scholarly journals Simplifying coefficients in a family of nonlinear ordinary differential equations

2019 ◽  
Vol 22 (2) ◽  
pp. 293-297 ◽  
Author(s):  
Feng Qi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Inversion Formula ◽  
Stirling Numbers ◽  
Combinatorial Analysis ◽  
Bell Polynomials ◽  
Nonlinear Ordinary Differential Equations ◽  
Faà Di Bruno Formula

By virtue of the Faá di Bruno formula, properties of the Stirling numbers and the Bell polynomials of the second kind, the binomial inversion formula, and other techniques in combinatorial analysis, the author finds a simple, meaningful, and signicant expression for coefficients in a family of nonlinear ordinary differential equations.

scholarly journals Simple forms for coefficients in two families of ordinary differential equations

2018 ◽  
Vol 6 (1) ◽  
pp. 7
Author(s):  
Feng Qi
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Inversion Formula ◽  
Stirling Numbers ◽  
Bell Polynomials ◽  
Faà Di Bruno Formula

In the paper, by virtue of the Faá di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion formula for the Stirling numbers of the first and second kinds, the author finds simple, meaningful, and significant forms for coefficients in two families of ordinary differential equations.

scholarly journals Simplifying Coefficients in Differential Equations Associated with Higher Order Bernoulli Numbers of the Second Kind

2017 ◽  
Author(s):  
Feng Qi ◽  
Da-Wei Niu ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Inversion Formula ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Higher Order ◽  
Bell Polynomials

In the paper, by virtue of the Faà di Bruno formula, some properties of the Bell polynomials of the second kind, and an inversion formula for the Stirling numbers of the first and second kinds, the authors establish meaningfully and significantly two identities which simplify coefficients in a family of ordinary differential equations associated with higher order Bernoulli numbers of the second kind.

scholarly journals Simplifying Two Families of Nonlinear Ordinary Differential Equations

2017 ◽  
Author(s):  
Feng Qi ◽  
Jing-Lin Wang ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stirling Numbers ◽  
Combinatorial Analysis ◽  
Nonlinear Ordinary Differential Equations

In the paper, by virtue of techniques in combinatorial analysis, the authors simplify two families of nonlinear ordinary differential equations in terms of the Stirling numbers of the first kind.

scholarly journals Simplification of Coefficients in Differential Equations Associated with Higher Order Frobenius-Euler Numbers

2018 ◽  
Vol 72 (1) ◽  
pp. 67-76 ◽  
Author(s):  
Feng Qi ◽  
Da-Wei Niu ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stirling Numbers ◽  
Higher Order ◽  
Bell Polynomials ◽  
Euler Numbers ◽  
Inversion Formulas ◽  
Faà Di Bruno Formula

Abstract In the paper, the authors apply Faà di Bruno formula, some properties of the Bell polynomials of the second kind, the inversion formulas of binomial numbers and the Stirling numbers of the first and the second kind, to significantly simplify coefficients in two families of ordinary differential equations associated with the higher order Frobenius–Euler numbers.

scholarly journals Simplification of Coefficients in Differential Equations Associated with Higher Order Frobenius–Euler Numbers

2017 ◽  
Author(s):  
Feng Qi ◽  
Da-Wei Niu ◽  
Bai-Ni Guo
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Stirling Numbers ◽  
Higher Order ◽  
Bell Polynomials ◽  
Euler Numbers ◽  
Inversion Formulas

In the paper, by virtue of the Fa`a di Bruno formula, some properties of the Bell polynomials of the second kind, and the inversion formulas of binomial numbers and the Stirling numbers of the first and second kinds, the authors simplify meaningfully and significantly coefficients in two families of ordinary differential equations associated with higher order Frobenius–Euler numbers.

scholarly journals The Generating Function of the Catalan Numbers and Lower Triangular Integer Matrices

2017 ◽  
Author(s):  
Feng Qi ◽  
Xiao-Long Qin ◽  
Yong-Hong Yao
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Generating Function ◽  
Catalan Numbers ◽  
Bell Polynomials ◽  
Nonlinear Ordinary Differential Equations ◽  
Inversion Theorem ◽  
Lower Triangular

In the paper, by the Faά di Bruno formula, several identities for the Bell polynomials of the second kind, and an inversion theorem, the authors simplify coefficients of two families of nonlinear ordinary differential equations for the generating function of the Catalan numbers and discover inverses of fifteen closely related lower triangular integer matrices.

scholarly journals Impact of double-diffusive convection and motile gyrotactic microorganisms on magnetohydrodynamics bioconvection tangent hyperbolic nanofluid

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 74-88 ◽  
Author(s):  
Tanveer Sajid ◽  
Muhammad Sagheer ◽  
Shafqat Hussain ◽  
Faisal Shahzad
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Transfer Rate ◽  
Mass Transfer Rate ◽  
Physical Nature ◽  
Working Fluid ◽  
Nonlinear Ordinary Differential Equations ◽  
Gyrotactic Microorganisms ◽  
Motile Gyrotactic Microorganisms ◽  
Double Diffusive

AbstractThe double-diffusive tangent hyperbolic nanofluid containing motile gyrotactic microorganisms and magnetohydrodynamics past a stretching sheet is examined. By adopting the scaling group of transformation, the governing equations of motion are transformed into a system of nonlinear ordinary differential equations. The Keller box scheme, a finite difference method, has been employed for the solution of the nonlinear ordinary differential equations. The behaviour of the working fluid against various parameters of physical nature has been analyzed through graphs and tables. The behaviour of different physical quantities of interest such as heat transfer rate, density of the motile gyrotactic microorganisms and mass transfer rate is also discussed in the form of tables and graphs. It is found that the modified Dufour parameter has an increasing effect on the temperature profile. The solute profile is observed to decay as a result of an augmentation in the nanofluid Lewis number.

A Multiple Interval Chebyshev-Gauss-Lobatto Collocation Method for Ordinary Differential Equations

2016 ◽  
Vol 9 (4) ◽  
pp. 619-639 ◽  
Author(s):  
Zhong-Qing Wang ◽  
Jun Mu
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Collocation Method ◽  
Initial Value Problems ◽  
Numerical Experiments ◽  
High Order Accuracy ◽  
Nonlinear Ordinary Differential Equations ◽  
Initial Value ◽  
Order Accuracy ◽  
Long Time

AbstractWe introduce a multiple interval Chebyshev-Gauss-Lobatto spectral collocation method for the initial value problems of the nonlinear ordinary differential equations (ODES). This method is easy to implement and possesses the high order accuracy. In addition, it is very stable and suitable for long time calculations. We also obtain thehp-version bound on the numerical error of the multiple interval collocation method underH1-norm. Numerical experiments confirm the theoretical expectations.

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