A Galerkin Approximation for the Initial-Value Problem for Linear Second-Order Differential Equations

1984 ◽  
Vol 15 (2) ◽  
pp. 93-108 ◽  
Author(s):  
John Gregory ◽  
Marvin Zeman
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Galerkin Approximation ◽  
Second Order ◽  
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.

scholarly journals On Second-Order Differential Equations with Nonsmooth Second Member

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
M. Milla Miranda ◽  
A. T. Lourêdo ◽  
L. A. Medeiros
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Hilbert Spaces ◽  
Initial Value Problem ◽  
Positive Function ◽  
Linear Operators ◽  
Initial Value ◽  
Second Order Differential Equations ◽  
Open Question ◽  
Abstract Framework

In an abstract framework, we consider the following initial value problem: u′′ + μAu + F(u)u = f  in  (0,T), u(0)=u0,u′(0)=u1, where μ is a positive function and f a nonsmooth function. Given u0, u1, and f we determine Fu in order to have a solution u of the previous equation. We analyze two cases of Fu. In our approach, we use the Theory of Linear Operators in Hilbert Spaces, the compactness Aubin-Lions Theorem, and an argument of Fixed Point. One of our two results provides an answer in a certain sense to an open question formulated by Lions in (1981, Page 284).

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