A local least-squares method for solving nonlinear partial differential equations of second order

2008 ◽  
Vol 111 (3) ◽  
pp. 351-375 ◽  
Author(s):  
Pascal Heider
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Least Squares ◽  
Least Squares Method ◽  
Nonlinear Partial Differential Equations ◽  
Second Order ◽  
Partial Differential

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 On the Mathematical Structure of Turbulence

1958 ◽  
Vol 8 ◽  
pp. 1084-1086
Author(s):  
J. Bass
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Mathematical Problem ◽  
Nonlinear Partial Differential Equations ◽  
Mathematical Structure ◽  
Second Order ◽  
Time Variable ◽  
Partial Differential ◽  
Three Space

One is able to formulate in a very general way the mathematical problem of turbulence in the following form.One has a system of second-order nonlinear partial differential equations with three space variables and one time variable. One looks for solutions sufficiently complex and nonstationary to represent irregular oscillations.

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