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The aim of this paper is to give a good strategy for solving some linear and non-linear partial differential equations in mechanics, physics, engineering and various other technical fields by Modified Reduced Differential Transform Method. In this article we use the method named with Laplace-Padé Reduced Differential Transform Method. This method is obtained by combining Laplace-Padé resummation method, which is a useful technique to find exact solutions, and the Reduced Differential Transform Method. We apply the method to the wave equations and give some examples to see its effectiveness and usefulness. The results and the findings showed that this method leads us to exact solutions with a few iterations or the approximate solutions with small errors.

In Part I of this paper, we proposed a well-posed generalized model for signal enhancement and restoration based on shock filters. A theoretical study of the Cauchy problem in the framework of generalized functions algebra was developed in detail. In Part II, we investigate the numerical aspects of the model. We derive an efficient, explicit numerical scheme in both one and two dimensions, and investigate the schemes' stability and convergence. Through experimental tests, we demonstrate the effectiveness of the numerical schemes when restoring and enhancing signals in various situations with a limited number of iterations. Moreover, we show the impact of the coefficients introduced in the model on the procedure's processing time.

International audience; This paper considers the control design of a nonlinear distributed parameter system in infinite dimension, described by the hyperbolic Partial Differential Equations (PDEs) of de Saint-Venant. The nonlinear system dynamic is formulated by a Multi-Models approach over a wide operating range, where each local model is defined around a set of operating regimes. A new Proportional Integral (PI) feedback is designed and performed through Bilinear Operator Inequality (BOI) and Linear Operator Inequality (LOI) techniques for infinite dimensional systems. The new results have been simulated and also compared to previous results in finite and infinite dimension, in order to illustrate the new theoretical contribution.

This paper presents a class of preconditioners for sparse systems arising from discretized partial differential equations (PDEs). In this class of preconditioners, we exploit the multilevel sequentially semiseparable (MSSS) structure of the system matrix. The off-diagonal blocks of MSSS matrices are of low-rank, which enables fast computations of linear complexity. In order to keep the low-rank property of the off-diagonal blocks, model reduction algorithm is necessary. We tested our preconditioners for 2D convection-diffusion equation, the computational results show the excellent performance of this approach.

Thermal energy or heat storage systems using chemical reaction to store and release energy operates in charging and discharging phases. For charging phase salt-hydrates as chemical components are decomposed by dehydration process with constant heating rate to obtain the anhydrous form and at the same time store the released heat (energy storage). Recently, discussions on thermal decomposition of several salt-hydrates were done (experimentally and numerically) [1,2]. A mathematical model of heat and mass transfer in fixed bed reactor for heat storage is proposed based on a set of partial differential equations (PDEs) controlling the balances or conservations of mass, extend conversion and energy in the reactor. These PDEs are numerically solved by means of the finite element method using Comsol Multiphysics 4.3a. The objective of this paper is to describe an adaptive modeling approach and establish a correct set of PDEs describing the physical system and appropriate parameters for simulating the thermal decomposition process. Kinetic behavior as stated by the international committee of thermal analysis [3] to understand transport phenomena and reactions mechanism in gas and solid phases is taking into account using the generalized Prout-Tompkins equation with modifications based on thermal analysis experiments. The kinetic model is then applied to two thermochemical materials CaCl2•6H2O and MgCl2•6H2O with experimental activation energies and a validation is made with thermogravimetric analysis – differential scanning calorimeter measurement results. This latter result after validation is used in the Comsol model to simulate the lab-scale reactor in charging mode.

