scholarly journals A verified method for solving piecewise smooth initial value problems

2013 ◽  
Vol 23 (4) ◽  
pp. 731-747 ◽  
Author(s):  
Ekaterina Auer ◽  
Stefan Kiel ◽  
Andreas Rauh
Keyword(s):  
Initial Value Problems ◽  
System Model ◽  
Future Research ◽  
Initial Value ◽  
Smooth Functions ◽  
Solution Processes ◽  
Right Hand ◽  
Future Research Directions ◽  
The Right ◽  
Derivatives Of

Abstract In many applications, there is a need to choose mathematical models that depend on non-smooth functions. The task of simulation becomes especially difficult if such functions appear on the right-hand side of an initial value problem. Moreover, solution processes from usual numerics are sensitive to roundoff errors so that verified analysis might be more useful if a guarantee of correctness is required or if the system model is influenced by uncertainty. In this paper, we provide a short overview of possibilities to formulate non-smooth problems and point out connections between the traditional non-smooth theory and interval analysis. Moreover, we summarize already existing verified methods for solving initial value problems with non-smooth (in fact, even not absolutely continuous) right-hand sides and propose a way of handling a certain practically relevant subclass of such systems. We implement the approach for the solver VALENCIA-IVP by introducing into it a specialized template for enclosing the first-order derivatives of non-smooth functions. We demonstrate the applicability of our technique using a mechanical system model with friction and hysteresis. We conclude the paper by giving a perspective on future research directions in this area.

scholarly journals A New Gronwall–Bellman Inequality in Frame of Generalized Proportional Fractional Derivative

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 747 ◽  
Author(s):  
Jehad Alzabut ◽  
Weerawat Sudsutad ◽  
Zeynep Kayar ◽  
Hamid Baghani
Keyword(s):  
Initial Data ◽  
Fractional Derivative ◽  
Initial Value Problems ◽  
Fractional Derivatives ◽  
Initial Value ◽  
Weighted Function ◽  
Fractional Derivatives And Integrals ◽  
Right Hand ◽  
The Right

New versions of a Gronwall–Bellman inequality in the frame of the generalized (Riemann–Liouville and Caputo) proportional fractional derivative are provided. Before proceeding to the main results, we define the generalized Riemann–Liouville and Caputo proportional fractional derivatives and integrals and expose some of their features. We prove our main result in light of some efficient comparison analyses. The Gronwall–Bellman inequality in the case of weighted function is also obtained. By the help of the new proposed inequalities, examples of Riemann–Liouville and Caputo proportional fractional initial value problems are presented to emphasize the solution dependence on the initial data and on the right-hand side.

scholarly journals Resolvent of nonautonomous linear delay functional differential equations

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
Joël Blot ◽  
Mamadou I. Koné
Keyword(s):  
Continuous Function ◽  
Differential Equations ◽  
Differentiable Function ◽  
Functional Differential Equations ◽  
Initial Value ◽  
Complete Proof ◽  
Right Hand ◽  
Linear Delay ◽  
Functional Differential ◽  
The Right

AbstractThe aim of this paper is to give a complete proof of the formula for the resolvent of a nonautonomous linear delay functional differential equations given in the book of Hale and Verduyn Lunel [9] under the assumption alone of the continuity of the right-hand side with respect to the time,when the notion of solution is a differentiable function at each point, which satisfies the equation at each point, and when the initial value is a continuous function.

Approximations by multivariate perturbed neural network operators

2017 ◽  
Vol 15 (03) ◽  
pp. 413-432 ◽  
Author(s):  
George A. Anastassiou
Keyword(s):  
Neural Network ◽  
Rate Of Convergence ◽  
Activation Function ◽  
Jackson Type ◽  
Partial Derivatives ◽  
Right Hand ◽  
Hidden Layer ◽  
The Right ◽  
Derivatives Of

This article deals with the determination of the rate of convergence to the unit of each of three newly introduced here multivariate perturbed normalized neural network operators of one hidden layer. These are given through the multivariate modulus of continuity of the involved multivariate function or its high-order partial derivatives and that appears in the right-hand side of the associated multivariate Jackson type inequalities. The multivariate activation function is very general, especially it can derive from any multivariate sigmoid or multivariate bell-shaped function. The right-hand sides of our convergence inequalities do not depend on the activation function. The sample functionals are of multivariate Stancu, Kantorovich and quadrature types. We give applications for the first partial derivatives of the involved function.

scholarly journals A Family of Modified Even Order Bernoulli-Type Multiquadric Quasi-Interpolants with Any Degree Polynomial Reproduction Property

