Nagrinėjama antrosios eilės diferencialinė lygtis su skirtingomis potencialo reikšmėmis. / We have the second order equation with different potential.

<p>This thesis presents a method for solving partial differential equations (PDEs) using articial neural networks. The method uses a constrained backpropagation (CPROP) approach for preserving prior knowledge during incremental training for solving nonlinear elliptic and parabolic PDEs adaptively, in non-stationary environments. Compared to previous methods that use penalty functions or Lagrange multipliers,</p><p>CPROP reduces the dimensionality of the optimization problem by using direct elimination, while satisfying the equality constraints associated with the boundary and initial conditions exactly, at every iteration of the algorithm. The effectiveness of this method is demonstrated through several examples, including nonlinear elliptic</p><p>and parabolic PDEs with changing parameters and non-homogeneous terms. The computational complexity analysis shows that CPROP compares favorably to existing methods of solution, and that it leads to considerable computational savings when subject to non-stationary environments.</p><p>The CPROP based approach is extended to a constrained integration (CINT) method for solving initial boundary value partial differential equations (PDEs). The CINT method combines classical Galerkin methods with CPROP in order to constrain the ANN to approximately satisfy the boundary condition at each stage of integration. The advantage of the CINT method is that it is readily applicable to PDEs in irregular domains and requires no special modification for domains with complex geometries. Furthermore, the CINT method provides a semi-analytical solution that is infinitely differentiable. The CINT method is demonstrated on two hyperbolic and one parabolic initial boundary value problems (IBVPs). These IBVPs are widely used and have known analytical solutions. When compared with Matlab's nite element (FE) method, the CINT method is shown to achieve significant improvements both in terms of computational time and accuracy. The CINT method is applied to a distributed optimal control (DOC) problem of computing optimal state and control trajectories for a multiscale dynamical system comprised of many interacting dynamical systems, or agents. A generalized reduced gradient (GRG) approach is presented in which the agent dynamics are described by a small system of stochastic dierential equations (SDEs). A set of optimality conditions is derived using calculus of variations, and used to compute the optimal macroscopic state and microscopic control laws. An indirect GRG approach is used to solve the optimality conditions numerically for large systems of agents. By assuming a parametric control law obtained from the superposition of linear basis functions, the agent control laws can be determined via set-point regulation, such</p><p>that the macroscopic behavior of the agents is optimized over time, based on multiple, interactive navigation objectives.</p><p>Lastly, the CINT method is used to identify optimal root profiles in water limited ecosystems. Knowledge of root depths and distributions is vital in order to accurately model and predict hydrological ecosystem dynamics. Therefore, there is interest in accurately predicting distributions for various vegetation types, soils, and climates. Numerical experiments were were performed that identify root profiles that maximize transpiration over a 10 year period across a transect of the Kalahari. Storm types were varied to show the dependence of the optimal profile on storm frequency and intensity. It is shown that more deeply distributed roots are optimal for regions where</p><p>storms are more intense and less frequent, and shallower roots are advantageous in regions where storms are less intense and more frequent.</p> / Dissertation

In this thesis we are going to study the energy shaping problem on controlled Lagrangian systems with degree of underactuation less than or equal to two. Energy shaping is a method of stabilization by designing a suitable feedback control force on the given controlled Lagrangian system so that the total energy of the feedback equivalent system has a non-degenerate minimum at the equilibrium. The feedback equivalent system can then be stabilized by a further dissipative force. Finding a feedback equivalent system requires solving a system of PDEs. The existence of solutions for this system of PDEs is guaranteed, under some conditions, in the case of one degree of underactuation. Higher degrees of underactuation, however, requires a more careful study on the system of PDEs, and we apply the formal theory of PDEs to achieve this purpose in the case of two degrees of underactuation. The thesis is divided into four chapters. First, we review the basic notion of energy shaping and state the results for the case of one degree of underactuation. We then devise a general scheme to solve the energy shaping problem with degree of underactuation equal to one, together with some examples to illustrate the general procedure. After that we review the tools from the formal theory of PDEs, as a preparation for solving the problem with two degrees of underactuation. We derive an equivalent involutive system of PDEs from which we can deduce the existence of solutions which suit the energy shaping requirement.

The theory of time scales is introduced by Stefan Hilger in his PhD thesis in 1998 in order to unify continuous and discrete analysis. In our thesis, by making use of the time scale calculus we study the oscillation of nonlinear matrix differential equations of second order. the first chapter is introductory in nature and contains some basic definitions and tools of the time scales calculus, while certain well-known results have been presented with regard to oscillation of the solutions of second order matrix equations and some new oscillation criteria for the same type equations have been established in the second chapter.

