Fig 2 - uploaded by Gülnur Çelik Kızılkan
Content may be subject to copyright.
Source publication
In this study, the step size strategies are obtained such that the local error is smaller than the desired error level in the numerical integration of a type of nonlinear equation system in interval [t0,T][t0,T]. The algorithms are given for calculating step sizes and numerical solutions according to these strategies. The algorithms are supported b...
Similar publications
It has long been assumed that over a sufficiently long period of time, changes in catchment water storage (ΔS) is a relatively minor term compared to other fluxes and can be neglected in the catchment water balance equation. However, the validity of this fundamental assumption has rarely been tested and the associated uncertainties in water balance...
This work presents an analysis of the error that is committed upon having obtained the approximate solution of the nonlinear Fredholm-Volterra-Hammerstein integral equation by means of a method for its numerical resolution. The main tools used in the study of the error are the properties of Schauder bases in a Banach space.
For front-force compact tension specimen (FFCT), based on the refined results of the relationship between crack tip opening displacement and load line crack opening displacement from Finite Element Analysis (FEA), the influences of material properties and plastic deformation near the crack tip have been analyzed. A simplified and accurate transform...
The beam hardening (BH) effect can influence medical interpretations in two notable ways. First, high attenuation materials, such as bones, can induce strong artifacts, which severely deteriorate the image quality. Second, voxel values can significantly deviate from the real values, which can lead to unreliable quantitative evaluation results. Some...
In this paper we study the finite element approximation of systems of p-Stokes type for p Є (1, ∞). We derive (in some cases optimal) error estimates for finite element approximation of the velocity and for the pressure in a suitable functional setting. The results are supported by numerical experiments.
Citations
... [12,16]). r * is called the practical parameter for the stepsize which chosen by user small enough [21,22]. With this criterion, less processing is needed and the given method run smoothly. ...
In this study, Schur stability, sensitivity and continuity theorems have been mentioned. In addition, matrix families, interval matrix and extend of the intervals also have been mentioned. The $\mathcal{I}_{\mathcal{L}}$ and $\mathcal{I}_{\mathcal{C}}$ iintervals of the matrix families have been determined so that the linear sums family $\mathcal{L}$ and convex combination family $\mathcal{C}$ are Schur stable. Samely, the $\mathcal{I}^{*}_{\mathcal{L}}$ and $\mathcal{I}_{\mathcal{C}}^{\ast }$ intervals have been determined and $\mathcal{L}$ and $\mathcal{C}$ are $\omega ^{\ast }-$Schur stable. Afterwards, the methods which based on continuity theorems and the algorithms which based on the methods have been given. Extended intervals have been obtained with the help of the methods and the algorithms. All definitions are supported by examples.
... The importance of the choice of step-size cannot be overemphasized in the numerical integration of stiff differential systems [45,46]. In fact, according to study [47], one of the most vital concepts in numerical integration of systems of differential equations is step-size selection because it is not practical to use constant step size in numerical integration. Proper step-size selection enhances accuracy, reduces computation time and minimizes the number of iterations. ...
Over the years, researches have shown that fixed (constant) step-size methods have been efficient in integrating a stiff differential system. It has however been observed that for some stiff differential systems, non-fixed (variable) step-size methods are required for efficiency and for accuracy to be attained. This is because such systems have solution components that decay rapidly and/or slowly than others over a given integration interval. In order to curb this challenge, there is a need to propose a method that can vary the step size within a defined integration interval. This challenge motivated the development of Non-Fixed Step-Size Algorithm (NFSSA) using the La-grange interpolation polynomial as a basis function via integration at selected limits. The NFSSA is capable of integrating highly stiff differential systems in both small and large intervals and is also efficient in terms of economy of computer time. The validation of properties of the proposed algorithm which include order, consistence, zero-stability, convergence, and region of absolute stability were further carried out. The algorithm was then applied to solve some samples mildly and highly stiff differential systems and the results generated were compared with those of some existing methods in terms of the total number of steps taken, number of function evaluation, number of failure/rejected steps, maximum errors, absolute errors, approximate solutions and execution time. The results obtained clearly showed that the NFSSA performed better than the existing ones with which we compared our results including the inbuilt MATLAB stiff solver, ode 15s. The results were also computationally reliable over long intervals and accurate on the abscissae points which they step on.
