Power Series Solutions and Special Functions

2015 ◽  
pp. 119-170
Author(s):  
Steven G. Krantz
Keyword(s):  
Power Series ◽  
Special Functions ◽  
Series Solutions ◽  
Power Series Solutions

A symmetry-preserving difference scheme and analytical solutions of a generalized higher-order beam equation

2021 ◽  
Vol 477 (2255) ◽  
Author(s):  
Shou-Fu Tian ◽  
Mei-Juan Xu ◽  
Tian-Tian Zhang
Keyword(s):  
Power Series ◽  
Vector Fields ◽  
Higher Order ◽  
Beam Equation ◽  
Series Solutions ◽  
Local Conservation ◽  
Power Series Method ◽  
Potential System ◽  
Power Series Solutions ◽  
The Difference

Under investigation in this work is a generalized higher-order beam equation, which is an important physical model and describes the vibrations of a rod. By considering Lie symmetry analysis, and using the power series method, we derive the geometric vector fields, symmetry reductions, group invariant solutions and power series solutions of the equation, respectively. The convergence analysis of the power series solutions are also provided with detailed proof. Furthermore, by virtue of the multipliers, the local conservation laws of the equation are derived as well. Furthermore, an effective and direct approach is proposed to study the symmetry-preserving discretization for the equation via its potential system. Finally, the invariant difference models of the generalized beam equation are successfully constructed. Our results show that it is very useful to construct the difference models of the potential system instead of the original equation.

The semi-analytic theory of standing waves

1987 ◽  
Vol 411 (1840) ◽  
pp. 123-137 ◽  
Keyword(s):  
Numerical Calculation ◽  
Power Series ◽  
Deep Water ◽  
Standing Waves ◽  
Affirmative Answer ◽  
Rigorous Proof ◽  
Series Solutions ◽  
Wave Problem ◽  
Power Series Solutions ◽  
Formal Power

The algorithm proposed by Schwartz & Whitney ( J. Fluid Mech . 107, 147–171 (1981)) for the numerical calculation of formal power series solutions of the classical standing-wave problem is vindicated by a rigorous proof that resonances do not occur in the calculations. A detailed account of a successful algorithm is given. The analytical question of the convergence of the power series whose coefficients have been calculated remains open. An affirmative answer would be a first demonstration of the existence of standing waves on deep water.

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