This thesis introduces a novel fast explicit diffusion (FED) scheme that uses varying time step sizes to efficiently solve parabolic problems. For the efficient solution of elliptic problems, we embed FED as well as an FED-based solver for linear systems of equations, the Fast-Jacobi method, in a cascadic coarse-to-fine approach. Both FED and Fast-Jacobi are very well-suited for simple, parallel implementations. FED is motivated from a decomposition of box filters, which is a well-known signal processing tool, in terms of explicit schemes for one-dimensional linear diffusion problems. The corresponding time step sizes violate the stability restrictions of usual explicit schemes in up to 50 percent of all cases. Unlike the known Super Time Stepping (STS) approach, it does not require an additional damping parameter due to the derivation based on signal processing. To improve the numerical stability, we present an FED Runge-Kutta scheme and a semi-iterative version of the Fast-Jacobi method. Regarding nonlinear problems, these approaches allow update strategies that can further improve both the accuracy and efficiency. Finally, we show that FED can be combined with extrapolation methods. This yields stable, higher order explicit schemes for the solution of parabolic problems. Diese Arbeit stellt ein neues schnelles explizites Diffusionsschema (FED) vor, das wechselnde Zeitschrittweiten nutzt, um parabolische Probleme effizient zu lösen. Für die effiziente Lösung von elliptischen Problemen bauen wir FED und auch einen auf FED basierenden Löser für lineare Gleichungssysteme, das schnelle Jacobi-Verfahren, in ein kaskadiertes Mehrgitterverfahren ein. Sowohl FED als auch das schnelle Jacobi-Verfahren sind sehr gut für einfache parallele Implementierungen geeignet. FED wird durch eine Zerlegung von aus der Signalverarbeitung bekannten Mittelwertfiltern im Sinne von expliziten Schemata für eindimensionale lineare Diffusionsprobleme motiviert. Die entsprechenden Zeitschrittweiten verletzen die Stabilitätsbedingungen gewöhnlicher expliziter Schemata in bis zu 50 Prozent der Fälle. Anders als das bekannte Superzeitschrittverfahren (STS) benötigt es dank der auf der Signalverarbeitung basierenden Herleitung keinen zusätzlichen Dämpfungsparameter. Zur Verbesserung der numerischen Stabilität präsentieren wir ein FED Runge-Kutta Schema und eine semi-iterative Version des schnellen Jacobi-Verfahrens. Im Bezug auf nichtlineare Probleme erlauben diese Verfahren Updatestrategien, die sowohl die Genauigkeit als auch die Effizienz weiter verbessern können. Schließlich zeigen wir, dass FED mit Extrapolationsverfahren kombiniert werden kann. Dies liefert stabile explizite Schemata höherer Ordnung zur Lösung parabolischer Probleme.

Copyright @ 2011 IEEE. Although the last decade has witnessed a great deal of improvements achieved for the microarray technology, many major developments in all the main stages of this technology, including image processing, are still needed. Some hardware implementations of microarray image processing have been proposed in the literature and proved to be promising alternatives to the currently available software systems. However, the main drawback of those proposed approaches is the unsuitable addressing of the quantification of the gene spot in a realistic way without any assumption about the image surface. Our aim in this paper is to present a new image-reconstruction algorithm using the cellular neural network that solves the Navier–Stokes equation. This algorithm offers a robust method for estimating the background signal within the gene-spot region. The MATCNN toolbox for Matlab is used to test the proposed method. Quantitative comparisons are carried out, i.e., in terms of objective criteria, between our approach and some other available methods. It is shown that the proposed algorithm gives highly accurate and realistic measurements in a fully automated manner within a remarkably efficient time.

From 19.07. to 24.07.2009, the Dagstuhl Seminar 09302 ``New Developments in the Visualization and Processing of Tensor Fields '' was held in Schloss Dagstuhl~--~Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.

This Dagstuhl Seminar was concerned with the visualization and processing of tensor fields, like its two predecessors: seminar 04172 organized by Hans Hagen and Joachim Weickert in April 2004, and the follow-up seminar 07022 in January 2007 with David Laidlaw and Joachim Weickert as organizers. Both earlier meetings were successful, resulting in well received books and triggering fruitful scientific interaction and exchange of experience across interdisciplinary boundaries. We believe that the 2009 seminar will prove to have been equally successful.