In this thesis, we study inverse problems of restoration of the unknown function in a boundary condition, where on the boundary of the domain there is a convective heat exchange with the environment. Besides the temperature of the domain, we seek either the temperature of the environment in Problem I and II, or the coefficient of external boundary heat emission in Problem III and IV. An additional information is given, which is the overdetermination condition, either on the boundary of the domain (in Problem III and IV) or on a time interval (in Problem I and II). If solution of inverse problem exists, then the temperature can be defined everywhere on the domain at all instants. The thesis consists of six chapters. In the first chapter, there is the introduction where the definition and applications of inverse problems are given and definition of the four inverse problems, that we will analyze in this thesis, are stated. In the second chapter, some definitions and theorems which we will use to obtain some conclusions about the corresponding direct problem of our four inverse problems are stated, and the conclusions about direct problem are obtained. In the third, fourth, fifth and sixth chapters we have the analysis of inverse problems I, II, III and IV, respectively.

Am Beispiel der Berechnung von Schockantwortspektren werden in dem Vortrag verschiedene Möglichkeiten aufgezeigt, um mit Mathcad den linearen Oszillator zu simulieren. Neben dem klassischen Lösungsblock von Mathcad wird gezeigt, wie sich auch in Kommandozeilenbefehlen Parameter für die transiente Differenzialgleichung unterbringen lassen. Der Vortrag zeigt, wie Schockspektren für beliebige Anregungen ermittelt werden können, indem die transiente Differezialgleichung für verschiedene Eigenfrequenzen aber für die identische Anregung berechnet wird. Neben der rein numerischen Simulation wird auch ein Ansatz vorgestellt, der sich direkt auf die Variation der Konstanten stützt.

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2010-02-19Bitstream added on 2014-06-13T18:06:55Z : No. of bitstreams: 1 pavan_jf_me_sjrp.pdf: 525723 bytes, checksum: 14295e01658745f42b4e6dd2b22c1791 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / No presente trabalho consideramos o sistema de equações diferenciais ordinároas x1 = afλ 1 (x1)+ bfµ 2 (x2) ˙ x2 = cfη 1 (x1)+ dfζ 2 (x2) (I) onde a,b,c e d são coeficientes constantes, λ, ,η e ζ são números racionais positivos numeradores e denominadores ímpares, as funções fi :(−h,h) → R, h> 0, são contínuas e satisfazem as condições fi(0)=0,i =1, 2e xifi(xi) > 0,para xi =0,i =1, 2. Associado ao sistema(I) consideramos a seguinte função V = α Z x1 0 fξ 1 (τ )dτ + Z x2 0 fθ 2 (τ )dτ, (II) onde ξ e θ são número racionais numeradores e denominadores ímpares. Nosso objetivo principal é encontar é encontrar sob quais condições dos parâmetros a,b,c,d e α> 0 a função V definidaem(II) é uma função de Liapunov estita para a solução nula dos sitema (I), o que leva a concluir a estabilidade assintótica da solução nula. / In this work we consider the system of ordinary differential equations x1 = afλ 1 (x1)+ bfµ 2 (x2) ˙ x2 = cfη 1 (x1)+ dfζ 2 (x2) (I) where a,b,c and d are constantco efficients, λ, ,η and ζ a repositive rational numbers with odd numerators and denominators ,and the functions fi :(−h,h) → R, h> 0,are continuous and satisfy the conditions fi(0)=0,i =1, 2and xifi(xi) > 0,for xi =0,i = 1, 2. Associated to the system(I) we consider the following function V = α Z x1 0 fξ 1 (τ )dτ + Z x2 0 fθ 2 (τ )dτ, (II) where ξ and θ are positive rational numbers with odd numerators and denominators and α is a positive constant. Our main goal is find under what conditions the parameters a,b,c,d and α> 0 the function V defined in(II) is a strict Liapunov function for the zero solution of the system (I), which leads us to conclude the asymptotic stability of zero solution.

Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2011-01-28Bitstream added on 2014-06-13T20:35:09Z : No. of bitstreams: 1 barros_aap_me_sjrp.pdf: 459604 bytes, checksum: 0a23c4af2e8f9afe3807f0dd603a1237 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo desse trabalho é estudar problemas de valor de contorno com três pontos lineares e não ineares, também conhecidos como problemas não Isto é feito, usando as funções de Green, usadas para resolver problemas de valor de contorno com dois pontos. / The aim of this work is to study boundary value problems with three points also known as non-classical problems. This is done using the Green's functions, which are used to solve two-point boundary value problems.

