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1
Content available Solution of fractional integro-differential equations using least squares and shifted legendre methods
Mahdy Amr M.S., Nagdy Abbas S., Mohamed Doaa Sh.
Journal of Applied Mathematics and Computational Mechanics
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2024
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Vol. 23, nr 1
59-70
EN
In recent years, fractional calculus (FC) has filled in a hole in traditional calculus in terms of the effect of memory, which lets us know things about the past and present and guess what will happen in the future. It is very important to have this function, especially when studying biological models and integral equations. This paper introduces developed mathematical strategies for understanding a direct arrangement of fractional integro-differential equations (FIDEs). We have presented the least squares procedure and the Legendre strategy for discussing FIDEs. We have given the form of the Caputo concept fractional order operator and the properties. We have presented the properties of the shifted Legendre polynomials. We have shown the steps of the technique to display the solution. Some test examples are given to exhibit the precision and relevance of the introduced strategies. Mathematical outcomes show that this methodology is a comparison between the exact solution and the methods suggested. To show the theoretical results gained, the simulation of suggested strategies is given in eye-catching figures and tables. Program Mathematica 12 was used to get all of the results from the techniques that were shown.
2
Content available Analysis of thin-walled beams with variable monosymmetric cross section by means of Legendre polynomials
Szybiński Józef, Ruta Piotr
Studia Geotechnica et Mechanica
|
2019
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Vol. 41, nr 1
1--12
EN
This article deals with the vibrations of a nonprismatic thin-walled beam with an open cross section and any geometrical parameters. The thin-walled beam model presented in this article was described using the membrane shell theory, whilst the equations were derived based on the Vlasov theory assumptions. The model is a generalisation of the model presented by Wilde (1968) in ‘The torsion of thin-walled bars with variable cross-section’, Archives of Mechanics, 4, 20, pp. 431–443. The Hamilton principle was used to derive equations describing the vibrations of the beam. The equations were derived relative to an arbitrary rectilinear reference axis, taking into account the curving of the beam axis and the axis formed by the shear centres of the beam cross sections. In most works known to the present authors, the equations describing the nonprismatic thin-walled beam vibration problem do not take into account the effects stemming from the curving (the inclination of the walls of the thin-walledcross section towards to the beam axis) of the analysed systems. The recurrence algorithm described in Lewanowicz’s work (1976) ‘Construction of a recurrence relation of the lowest order for coefficients of the Gegenbauer series’, Applicationes Mathematicae, XV(3), pp. 345–396, was used to solve the derived equations with variable coefficients. The obtained solutions of the equations have the form of series relative to Legendre polynomials. A numerical example dealing with the free vibrations of the beam was solved to verify the model and the effectiveness of the presented solution method. The results were compared with the results yielded by finite elements method (FEM).
3
Content available Legendre polynomials application for expanding functions in the series by these polynomials
Ignaczak P., Czajkowski A. A., Skorny G. P., Udała R.
Problemy Nauk Stosowanych
|
2016
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T. 5
57--64
EN
Introduction and aim: Selected elementary material about Legendre polynomials have been shown in the paper. The algorithm of expanding functions in the series by Legendre polynomials has been elaborated in the paper. Material and methods: The selected knowledge about Legendre polynomials have been taken from the right literature. The analytical method has been used in this paper. Results: Has been shown the theorem describing expanding functions in a series by using Legendre polynomials. It have been shown selected examples of expanding functions in a series by applying Legendre polynomials. Conclusion: The function f(z) can be expand in the interval ‹-1,1› in a series according to Legendre polynomials where the unknown coefficients can be determined using the method of undetermined coefficients.
PL
Wstęp i cel: W pracy pokazuje się wybrane podstawowe wiadomości o wielomianach Legendre’a. W artykule opracowano algorytm rozwijania funkcji w szereg według wielomianów Legendre’a. Materiał i metody: Wybrane wiadomości o wielomianach Legendre’a zaczerpnięto z literatury przedmiotu. W pracy zastosowano metodę analityczną. Wyniki: W pracy pokazano twierdzenie dotyczące rozwijania funkcji w szereg według wielomianów Legendre’a. Pokazano wybrane przykłady rozwijania funkcji w szereg według wielomianów Legendre’a Wniosek: Funkcja f(z) może być w przedziale ‹-1,1› rozwinięta w szereg według wielomianów Legendre’a, gdzie nieznane współczynniki można wyznaczyć stosując metodę współczynników nieoznaczonych.