Diffusion is a phenomenon in which particles move from regions of higher density to regions of lower density. Many physical systems, in fields as diverse as plant biology and finance, are known to involve diffusion phenomena. Typically, diffusion systems are modeled by partial differential equations (PDEs), which include certain parameters. These parameters characterize a given diffusion system. Therefore, for both modeling and simulation of a diffusion system, one has to either know or determine these parameters. Moreover, as PDEs are infinite order dynamic systems, for computational purposes one has to approximate them by a finite order model. In this thesis, we investigate these two issues of model reduction and parameter estimation by considering certain specific cases of heat diffusion systems. We first address model reduction by considering two specific cases of heat diffusion systems. The first case is a one-dimensional heat diffusion across a homogeneous wall, and the second case is a two-dimensional heat diffusion across a homogeneous rectangular plate. In the one-dimensional case we construct finite order approximations by using some well known PDE solvers and evaluate their effectiveness in approximating the true system. We also construct certain other alternative approximations for the one-dimensional diffusion system by exploiting the different modal structures inherently present in it. For the two-dimensional heat diffusion system, we construct finite order approximations first using the standard finite difference approximation (FD) scheme, and then refine the FD approximation by using its asymptotic limit. As for parameter estimation, we consider the same one-dimensional heat diffusion system, as in model reduction. We estimate the parameters involved, first using the standard batch estimation technique. The convergence of the estimates are investigated both numerically and theoretically. We also estimate the parameters of the one-dimensional heat diffusion system recursively, initially by adopting the standard recursive prediction error method (RPEM), and later by using two different recursive algorithms devised in the frequency domain. The convergence of the frequency domain recursive estimates is also investigated.

This thesis describes initial language extensions to the Modelica language to define a more general language called PDEModelica, with built-in support for modeling with partial differential equations (PDEs). Modelica® is a standardized modeling language for objectoriented, equation-based modeling. It also supports component-based modeling where existing components with modified parameters can be combined into new models. The aim of the language presented in this thesis is to maintain the advantages of Modelica and also add partial differential equation support. Partial differential equations can be defined using a coefficient-based approach, where a predefined PDE is modified by changing its coefficient values. Language operators to directly express PDEs in the language are also discussed. Furthermore, domain geometry description is handled and language extensions to describe geometries are presented. Boundary conditions, required for a complete PDE problem definition, are also handled. A prototype implementation is described as well. The prototype includes a translator written in the relational meta-language, RML, and interfaces to external software such as mesh generators and PDE solvers, which are needed to solve PDE problems. Finally, a few examples modeled with PDEModelica and solved using the prototype are presented. Report code: LiU-Tek-Lic-2002:63.

Partial differential equations (PDEs) with boundary conditions (Dirichlet or Neumann) defined on boundaries with simple geometry hare been successfully treated using sigmoidal multilayer perceptrons in previous works. This article deals,vith the case of complex: boundary geometry, where the boundary is determined by a number of points that belong to it and are closely located, so as to offer a reasonable representation. Two networks are employed: a multilayer perceptron and a radial basis function network. The later is used to account for the exact satisfaction of the boundary conditions, The method has been successfully tested on two-dimensional and three-dimensional PDEs and has yielded accurate results. Ieee Transactions on Neural Networks

Signal enhancement and restoration is one of the fields that make extensive use of PDE theory. More specifically, some authors have proposed successive improved shock filters based on non-linear hyperbolic equations. These models yield satisfactory results; however, a wider range of degrees of freedom when handling the model parameters (coefficients and components) would be of great interest because it would increase the model's efficiency and facilitate adaptation to specific situations. Naturally, the key challenge in proceeding thus is to ensure that the problem remains well-posed. In this paper, we propose a more general shock filter that introduces new parameters to control the shock speed. Interpreting the proposed model in a framework of generalized functions algebra, we prove existence and uniqueness solution results.