was presented recently. In our current method, the equations have first been transformed using the Laplace transformation. The expressions for the producing wells are combined with auxiliary sources outside the reservoir. The core of the semi-analytic method involves an adjustment of the positions and strengths of these sources in order to approximate the boundary conditions at the reservoir boundaries. The solution that is obtained is transformed back into the time domain using a Stehfest algorithm. The new approach has been validated with numerical tools, including both reservoir simulators and well-test interpretation software. Validations were performed with artificial cases using both single-well and multiple-well production tests. The results of these tests were excellent.

YÖK Tez ID: 185137 ÖZETİNTEGRAL DÖNÜŞÜMLERİ VE BAZI UYGULAMALARIBİÇER, CenkerKırıkkale ÜniversitesiFen Bilimleri EnstitüsüMatematik Anabilim Dalı, Yüksek Lisans TeziDanışman : Prof. Dr. Kerim KOCAŞubat 2006, 131 sayfaTez üç temel bölümden oluşmaktadır. Birinci bölümde amaç ve konunungelişimi hakkında ön bilgiler verilmiştir. İkinci bölümde integral dönüşümlerihakkında temel bilgiler ve Fourier dönüşümünün temelini oluşturan periyodik olaylariçin Fourier serileri incelenmiştir. Üçüncü bölümde ise özel integral çekirdeklerinesahip Fourier, Laplace ve Mellin dönüşümleri ortaya konulmuştur. Ayrıca bubölümde integral dönüşümlerin fiziksel ve istatistiksel bazı uygulamaları üzerindedurulmuş ve konuya ilişkin örnekler verilmiştir. ABSTRACTINTEGRAL TRANSFORMS AND SOME APPLICATIONSBİÇER, CenkerKırıkkale UniversityGraduate School Of Natural and Applied SciencesDeparment of Mathematics, M. Sc. ThesisSupervisor : Prof. Dr. Kerim KOCAFebruary 2006, 131 pagesThis thesis consists of three chapters. In the first chapter, our objectives arelisted and a short background on the subject is given. In the second chapter, somefundamental knowledge on integral transformations are stated and Fourier series ofperiodic events which are the basis of Fourier transformation are analyzed. In thethird chapter, the transformations those have special integral kernels such as Fourier,Laplace and Mellin transformations are presented. Furthermore some examples onthe physical an statistical applications of the integral transformations are given.Key Words: Integral transformations, Fourier transformations, Laplacetransformations, Mellin transformations, Characteristic function,Distributions, Moments.ii

We consider the integral equation p(t) = ∫ 0 t K(t−τ)p(τ)dτ+r(t) where both K(t) and r(t) behave as exp(αt3) as t → ∞ (α > 0). So straightforward application of the Laplace transform technique is not possible. By introducing a complex parameter the equation is solved in the complex domain. Analytic continuation with respect to this parameter yields the desired solution. For a particular example (which arose in a statistical problem on estimating monotone densities) we describe the construction of the explicit solution of the equation.