... It is easy to verify that, in this case, vector-function v(u) (5), defined as (6) v i (u) = u i (v k (u)), i ∈ 1, n, i = k, v k (u) = u −1 k (u), satisfies system (4) ∀u ∈ (u k (t 0 ), u k (t 1 )). As one can notice from the example above, the k-swap operator results in the unknown function u k (·) and the independent variable t being "swapped" (hence, the term), in the sense that in system (4) the former becomes the new independent variable u, whereas the latter turns into a new unknown function v k (u). ...
... As for the latter problem -it also can be pretty much easily fixed by "decimating" the dense regions (i.e., by removing mesh points if they are closer than some acceptable threshold) as well as by generating new mesh points to fill the gaps in sparse regions (via linear or nonlinear interpolation). In general, when refining the mesh (by adding/removing points) one can consider using any of the known step size selection strategies appropriate for the "base" method in hands (see, for example, [6]). The strategies, of course, should be applied with respect to the actual independent variables on each sub-interval. ...
A new approach for solving stiff boundary value problems for systems of ordinary differential equations is presented. Its idea essentially generalizes and extends that from arXiv:1601.04272v8. The approach can be viewed as a methodology framework that allows to enhance "stiffness resistance" of pretty much all the known numerical methods for solving two-point BVPs. The latter is demonstrated on the example of the {\it trapezoidal scheme} with the corresponding C++ source code available at \url{https://github.com/imathsoft/MathSoftDevelopment}.
... Although it is known that the solution of Cauchy problem (1.1) does not exist at t = 4 point, it is seen that the solution of Cauchy problem (1.1) exist at t = 4 point at graphic given in Figure 1.2. The step size strategies given in [5,6,7,8,9,10,11,12] have been produced to overcome aforementioned problems. In this study, an interactive web interface is designed for the step size strategies given in [5,6,7,8,9,10,11,12]. ...
... The step size strategies given in [5,6,7,8,9,10,11,12] have been produced to overcome aforementioned problems. In this study, an interactive web interface is designed for the step size strategies given in [5,6,7,8,9,10,11,12]. In Section 2, preliminaries on the solution of Cauchy problems have been given. ...
... There are many studies about the variable step size strategies in literature (for example, [18,19,24,30]). We have designed a web interface for different step size strategies in studies [5,6,7,8,9,10,11,12]. We let give that strategies now. ...
In this study, it has been designed an interactive web interface which provides the online use of step size strategies to obtain the numerical solutions of Cauchy problems. This web interface has been created by using Django web framework of Python programming language.
The study aims to characterize an interactive and clinically applicable population pharmacokinetic-pharmacodynamic-pharmacodynamic (PK-PD-PD) model describing follicle-stimulating hormone (FSH)-Inhibin B-oocyte relationship in women undergoing assisted reproduction with invitro-fertilization (IVF) or intracytoplasmic sperm injection (ICSI). The study was a prospective analysis of 25 healthy women undergoing IVF/ICSI using GnRH antagonist protocol. The developed model utilized the FSH PK profiles to predict both inhibin B (1st PD endpoint) and the oocyte retrieval (2nd PD endpoint). The modeling framework involved 2 stages; first, the FSH-Inhibin B model was developed by simultaneous approach, and applied to estimate the individual area under the Inhibin B time curve (AUCInhb ) at the end of stimulation cycles that varied in length in each women. At the second stage, the estimated AUCInhb was introduced as a link covariate to predict oocyte retrieval and response category. Population FSH-Inhibin B model was described as three submodels; PK (exogenous), endogenous, and inhibin B PD models. Weight was the main determinant of both endogenous and exogenous FSH exposures. GnRH antagonist therapy was a significant time-varying covariate when tested against endogenous FSH production rate (P<0.001). AUCinhb could be predicted with women's age and weight. Log-transformed AUCInhb was significant covariate when tested against oocyte retrieval (p<0.001). Simulations concluded a target AUCInhb of 144-303 ng.ml/hr for optimal ovarian response. GnRH antagonist was better started on day 7 of the cycle. Covariate based dosing suggests lower rFSH requirements in thin and/or young population. An interactive web application 'GonadGuide' was developed to facilitate the application into the clinical practice. This article is protected by copyright. All rights reserved.
The topic of differential equations is one of the most important subjects in
mathematics and other sciences. There are several methods to solve ordinary
differential equations. However, there are no general methods to solve partial
differential equations. One of the most powerful known tools to solve these equations
is the integral transform method. The multi-variable Laplace transforms can be used
to solve such equations. However, the theory of the multiple-variable Laplace
transform is not completed yet. On the other hand, the fractional calculus is one of
the most rapidly growing field of research. It has been gaining the attention of many
scientists because of the interesting results obtained when these researchers utilized
the fractional derivatives for the sake of modelling real world problems.