Submitted by Wesley Felipe Ferreira Mora Gil null (wes_moragil@hotmail.com) on 2018-04-03T13:14:51Z No. of bitstreams: 1 dissertacao weley versao final.pdf: 3054931 bytes, checksum: 321569a638824c9d0f4d6a48a3d1dee2 (MD5) / Approved for entry into archive by ROSANGELA APARECIDA LOBO null (rosangelalobo@btu.unesp.br) on 2018-04-05T13:02:04Z (GMT) No. of bitstreams: 1 gil_wffm_me_bot.pdf: 3054931 bytes, checksum: 321569a638824c9d0f4d6a48a3d1dee2 (MD5) / Made available in DSpace on 2018-04-05T13:02:04Z (GMT). No. of bitstreams: 1 gil_wffm_me_bot.pdf: 3054931 bytes, checksum: 321569a638824c9d0f4d6a48a3d1dee2 (MD5) Previous issue date: 2018-02-16 / Câncer pode ser definido como um crescimento desordenado de células que não permanecem em uma região limitada, invadindo outros tecidos e órgãos. Indicadores mostram que a mortalidade por câncer vem aumentando, por esse motivo é imprescindível a busca por novos tratamentos. A imunoterapia surge como uma modalidade de tratamento promissora, a qual utiliza-se do sistema imunológico no combate ao câncer. Outra tendência na oncologia é a combinação de diferentes modalidades de tratamentos. Neste trabalho, propomos um modelo matemático de equações diferenciais ordinárias, com o intuito de analisar como o tratamento imunoterápico e quimioterápico podem auxiliar um ao outro no tratamento do câncer. Utilizamos um software matemático para a construção dos retratos de fase e o método Runge-Kutta de quarta ordem para as simulações numéricas. As simulações mostraram que a imunoterapia e a quimioterapia podem levar à eliminação das células e uma sobrevida maior após o tratamento. É exibido também que citotoxicidade da quimioterapia é fundamental para o sucesso do tratamento. / Cancer may be defined as a uncontrolled growth of cells that do not remain in a limited region, invading other tissues and organs. Indicators show that mortality from cancer is increasing, so the search for new treatments is essential. Immunotherapy appears as a promising treatment modality, which uses the immunological system in the fight against cancer. Another trend in oncology is the combination of different treatment modalities. In this work, we propose a mathematical model of ordinary differential equations, in order to analyze how immunotherapeutic and chemotherapeutic treatment can help one another reciprocally. We use a mathematical software for the construction of the phase portraits and the method fourth-order Runge-Kutta for numerical simulations. The simulations have shown indications that immunotherapy may assist the chemotherapy by causing cure or by allowing a longer overlife after treatment. It is also shown that cytotoxicity of chemotherapy is critical to successful treatment.

Orientador: Paulo Fernando de Arruda Mancera / Resumo: Câncer pode ser definido como um crescimento desordenado de células que não permanecem em uma região limitada, invadindo outros tecidos e órgãos. Indicadores mostram que a mortalidade por câncer vem aumentando, por esse motivo é imprescindível a busca por novos tratamentos. A imunoterapia surge como uma modalidade de tratamento promissora, a qual utiliza-se do sistema imunológico no combate ao câncer. Outra tendência na oncologia é a combinação de diferentes modalidades de tratamentos. Neste trabalho, propomos um modelo matemático de equações diferenciais ordinárias, com o intuito de analisar como o tratamento imunoterápico e quimioterápico podem auxiliar um ao outro no tratamento do câncer. Utilizamos um software matemático para a construção dos retratos de fase e o método Runge-Kutta de quarta ordem para as simulações numéricas. As simulações mostraram que a imunoterapia e a quimioterapia podem levar à eliminação das células e uma sobrevida maior após o tratamento. É exibido também que citotoxicidade da quimioterapia é fundamental para o sucesso do tratamento. / Abstract: Cancer may be defined as a uncontrolled growth of cells that do not remain in a limited region, invading other tissues and organs. Indicators show that mortality from cancer is increasing, so the search for new treatments is essential. Immunotherapy appears as a promising treatment modality, which uses the immunological system in the fight against cancer. Another trend in oncology is the combination of different treatment modalities. In this work, we propose a mathematical model of ordinary differential equations, in order to analyze how immunotherapeutic and chemotherapeutic treatment can help one another reciprocally. We use a mathematical software for the construction of the phase portraits and the method fourth-order Runge-Kutta for numerical simulations. The simulations have shown indications that immunotherapy may assist the chemotherapy by causing cure or by allowing a longer overlife after treatment. It is also shown that cytotoxicity of chemotherapy is critical to successful treatment. / Mestre