scholarly journals The Polynomial Least Squares Method for Nonlinear Fractional Volterra and Fredholm Integro-Differential Equations

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2324
Author(s):  
Bogdan Căruntu ◽  
Mădălina Paşca
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Analytical Solutions ◽  
Efficient Method ◽  
General Class ◽  
Least Squares Method ◽  
Test Problems ◽  
Approximate Analytical Solutions

We present a relatively new and very efficient method to find approximate analytical solutions for a very general class of nonlinear fractional Volterra and Fredholm integro-differential equations. The test problems included and the comparison with previous results by other methods clearly illustrate the simplicity and accuracy of the method.

scholarly journals Approximate Solutions for a Class of Nonlinear Fredholm and Volterra Integro-Differential Equations Using the Polynomial Least Squares Method

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2692
Author(s):  
Bogdan Căruntu ◽  
Mădălina Sofia Paşca
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Analytical Solutions ◽  
General Class ◽  
Least Squares Method ◽  
Approximate Solutions ◽  
Test Problems ◽  
Numerical Examples ◽  
Approximate Analytical Solutions ◽  
Very High

We apply the polynomial least squares method to obtain approximate analytical solutions for a very general class of nonlinear Fredholm and Volterra integro-differential equations. The method is a relatively simple and straightforward one, but its precision for this type of equations is very high, a fact that is illustrated by the numerical examples presented. The comparison with previous approximations computed for the included test problems emphasizes the method’s simplicity and accuracy.

scholarly journals Approximate Analytical Solutions of the Fractional-Order Brusselator System Using the Polynomial Least Squares Method

2015 ◽  
Vol 2015 ◽  
pp. 1-5 ◽  
Author(s):  
Constantin Bota ◽  
Bogdan Căruntu
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Fractional Order ◽  
Analytical Solutions ◽  
Least Squares Method ◽  
Nonlinear Differential Equations ◽  
Fractional Order System ◽  
New Method ◽  
Order System ◽  
Approximate Analytical Solutions

The paper presents a new method, called the Polynomial Least Squares Method (PLSM). PLSM allows us to compute approximate analytical solutions for the Brusselator system, which is a fractional-order system of nonlinear differential equations.

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 Approximate Periodic Solutions for Oscillatory Phenomena Modelled by Nonlinear Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Constantin Bota ◽  
Bogdan Căruntu ◽  
Olivia Bundău
Keyword(s):  
Differential Equations ◽  
Least Squares ◽  
Periodic Solutions ◽  
General Class ◽  
Least Squares Method ◽  
Nonlinear Differential Equations ◽  
Duffing Oscillator ◽  
Nonlinear Problems ◽  
Van Der Pol Oscillator ◽  
Van Der Pol

We apply the Fourier-least squares method (FLSM) which allows us to find approximate periodic solutions for a very general class of nonlinear differential equations modelling oscillatory phenomena. We illustrate the accuracy of the method by using several significant examples of nonlinear problems including the cubic Duffing oscillator, the Van der Pol oscillator, and the Jerk equations. The results are compared to those obtained by other methods.

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.

Regionale Bildungsrenditen – von der Methode zur Rendite

2012 ◽  
Vol 63 (1) ◽  
Author(s):  
Julia Reilich
Keyword(s):  
Least Squares ◽  
Empirical Analysis ◽  
Efficient Method ◽  
Returns To Education ◽  
Least Squares Method ◽  
Federal State ◽  
Average Return ◽  
German States ◽  
East German ◽  
Three Stage Least Squares

SummaryThe paper shows an empirical analysis of regional returns to education in Germany. Therefore the returns to education are estimated for Germany as a whole and for the eastern and western part seperately. Additionally the analysis examines the returns for each federal state. For the estimations it is necessary to identify an efficient method and the best estimable model. It can be shown that the Three-Stage-Least-Squares-method yields the best results. For Germany the regression shows an average return for an additional year of education of about 10 %. By comparing the eastern and western part, it can be shown that in the east german states - the New Länder - employees get a much higher return. A further separation of states verifies the heterogeneities of the regions.

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