scholarly journals A finite element approximation for the initial-value problem for nonlinear second-order differential equations

1985 ◽  
Vol 111 (1) ◽  
pp. 90-104 ◽  
Author(s):  
John Gregory ◽  
Marvin Zeman ◽  
Mohsen Badiey
Keyword(s):  
Finite Element ◽  
Differential Equations ◽  
Initial Value Problem ◽  
Finite Element Approximation ◽  
Second Order ◽  
Element Approximation ◽  
Initial Value ◽  
Second Order Differential Equations

Superexponentials: A Generalization of Hyperbolic and Trigonometric Functions

2017 ◽  
Vol 3 (1) ◽  
pp. 7
Author(s):  
Alfonso F. Agnew ◽  
Brandon Gentile ◽  
John H. Mathews
Keyword(s):  
Power Series ◽  
Differential Equations ◽  
Initial Value Problem ◽  
Second Order ◽  
Series Expansions ◽  
Roots Of Unity ◽  
Trigonometric Functions ◽  
Initial Value ◽  
Second Order Differential Equations ◽  
Power Series Expansions

We construct and explore the properties of a generalization of hy- perbolic and trigonometric functions we cal l superexponentials. The general ization is based on the characteristic second-order differential equations (DE) these functions satisfy, and leads to functions satisfying analogous mth order equations and having many properties analogous to the usual hyperbolic and trigonometric functions. Roots of unity play a key role in providing the periodicity resulting in various properties. We also show how these functions solve the general initial value problem for the differential equations y(n) = y, and a look at the power series expansions reveal surprisingly simple patterns that clarify the properties of the superexponentials.

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