4
Content available Związek rekurencyjny oraz zależności i równanie różniczkowe dla wielomianów Legendre’a
Czajkowski A. A.
Problemy Nauk Stosowanych
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2014
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T. 2
59--68
PL
W pracy przedstawiono związek rekurencyjny, zależności różniczkowe i równanie różniczkowe dla wielomianów Legendre’a. Celem rozważań było przeprowadzenie dowodów omawianych własności. Materiał i metody: Materiał stanowiły wybrane zależności rekurencyjne i równanie różniczkowe uzyskane z literatury przedmiotu. W przeprowadzonych dowodach zastosowano metodę dedukcji. Wyniki: Pokazano dowód twierdzenia o funkcji tworzącej dla wielomianów Legendre’a stosując metodę residuum funkcji. Przeprowadzono dowód związku rekurencyjnego, czterech zależności różniczkowych oraz równania różniczkowego dla wielomianów Legendre’a. Wnioski: Pochodną wielomianu Legendre’a wyrażoną przez wielomiany Legendre’a można określić z równania (1–z2)P'n(z) = nPn-1(z) – nzPn(z) dla n = 1, 2, … . Wielomian Legendre’a u=Pn(z) jest całką szczególną równania [(1-z2)u']'+n(n+1)u =0 dla n = 0, 1, 2,
EN
Introduction and aim: The paper presents a recurrence formula, some differential compounds and differential equation for Legendre polynomials. The aim of the discussion was to give some proofs of presented dependences. Material and methods: Selected material based on a recurrence formula, some differential compounds and differential equation has been obtained from the right literature. In presented proofs of theorems was used a deduction method. Results: Has been shown some proof of the theorem of the generating function for Legendre polynomials by using the method of function residue. It has been done the proof of recurrence formula, some proofs of four differential compounds and differential equation for Legendre polynomials. Conclusions: Some derivative of Legendre polynomial expressed by Legendre polynomials can be determined from the equation (1–z2)P'n(z) = nPn-1(z) – nzPn(z) for n = 1, 2, … . Legendre polynomial u=Pn(z) is the particular integral solution of the equation [(1-z2)u']'+n(n+1)u =0 for n = 0, 1, 2, … .
5
Content available remote Quantitative comparison of cylindricity profiles measured with different methods using Legendre-Fourier coefficients
Adamczak S., Janecki D., Makieła W., Stępień K.
Metrology and Measurement Systems
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2010
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Vol. 17, nr 3
397-403
EN
The paper discusses a method of quantitative comparison of cylindricity profiles measured with different strategies. The method is based on applying so-called Legendre-Fourier coefficients. The comparison is carried out by computing the correlation coefficient between the profiles. It is conducted by applying a normalized cross-correlation function and it requires approximation of cylindrical surfaces using the Legendre-Fourier method. As the example two sets of measurement data are employed: the first from the CMM and the second one from the traditional radial measuring instrument. The measuring data are compared by analyzing the values of selected cylindricity parameters and calculating the coefficient of correlation between profiles.
6
Content available remote On sequences of the white noises
Beśka M., Ciesielski Z.
Probability and Mathematical Statistics
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2006
|
Vol. 26, Fasc. 1
201--209
EN
The aim of the paper is to prove the strong law of large numbers for Gaussian functionals (Theorem 3.1). The functionals are of the form f (Xi), where / is integrable with respect to the Gaussian noise and the random vectors Xi are coordinatewise suitable correlated. In the last section we comment on the possibility of building noise analysis corresponding to the Legendre orthogonal polynomials analogous to the Wiener white noise theory based on Hermite orthogonal polynomials (Mehler’s kernel).
7
Content available remote Synthesis of near-optimal closed-loop control of the electric DC drive system with disturbance and constraints
Pełczewski P., Pełczewski J.
