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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

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2021

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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.

2

Non integer order, discrete, state space model of heat transfer process using Grünwald-Letnikov operator

Oprzędkiewicz K., Dziedzic K., Więckowski Ł.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2019

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Vol. 67, nr 5

905--914

EN

The paper is intented to show a new, state space, discrete, non integer order model of a one-dimensional heat transfer process. The proposed model derives directly from time continuous, state space model and it uses the discrete Grünwald-Letnikov operator to express the fractional order difference with respect to time. Stability and spectrum decomposition for the proposed model are recalled, the accuracy and convergence are analyzed too. The convergence of the proposed model does not depend on parameters of heater and measuring sensors. The dimension of the model assuring stability and predefined rate of convergence and stability is estimated. Analytical results are confirmed by experiments.

3

Remarks on five equivalent forms of the fractional – order backward – difference

Ostalczyk P.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2014

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Vol. 62, nr 2

271--278

EN

In this paper a fractional-order backward-difference/sum (FOBD/S) equivalent formulae are considered. From the Gr¨unwald- Letnikov (GL - FOBD) definition formula and its Horner equivalent form one derives the Riemann-Liouville FOBD (RL - FOBD). Also the Caputo and polynomial-like forms are defined. All forms may be useful in real-time calculations (in the evaluation of digital control strategies) due to the reduction of fractional orders. The investigations are illustrated by a numerical example.

4

Fractional-order backward-difference Grunvald-Letnikov and Horner simplified forms evaluation accuracy analysis

Ostalczyk P., Duch P., Sankowski D.

Automatyka / Akademia Górniczo-Hutnicza im. Stanisława Staszica w Krakowie

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2011

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T. 15, z. 3

443-453

EN

The aim of this paper is to investigate two equivalent forms of the fractional-order backward difference (FOBD) simplified formulae evaluation. The first form is known as the Grunvald-Letnikov form the second one as the Horner form. The simplifications are forced by microprocessor systems requirements. Two simplified forms are analyzed. The investigations are illustrated by numerical examples of simplified FOBD evaluation results.

PL

Celem pracy jest analiza dokładności obliczeń uproszczonych postaci Grunvalda-Letnikowa różnicy wstecznej niecałkowitych rzędów. Podane zostały równoważne definicje obu postaci oraz ich uproszczone formy wynikające z ograniczeń narzuconych przez systemy mikroprocesorowe. Omówiono podstawowe właściwości rozważanych postaci różnicy wstecznej niecałkowitego rzędu. Rozważania ilustrują przykłady obliczeniowe pokazujące dokładność obliczeń obu uproszczonych postaci różnic.