scholarly journals Polynomial Least Squares Method for Fractional Lane–Emden Equations

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 479 ◽  
Author(s):  
Bogdan Căruntu ◽  
Constantin Bota ◽  
Marioara Lăpădat ◽  
Mădălina Paşca
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Least Squares Method ◽  
Polynomial Solution ◽  
Approximate Polynomial

This paper applies the Polynomial Least Squares Method (PLSM) to the case of fractional Lane-Emden differential equations. PLSM offers an analytical approximate polynomial solution in a straightforward way. A comparison with previously obtained results proves how accurate the method is.

Analytic approximate solutions for a class of variable order fractional differential equations using the polynomial least squares method

2017 ◽  
Vol 20 (4) ◽  
Author(s):  
Constantin Bota ◽  
Bogdan Căruntu
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Fractional Differential Equations ◽  
Least Squares Method ◽  
Approximate Solutions ◽  
Variable Order ◽  
Polynomial Solutions ◽  
Fractional Differential ◽  
Approximate Polynomial

AbstractIn this paper a new way to compute analytic approximate polynomial solutions for a class of nonlinear variable order fractional differential equations is proposed, based on the Polynomial Least Squares Method (PLSM). In order to emphasize the accuracy and the efficiency of the method several examples are included.

scholarly journals Numerical solution of the first order nonlinear differen-tial equations with the mixed nonlinear conditions by using PLSM(comparison with Bernstein polynomials method)

2019 ◽  
Vol 29 ◽  
pp. 01014
Author(s):  
Marioara Lăpădat ◽  
Mohsen Razzaghi ◽  
Mădălina Sofia Paşca
Keyword(s):  
Differential Equations ◽  
Numerical Solution ◽  
Least Squares ◽  
Ordinary Differential Equations ◽  
Least Squares Method ◽  
Bernstein Polynomials ◽  
Approximate Solutions ◽  
Nonlinear Ordinary Differential Equations ◽  
First Order ◽  
Approximate Polynomial

We use the Polynomial Least Squares Method (PLSM), which allows us to compute analytical approximate polynomial solutions for nonlinear ordinary differential equations with the mixed nonlinear conditions. The accuracy of the method is illustrated by a comparison with approximate solutions previously computed using Bernstein polynomials method.

scholarly journals Experimentalanalytical identification of a mathematical model of a dc motor using the least squares method

2019 ◽  
Vol 21 (4) ◽  
pp. 113-122
Author(s):  
N. A. Malev ◽  
A. I. Mukhametshin ◽  
O. V. Pogoditsky ◽  
A. G. Gorodnov
Keyword(s):  
Mathematical Model ◽  
Least Squares ◽  
Least Squares Method ◽  
Dynamic Properties ◽  
Dc Motor ◽  
High Accuracy ◽  
Electrical Machines ◽  
Analytical Assessment ◽  
Approximate Polynomial ◽  
Process Graphs

The urgency of the problem lies in the formation of mathematical models of electromechanical converters corresponding to the objects of study with high accuracy. An experimental-analytical assessment of transient modes of a DC motor based on an installation for the study of electrical machines has been carried out. Based on the results obtained, an approximation of transient process graphs was carried out using the least squares method and an approximate polynomial of the corresponding order was selected with the closest imminence to the dynamic properties of the object under study.

scholarly journals A Least Squares Differential Quadrature Method for a Class of Nonlinear Partial Differential Equations of Fractional Order

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1336
Author(s):  
Constantin Bota ◽  
Bogdan Căruntu ◽  
Dumitru Ţucu ◽  
Marioara Lăpădat ◽  
Mădălina Sofia Paşca
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Least Squares ◽  
Least Squares Method ◽  
Differential Quadrature Method ◽  
Nonlinear Partial Differential Equations ◽  
Differential Quadrature ◽  
Test Problems ◽  
Quadrature Method ◽  
Partial Differential

