Solving Systems of Linear Fractional Integro-Differential Equations with Delay using Least Squares Method and Chebyshev Polynomials

2021 ◽  
Vol 15 (5) ◽  
pp. 621-629
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Chebyshev Polynomials ◽  
Least Squares Method ◽  
Equations With Delay

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 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.

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).

ESTIMATION OF PARAMETERS OF ECONOMIC GROWTH AUTOREGRESSIVE MODELS

2021 ◽  
Vol 1 (195) ◽  
pp. 33-37
Author(s):  
D.V. Ivanov ◽  
Keyword(s):  
Economic Growth ◽  
Differential Equations ◽  
Least Squares ◽  
Statistical Data ◽  
Least Squares Method ◽  
Growth Models ◽  
Estimation Of Parameters ◽  
Additive Interference ◽  
Constant Coefficients ◽  
Economic Growth Models

The construction of time series models based on statistical data is one of the central tasks of modern econometric studies. Numerous economic models based on differential equations are used to explain the patterns of economic growth. The article is devoted to the estimation of parameters of economic growth models based on solutions of homogeneous differential equations with constant coefficients. A comparative analysis of methods for estimating the parameters of autoregression of economic dynamics series with additive interference in the output signal is carried out. The simulation results showed that the full least squares method gives the most accurate estimates. The most commonly used least squares method gives the worst estimates.

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