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Ruifeng Wu ◽  
Huilai Li ◽  
Tieru Wu
Keyword(s):  
Initial Value Problems ◽  
Bernoulli Polynomials ◽  
Interpolation Scheme ◽  
Degree Polynomial ◽  
Initial Value ◽  
Linear Combinations ◽  
Shape Preserving ◽  
Runge Kutta Method ◽  
Even Order ◽  
Derivatives Of

By using the polynomial expansion in the even order Bernoulli polynomials and using the linear combinations of the shifts of the functionf(x)(x∈ℝ)to approximate the derivatives off(x), we propose a family of modified even order Bernoulli-type multiquadric quasi-interpolants which do not require the derivatives of the function approximated at each node and can satisfy any degree polynomial reproduction property. Error estimate indicates that our operators could provide the desired precision by choosing a suitable shape-preserving parametercand a nonnegative integerm. Numerical comparisons show that this technique provides a higher degree of accuracy. Finally, applying our operators to the fitting of discrete solutions of initial value problems, we find that our method has smaller errors than the Runge-Kutta method of order 4 and Wang et al.’s quasi-interpolation scheme.

Marketing Educational Programs through Technology and the Right Philosophies

2011 ◽  
Vol 1 (1) ◽  
pp. 50-59 ◽  
Author(s):  
Victor X. Wang
Keyword(s):  
Web 2.0 ◽  
Educational Programs ◽  
Academic Departments ◽  
Real Case ◽  
Future Research ◽  
Web 2.0 Technologies ◽  
Research Directions ◽  
Different Types ◽  
Future Research Directions ◽  
The Right

This article uses a real case to illustrate that marketing educational programs through technology is affected by one’s philosophies positively or negatively, depending on the philosophies adopted. Seven philosophies are discussed in relationship to marketing educational programs via technology. Connections between the seven philosophies and different types of universities/colleges are drawn. Future research directions also revolve around marketing educational programs, Web 2.0 technologies and one’s philosophies. It is recommended that universities/colleges seriously consider hiring those administrators who have adopted the right philosophies as the wrong philosophies will work against the mission of certain academic departments, hence the whole university or college.

Nanocomposites in the Food Packaging Industry

2021 ◽  
pp. 122-146
Author(s):  
Dheeraj Kumar ◽  
Md. Farrukh ◽  
Nadeem Faisal
Keyword(s):  
Food Packaging ◽  
Food Product ◽  
Barrier Properties ◽  
Future Research ◽  
Nano Sensors ◽  
Future Research Directions ◽  
Quality Of Food ◽  
Packaging Industry ◽  
Derivatives Of

The recent innovations in nanomaterials for the food packaging industry over the conventional food packaging material have made for a better quality of food product. The use of biodegradable materials is environmentally friendly and suitable for maintaining the quality of food. The chapter focuses on nano-composite materials that enhance the antimicrobial, mechanical, thermal, as well as barrier properties against the migrating element in the food packaging system. Bio-composite derivatives such as PLA, PCL, starch and cellulose, protein derivatives of nanocomposite materials have also been discussed in the chapter along with nano-sensors. Aspects of safety for human and environments and need for regulations of hazard assessment for safety purpose for the food packaging have also been discussed in this chapter. The chapter concludes by discussing the use of nanomaterials applications in food packaging for developing countries, forming some conclusions and leaving readers with thoughts for future research directions.

An Integrated Model of Success in IT Outsourcing Relationships

2014 ◽  
pp. 142-167
Author(s):  
Francois Duhamel ◽  
Isis Gutiérrez-Martínez ◽  
Sergio Picazo-Vela ◽  
Luis Felipe Luna-Reyes
Keyword(s):  
Knowledge Sharing ◽  
Empirical Test ◽  
It Outsourcing ◽  
Future Research ◽  
Organizational Boundaries ◽  
Factors Affecting ◽  
The Public ◽  
Future Research Directions ◽  
The Right ◽  
Outsourcing Contracts

Possible remedies for the failure of IT outsourcing in the public sector include the improvement of knowledge-sharing processes over organizational boundaries between partners, who may learn more about the problems that occur while looking at possible solutions together. Ensuring the right flow of knowledge in the two directions is central to the success of IT outsourcing operations. However, these solutions do not fully acknowledge the different interrelationships between the main factors affecting knowledge transfer in outsourcing relationships in a dynamic way. In this chapter, the authors apply previous research on modeling knowledge-sharing across boundaries to IT outsourcing contracts during the transition phase where both partners initiate an IT outsourcing relationship. Simulation experiments suggest that four reinforcing processes play key roles in the progress of the outsourcing relationship: trust, outsourcers' and providers´ knowledge, commitment, and interfacing. The authors propose future research directions to conduct empirical test of the conceptual model in the context of the Mexican Public Administration.

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