scholarly journals Rosenbrock Type Methods for Solving Non-Linear Second-Order in Time Problems

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2225
Author(s):  
Maria Jesus Moreta
Keyword(s):  
Computational Cost ◽  
Intermediate Stage ◽  
Second Order ◽  
First Order ◽  
New Class ◽  
Nyström Methods ◽  
Non Linear ◽  
Second Order In Time ◽  
Linear Problems ◽  
Runge Kutta

In this work, we develop a new class of methods which have been created in order to numerically solve non-linear second-order in time problems in an efficient way. These methods are of the Rosenbrock type, and they can be seen as a generalization of these methods when they are applied to second-order in time problems which have been previously transformed into first-order in time problems. As they also follow the ideas of Runge–Kutta–Nyström methods when solving second-order in time problems, we have called them Rosenbrock–Nyström methods. When solving non-linear problems, Rosenbrock–Nyström methods present less computational cost than implicit Runge–Kutta–Nyström ones, as the non-linear systems which arise at every intermediate stage when Runge–Kutta–Nyström methods are used are replaced with sequences of linear ones.

A new class of second-order non-linear optical material: stilbazolium benzimidazolate covalently bound to polymer backbone

1997 ◽  
Vol 7 (8) ◽  
pp. 1389-1393 ◽  
Author(s):  
Nobukatsu Nemoto ◽  
Jiro Abe ◽  
Fusae Miyata ◽  
Yasuo Shirai ◽  
Yu Nagase
Keyword(s):  
Second Order ◽  
Optical Material ◽  
Polymer Backbone ◽  
New Class ◽  
Non Linear ◽  
Linear Optical ◽  
Covalently Bound

scholarly journals A Class of Explicit Integrators with o-grid Interpolation for Solving Non-linear Systems of First Order ODEs

2020 ◽  
pp. 106-118
Author(s):  
U. W. Sirisena ◽  
S. I. Luka ◽  
S. Y. Yakubu
Keyword(s):  
Research Work ◽  
Stability And Convergence ◽  
Initial Value ◽  
First Order ◽  
Non Linear ◽  
Improved Stability ◽  
Linear Problems ◽  
The Stability ◽  
Stability And Accuracy ◽  
Grid Interpolation

This research work is aimed at constructing a class of explicit integrators with improved stability and accuracy by incorporating an off-gird interpolation point for the purpose of making them effcient for solving stiff initial value problems. Accordingly, continuous formulations of a class of hybrid explicit integrators are derived using multi-step collocation method through matrix inversion technique, for step numbers k = 2; 3; 4: The discrete schemes were deduced from their respective continuous formulations. The stability and convergence analysis were carried out and shown to be A(α)-stable and convergent respectively. The discrete schemes when implemented as block integrators to solve some non-linear problems, it was observed that the results obtained compete favorably with the MATLAB ode23 solver.

scholarly journals Smart Core and Surface Temperature Estimation Techniques for Health-conscious Lithium-ion Battery Management Systems: A Model-to-Model Comparison

2021 ◽  
Author(s):  
Sumukh Surya ◽  
Akash Samanta ◽  
Sheldon Williamson
Keyword(s):  
Surface Temperature ◽  
Computational Cost ◽  
Lithium Ion ◽  
Second Order ◽  
Battery Management ◽  
Order Model ◽  
Temperature Estimation ◽  
Thermal Models ◽  
First Order ◽  
Estimation Scheme

