Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1244.34052
Pašić, Mervan; Tanaka, Satoshi
Fractal oscillations of self-adjoint and damped linear differential equations of second-order. (English)
[J] Appl. Math. Comput. 218, No. 5, 2281-2293 (2011). ISSN 0096-3003

Authors' abstract: For a prescribed real number $s \in $ [1, 2), we give some sufficient conditions on the coefficients $p(x)$ and $q(x)$ such that every solution $y = y(x), ~y \in C^{2}((0, T$]) of the linear differential equation $$(p(x)y^{\prime})^{\prime} + q(x)y = 0 \text{ on }(0, T]$$ is bounded and fractal oscillatory near $x = 0$ with the fractal dimension equal to $s$. This means that $y$ oscillates near $x = 0$ and the fractal (box-counting) dimension of the graph $\Gamma (y)$ of $y$ is equal to $s$ as well as the $s$ dimensional upper Minkowski content (generalized length) of $\Gamma (y)$ is finite and strictly positive. It verifies that $y$ admits similar kind of the fractal geometric asymptotic behaviour near $x = 0$ like the chirp function $y_{ch}(x) = a(x)S(\varphi (x))$, which often occurs in the time -- frequency analysis and its various applications. Furthermore, this kind of oscillations is established for the Bessel, chirp and other types of damped linear differential equations given in the form $$y''+ (\mu /x)y^{\prime} + g(x)y = 0,~ x \in (0, T].$$ In order to prove the main results, we state a new criterion for fractal oscillations near $x = 0$ of real continuous functions.
[Qingkai Kong (DeKalb)]

MSC 2000:: *34C10 Qualitative theory of oscillations of ODE: Zeros, etc.
34C11 Qualitative theory of solutions of ODE: Growth, etc.

Keywords: linear equations; asymptotic behaviour of solutions; oscillations; chirps; fractal curves; fractal dimension; Minkowski content; Bessel equation

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]