Maravall used the Laplace transform method to obtain the explicit solution of
certain of ordinary differential equations with fractional derivatives. Oldham and
Spanier also used the Laplace transform method to solve another type of
homogeneous ordinary differential equations of fractional order. The Laplace
transform was also used by Dorta, Seitkazieva, Miller and Ross to find solutions of
such equations. Many authors like Gerasimov, Fedosov, Yanenko, Biacino,
Miseredino and Fujita attempted to solve partial differential equations with fractional
orders. But no one, to our knowledge attempted to solve fractional order partial
differential equations using the double Laplace transform.
In this work, we establish the double Laplace formulas for the partial fractional
derivatives and apply these formulas to solve a fractional heat equation with certain
initial and boundary conditions.
Analysis of complex failure scenarios and mitigation procedures of an industrial plant is one of the most important activity for the safety of the factory, the personnel and the surrounding areas. The dependability assessment of such systems is fulfilled by risk experts who, adopting well-known Reliability, Availability, Maintenance and Safety (RAMS) techniques, design and solve the stochastic failure model of the system. Traditional techniques like Fault Tree Analysis (FTA) or Reliability Block Diagrams (RBD) are of easy implementation but unrealistic, due to their simplified hypotheses that assume the components malfunction to be independent from each other and from the system working conditions.
Dynamic Probabilistic Risk Assessment (DPRA) is the umbrella framework encompassing new mathematical and simulation formalisms aiming to relax the constraints of traditional techniques and increase the accuracy of dependability assessment. At the state of the art, DPRA cannot boast a well-defined methodology because the nature of a dynamic reliability problem can be so complex to require an ad-hoc modelling and resolution. Moreover, one of the main issues encountered by risk-practitioners is that there is a small support in terms of available tools or expert systems, specifically designed for DPRA problems.
To tackle this lack, this paper presents the conception of general framework for the modelling and the simulation of a Stochastic Hybrid Fault Tree Automaton (SHyFTA), one of the most promising DPRA methodologies, able to combine Dynamic Fault Tree (DFT) with the deterministic model of the system process.
The logic of the repairable DFT gates and the concepts for the implementation of a simulation engine combining Discrete Event Simulation (DES) and Time Driven Simulation (TDS) are illustrated and, a Matlab® toolbox library (SHyFTOO) has been coded and tested with a thorough validation campaign. Finally, a common case study in industrial engineering has been modelled and analyzed under different stand-by configurations in order to demonstrate the modelling flexibility of the toolbox.
Dependability assessment is a crucial activity to ensure the correct operation of complex systems. The output of dependability assessment activities include the quantification of reliability, availability, maintenance and safety related metrics. These metrics can assist in the identification of the system weak points or in the conception of mitigation strategies to increase the system dependability level. The development of advanced computer-aided methodologies to support dependability assessment activities is essential to automate and reduce the efforts implied by this process and similarly, the development of accurate dependability assessment methods is very important to increase the quality of the results. In this context, it is possible to identify different contributions that improve the dependability assessment through general-purpose modeling methodologies. However, existing solutions are ad-hoc applications specified with low-level stochastic formalisms and this complicates their adoption in the industry. Accordingly, this paper presents Stochastic Hybrid Fault Tree Automaton (SHyFTA) based simulation algorithm that allows the accurate dependability analysis of repairable multi-state systems. SHyFTA integrates the stochastic and deterministic operation of the system under study as well as their interactions. The algorithm is formalized through an object-oriented software architecture, which is developed as a software library for the modeling and simulation of repairable SHyFTA models. Following the proposed architecture, a Matlab® implementation of this library, SHyFTOO, has been developed and validated with a thorough test campaign. In order to provide a guideline to the end-users and show the potential of the SHyFTOO library, the case study of a feed-water pumping system is implemented in detail and it is used to evaluate different preventive maintenance policies. The SHyFTOO library can open the way to further investigations that address the interactions between the failure behavior and the functional operation of a system and their combined effect on system dependability.
In this work, a MATLAB program was developed to compute the simulation of the transient stability with variable step-size integration. The program uses the alternating solution method, with the implicit trapezoidal rule for numerical integration. The proposed step-size variation is performed automatically by a heuristic algorithm, allowing a trade-off between precision and computational performance. The results were properly compared with the commercial software ANATEM, in order to validate all the work.
