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1
Content available The discrete, fractional order model of a two-dimensional temperature field using Grünwald-Letnikov definition
Oprzędkiewicz Krzysztof
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2024
|
Vol. 72, nr 4
art. no. e150332
EN
In the paper a new, fractional order, discrete model of a two-dimensional temperature field is addressed. The proposed model uses Grünwald-Letnikov definition of the fractional operator. Such a model has not been proposed yet. Elementary properties of the model: practical stability, accuracy and convergence are analysed. Analytical conditions of stability and convergence are proposed and they allow to estimate the orders of the model. Theoretical considerations are validated using exprimental data obtained with the use of a thermal imaging camera. Results of analysis supported by experiments point that the proposed model assures good accuracy and convergence for low order and relatively short memory length.
2
Content available The positivity of the fractional order model of a two-dimensional temperature field
Oprzędkiewicz Krzysztof
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2023
|
Vol. 71, nr 4
art. no. e145675
EN
The paper presents analysis of the positivity for a two-dimensional temperature field. The process under consideration is described by the linear, infinite-dimensional, noninteger order state equation. It is derived from a two-dimensional parabolic equation with homogenous Neumann boundary conditions along all borders and homogenous initial condition. The form of control and observation operators is determined by the construction of a real system. The internal and external positivity of the model are associated to the localization of heater and measurement. It has been proven that the internal positivity of the considered system can be achieved by the proper selection of attachment of a heater and place of a measurement as well as the dimension of the finite-dimensional approximation of the considered model. Conditions of the internal positivity associated with construction of real experimental system are proposed. The postivity is analysed separately for control and output of the system. This allows one to analyse the positivity of thermal systems without explicit control. Theoretical considerations are numerically verified with the use of experimental data. The proposed results can be applied i.e. to point suitable places for measuring of a temperature using a thermal imaging camera.
3
Content available Fractional order, discrete model of heat transfer process using time and spatial Grünwald-Letnikov operator
Oprzędkiewicz Krzysztof
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2021
|
Vol. 69, nr 1
art. no. e135843
EN
In the paper a new, state space, fully discrete, fractional model of a heat transfer process in one dimensional body is addressed. The proposed model derives directly from fractional heat transfer equation. It employes the discrete Grünwald-Letnikov operator to express the fractional order differences along both coordinates: time and space. The practical stability and numerical complexity of the model are analysed. Theoretical results are verified using experimental data.
4
Content available Fractional discrete-continuous model of heat transfer process
Oprzędkiewicz Krzysztof, Dziedzic Klaudia
Archives of Control Sciences
|
2021
|
Vol. 31, No. 2
287--306
EN
The paper proposes a new, state space, finite dimensional, fractional order model of a heat transfer in one dimensional body. The time derivative is described by Caputo operator. The second order central difference describes the derivative along the length. The analytical formulae of the model responses are proved. The stability, convergence, and positivity of the model are also discussed. Theoretical results are verified by experiments.
5
Content available Non integer order, state space model of heat transfer process using Atangana-Baleanu operator
Oprzędkiewicz K.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2020
|
Vol. 68, nr 1
43--50
EN
In the paper a new, state space, non integer order model of an one-dimensional heat transfer process is proposed. The model uses a new operator with Mittag-Leffler kernel, proposed by Atangana and Beleanu. The non integer order spatial derivative is expressed by Riesz operator. Analytical formula of the step response is given, the convergence of the model is discussed too. Theoretical results are verified by experiments.
6
Content available A memory-efficient noninteger-order discrete–time state–space model of a heat transfer process
Oprzędkiewicz K., Mitkowski W.
International Journal of Applied Mathematics and Computer Science
|
2018
|
Vol. 28, no. 4
649--659
EN
A new, state space, discrete-time, and memory-efficient model of a one-dimensional heat transfer process is proposed. The model is derived directly from a time-continuous, state-space semigroup one. Its discrete version is obtained via a continuous fraction expansion method applied to the solution of the state equation. Fundamental properties of the proposed model, such as decomposition, stability, accuracy and convergence, are also discussed. Results of experiments show that the model yields good accuracy in the sense of the mean square error, and its size is significantly smaller than that of the model employing the well-known power series expansion approximation.
7
Content available Non integer order, state space model of heat transfer process using Caputo-Fabrizio operator
Oprzędkiewicz K.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2018
|
Vol. 66, nr 3
249--255
EN
The paper is intended to show a new state space, non integer order model of an one-dimensional heat transfer process. The proposed model derives directly from time continuous, state space semigroup model. The fractional order derivative with respect to time is by a new operator proposed by Caputo and Fabrizio, the non integer order spatial derivative is expressed by Riesz operator. The Caputo-Fabrizio operator can be directly implementated using MATLAB, because it does not require us to apply any approximation. Analytical formulae of step response are given, the system decomposition was discussed also. Main results from the paper show that the use of Caputo Fabrizio operator allows us to obtain the simple in implementation and analysis model of the considered heat transfer process. The accuracy of the proposed model in the sense of a MSE cost function is satisfying.
8
Content available Modeling heat distribution with the use of a non-integer order, state space model
Oprzędkiewicz K., Gawin E., Mitkowski W.
International Journal of Applied Mathematics and Computer Science
|
2016
|
Vol. 26, no. 4
749--756
EN
A new, state space, non-integer order model for the heat transfer process is presented. The proposed model is based on a Feller semigroup one, the derivative with respect to time is expressed by the non-integer order Caputo operator, and the derivative with respect to length is described by the non-integer order Riesz operator. Elementary properties of the state operator are proven and a formula for the step response of the system is also given. The proposed model is applied to the modeling of temperature distribution in a one dimensional plant. Results of experiments show that the proposed model is more accurate than the analogical integer order model in the sense of the MSE cost function.
9
Content available remote Natural convection numerical model based on GFD method
Kulawik A.
Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej
|
2009
|
Vol. 8, nr 1
91-95
EN
Classical finite difference method (FDM) is typically used for solving Navier-Stokes (N-S) equation. However to obtain a solution on irregular grid of points other methods have to be applied. Generalized finite difference method (GDFM) is one of the methods that my be used to solve mentioned problem. In this paper N-S and heat transfer equation have been solved using the GFDM. Results of numerical solutions for cooling processes with convection move in two-dimensional region are presented.
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