Archives of Electrical Engineering
|
2006
|
Vol. 55, nr 1
3-16
EN
The paper deals with synthesis of the quadratic near-optimal closed-loop control of electric DC drive system with inequality constraints imposed on the control signal and on state-vector's components. The system is subject to the known deterministic disturbance (the external torque on motor's shaft). For such drive system the near-optimal, open-loop control was by application of Legendre polynomials method. On the basis of that the synthesis procedure of near-optimal, closed-loop, control system was presented. The application of such procedure was demonstrated by numerical example.
PL
W pracy przedstawiono sposób syntezy zamkniętego układu sterowania bliskiego optymalnemu z kwadratowym wskaźnikiem jakości dla napędu elektrycznego prądu stałego z ograniczeniami nierównościowymi nałożonymi na sygnał sterujący i na składowe wektora stanu. Na układ działa znane zdeterminowane zakłócenie (moment zewnętrzny na wale silnika). Przy zastosowaniu metody wielomianów Legendre'a wyznaczono sterowanie bliskie optymalnemu realizowane w systemie otwartym. Na podstawie tych wyników podano sposób syntezy bliskiego optymalnemu układu sterowania w systemie zamkniętym.
8
Content available remote Synthesis of the Quadratic Near-optimal Closed of a Linear System with Disturbances
Pełczewski J.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2002
|
Vol. 50, Nr 2
153--159
EN
The paper deals with the synthesis procedure of a closed-loop quadratic, near-optimal control of a linear, time-invariant system, subject to known, deterministic, external disturbances. On the basis of the approximate, open-loop control obtained for that system by application of Legendre polynomials method the state-feedback matrix and the input signal vector were found.
9
Content available remote Application of Legendre polynomials method in optimal control of electric DC drive system
Pełczewski J., Pełczewski P.
Archives of Electrical Engineering
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2002
|
Vol. 51, nr 3
269-280
EN
The paper deals with the quadratic optimal control for the electric DC-drive, subject to the known, deterministic, external disturbance. For safety reasons the constraints are imposed on the supplying voltage, on the armature current and on the motor's angular velocity. The classical theory of the linear quadratic optimal control problem, concerning the undisturbed and unconstrained systems is not valid for the considered drive. By application of Legendre polynomials method the approximate, near-optimal control was found.
PL
W pracy przedstawiono zagadnienie sterowania optymalnego z kwadratowym wskaźnikiem jakości dla ukłądu napędu elektrycznego prądu stałego, poddanego działaniu znanego, zdeterminowanego zakłócenia zewnętrznego. Ze względów bezpieczeństwa zostały nałożone ograniczenia na napięcie zasilające, na prąd twornika oraz na prędkość kątową wału silniku. Klasyczna teoria sterowania optymalnego z kwadratowym wskaźnikiem jakości, dotycząca układów liniowych bez ograniczeń i nie poddanych działaniu zakłóceń, nie może być wykorzystana w przypadku rozpatrywanego napędu. Dzięki zastosowaniu metody wielomianów Legenfre'a uzyskano przybliżone, bliskie optymalnemu rozwiązanie.
10
Integrals of legendre polynomials and solution of some partial differential equations
Belinsky R.
Journal of Applied Analysis
|
2000
|
Vol. 6, nr 2
259-282
EN
We show a connection between the polynomials whose inflection points coincide with their interior roots (let us write shorter PIPCIR), Legendre polynomials, and Jacobi polynomials, and study some properties of PIPCIRs (Part I). In addition, we give new formulas for some classical orthogonal polynomials. Then we use PIPCIRs to solve some partial differential equations (Part II).
11
Content available Legendre polynomials method in time-optimal control of linear single-input
Pełczewski J.
Control and Cybernetics
|
1998
|
Vol. 27, no 1
105-117
EN
The paper deals with application of shifted Legendre polynomials in the time-optimal control problem for a linear, time invariant, undisturbed, single-input system. It was assumed that the normality condition of the time-optimal control is satisfied, the state matrix is nonsingular, and all its eigenvalues are real nonpositive. The method of evaluating the approximate swiching instants of the bang-bang control is presented. The proposed computational procedure is based on the solution of algebraic matrix equation, which corresponds to the differential state equation, and was obtained according to the propertyies of Legendre polynomials.
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