Estimation of core and surface temperature is one of the crucial functionalities of the lithium-ion Battery Management System (BMS) towards providing effective thermal management, fault detection and operational safety. While, it is impractical to measure core temperature using physical sensors, implementing a complex estimation strategy in on-board low-cost BMS is challenging due to high computational cost and the cost of implementation. Typically, a temperature estimation scheme consists of a heat generation model and a heat transfer model. Several researchers have already proposed ranges of thermal models having different levels of accuracy and complexity. Broadly, there are first-order and second-order heat capacitor-resistor-based thermal models of lithium-ion batteries (LIBs) for core and surface temperature estimation. This paper deals with a detailed comparative study between these two models using extensive laboratory test data and simulation study to access suitability in online prediction and onboard BMS. The aim is to guide whether it’s worth investing towards developing a second-order model instead of a first-order model with respect to prediction accuracy considering modelling complexity, experiments required and the computational cost. Both the thermal models along with the parameter estimation scheme are modelled and simulated using MATLAB/Simulink environment. Models are validated using laboratory test data of a cylindrical 18650 LIB cell. Further, a Kalman Filter with appropriate process and measurement noise levels are used to estimate the core temperature in terms of measured surface and ambient temperatures. Results from the first-order model and second-order models are analyzed for comparison purposes.

scholarly journals Numerical Solution of Second-Order Fuzzy Differential Equation Using Improved Runge-Kutta Nystrom Method

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Faranak Rabiei ◽  
Fudziah Ismail ◽  
Ali Ahmadian ◽  
Soheil Salahshour
Keyword(s):  
Differential Equation ◽  
Computational Cost ◽  
Second Order ◽  
Fuzzy Differential Equation ◽  
Nyström Method ◽  
Numerical Examples ◽  
Time Consumption ◽  
Nystrom Method ◽  
Fuzzy Function ◽  
Runge Kutta

We develop the Fuzzy Improved Runge-Kutta Nystrom (FIRKN) method for solving second-order fuzzy differential equations (FDEs) based on the generalized concept of higher-order fuzzy differentiability. The scheme is two-step in nature and requires less number of stages which leads to less number of function evaluations in comparison with the existing Fuzzy Runge-Kutta Nystrom method. Therefore, the new method has a lower computational cost which effects the time consumption. We assume that the fuzzy function and its derivative are Hukuhara differentiable. FIRKN methods of orders three, four, and five are derived with two, three, and four stages, respectively. The numerical examples are given to illustrate the efficiency of the methods.

Semi-Analytical Solution of a Non-Linear Heat Transfer Model for a Tubular Co-Current Moving-Bed Reactor with a First-Order Chemical Reaction in the Solid Phase

2021 ◽  
Author(s):  
Lucas Ivan de Souza Vereza Medeiros ◽  
Sávio Leandro Bertoli ◽  
Marcel Jefferson Gonçalves ◽  
Tuany Gabriela Hoffmann ◽  
Betina Louise Angioletti ◽  
...  
Keyword(s):  
Heat Transfer ◽  
Chemical Reaction ◽  
Solid Phase ◽  
Transfer Model ◽  
Moving Bed ◽  
First Order ◽  
Order Chemical Reaction ◽  
Non Linear ◽  
Linear Problems ◽  
First Order Chemical Reaction

Abstract The development of mathematical models plays a fundamental role in the design, optimization and control of processes. Regarding heat transfer in moving bed reactors, the chemical reaction implies in the inclusion of a non-homogeneous and non-linear term in model equations, making the analytical integration a very difficult task. Up to date, there is not an analytic and/or a semi-analytic solution to a heat transfer model of a moving bed reactor (MBR) with isothermal walls to distributed parameter in the solid phase. Therefore, starting from analytical solutions of the associated homogeneous (linear) problems and through the spectral expansion of the non-homogeneous vector, this work presents strategies for determining semi-analytical solutions of non-homogeneous and non-linear problems. A MBR with a first-order chemical reaction in the solid phase - kaolinite dehydroxylation in the kaolinite flash calcination process - is selected as the case study; however, the strategies can easily be applied to other non-linear models. Results for conversion, and fluid and particle temperatures, are given for different parameter values. The solutions perform stable, fast and accurate. When compared with a hybrid Finite Difference and Finite Analytic (FD\&FA) numerical method, the solution showed a very good agreement.

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