In this paper a new method called the least squares differential quadrature method (LSDQM) is introduced as a straightforward and efficient method to compute analytical approximate polynomial solutions for nonlinear partial differential equations with fractional time derivatives. LSDQM is a combination of the differential quadrature method and the least squares method and in this paper it is employed to find approximate solutions for a very general class of nonlinear partial differential equations, wherein the fractional derivatives are described in the Caputo sense. The paper contains a clear, step-by-step presentation of the method and a convergence theorem. In order to emphasize the accuracy of LSDQM we included two test problems previously solved by means of other, well-known methods, and observed that our solutions present not only a smaller error but also a much simpler expression. We also included a problem with no known exact solution and the solutions computed by LSDQM are in good agreement with previous ones.

scholarly journals Statistical Inference for Stochastic Differential Equations with Small Noises

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Liang Shen ◽  
Qingsong Xu
Keyword(s):  
Differential Equations ◽  
Normal Distribution ◽  
Least Squares ◽  
Stochastic Differential Equations ◽  
Asymptotic Distribution ◽  
Asymptotic Property ◽  
Stable Distribution ◽  
Least Squares Method ◽  
Regularity Conditions ◽  
Drift Parameter

This paper proposes the least squares method to estimate the drift parameter for the stochastic differential equations driven by small noises, which is more general than pure jumpα-stable noises. The asymptotic property of this least squares estimator is studied under some regularity conditions. The asymptotic distribution of the estimator is shown to be the convolution of a stable distribution and a normal distribution, which is completely different from the classical cases.

Estimation of Parameters of Hyperbolic Functions with Additive Noise

2021 ◽  
Vol 105 ◽  
pp. 302-308
Author(s):  
Dmitriy V. Ivanov ◽  
Ilya L. Sandler ◽  
Natalya V. Chertykovtseva
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Measurement Errors ◽  
Least Squares Method ◽  
Ordinary Least Squares ◽  
Least Square ◽  
Hyperbolic Function ◽  
Estimation Of Parameters ◽  
Hyperbolic Functions ◽  
Hyperbolic Sine

Hyperbolic functions are widely used to write solutions to ordinary differential equations and partial differential equations. These functions are nonlinear in parameters, which makes it difficult to estimate the parameters of these functions. In the paper, two-step algorithms for estimating the parameters of hyperbolic sine and cosine (sinh and cosh) in the presence of measurement errors are proposed. At the first step, the hyperbolic function is transformed into a linear difference equation (autoregression) of the second order. Estimation in the presence of noise of observation of autoregression parameters using ordinary least square (OLS) gives biased estimates. Modifications of the two-stage estimation algorithm based on the use of the method of total least squares (TLS) and the method of extended instrumental variables (EIV), hyperbolic sine and cosine in the presence of errors in measurements are proposed. Numerical experiments have shown that the accuracy of the parameter estimation using the proposed modifications is higher than the accuracy of the estimate obtained using the ordinary least squares method (OLS).

scholarly journals Integrate-and-Differentiate Approach to Nonlinear System Identification

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 2999
Author(s):  
Artur I. Karimov ◽  
Ekaterina Kopets ◽  
Erivelton G. Nepomuceno ◽  
Denis Butusov
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Mechanical System ◽  
Governing Equation ◽  
Least Squares Method ◽  
Nonlinear System Identification ◽  
Identification Accuracy ◽  
Accelerometer Data ◽  
Iteratively Reweighted Least Squares ◽  
Linear Differential

In this paper, we consider a problem of parametric identification of a piece-wise linear mechanical system described by ordinary differential equations. We reconstruct the phase space of the investigated system from accelerometer data and perform parameter identification using iteratively reweighted least squares. Two key features of our study are as follows. First, we use a differentiated governing equation containing acceleration and velocity as the main independent variables instead of the conventional governing equation in velocity and position. Second, we modify the iteratively reweighted least squares method by including an auxiliary reclassification step into it. The application of this method allows us to improve the identification accuracy through the elimination of classification errors needed for parameter estimation of piece-wise linear differential equations. Simulation of the Duffing-like chaotic mechanical system and experimental study of an aluminum beam with asymmetric joint show that the proposed approach is more accurate than state-of-the-art solutions.

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