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Full-Text Articles in Physical Sciences and Mathematics
Elgenvalues Of Fibonacci-Like Sequences, Elyssa Hurst
Elgenvalues Of Fibonacci-Like Sequences, Elyssa Hurst
Masters Theses & Specialist Projects
The familiar Fibonacci sequence 1,1,2,3,5,8,13,... can be described by the recurrence relation x(0) = 1, x(1) = 1, x(n) = x(n-1) + x(n-2). For this relation, as n → oo, x(n+1) → 1 +√5 x(n) 2 ' which is the familiar golden ratio. This value is also the dominant eigenvalue of the above recurrence relation. In this series, we consider the dominant eigenvalue of some Fibonacci-like sequence of the form x(n) = ∑n-1/k+1 ak Zk (n-k) where the Zk's are independent random variables with Zk = {+1 with probability p - 1 with probability q, with p + q = …
Power Series Solutions To Ordinary Differential Equations, John Lagrange
Power Series Solutions To Ordinary Differential Equations, John Lagrange
Masters Theses & Specialist Projects
In this thesis, the reader will be made aware of methods for finding power series solutions to ordinary differential equations. In the case that a solution to a differential equation may not be expressed in terms of elementary functions, it is practical to obtain a solution in the form of an infinite series, since many differential equations which yield such a solution model an actual physical situation. In this thesis, we introduce conditions that guarantee existence and uniqueness of analytic solutions, both in the linear and nonlinear case. Several methods for obtaining analytic solutions are introduced as well. For the …