A program for predicting the intervals of oscillations in the solutions of ordinary second-order linear homogeneous differential equations
- Dimitris M Christodoulou^{1},
- James Graham-Eagle^{1} and
- Qutaibeh D Katatbeh^{2}Email author
https://doi.org/10.1186/s13662-016-0774-x
© Christodoulou et al. 2016
Received: 25 October 2015
Accepted: 27 January 2016
Published: 15 February 2016
Abstract
We derive a new criterion for deducing the intervals of oscillatory behavior in the solutions of ordinary second-order linear homogeneous differential equations from their coefficients. The validity of the method depends on one’s ability to transform a given differential equation to its simplest possible form, so a program must be executed that involves transformations of both variables before the criterion can be applied. The payoff of the program is the detection of oscillations precisely where they may occur in finite or infinite intervals of the independent variable. We demonstrate how the oscillation-detection program can be carried out for a variety of well-known differential equations from applied mathematics and mathematical physics.
Keywords
MSC
1 Introduction
It is rather obvious that the above difficulties with the term \(-b'(x)/2\) do not materialize in cases where \(b(x)\) is constant since then \(b' = 0\) and the canonical coefficient \(q(x)\) is not ‘contaminated’ by the term \(-b'(x)/2\). This observation shows us how to circumvent the difficulties associated with \(b'(x)\): Equations of the form (2) should be cast to their canonical form, and then this form should be cast into a new form in which the coefficient of the first derivative is constant. Then, another transformation to the canonical form will fold a constant-damping term into the \(q(x)\) term of equation (5) that will clearly oppose any oscillatory tendency without introducing contamination from \(b'(x)\). Once the competition between oscillation and damping has been established in that new \(q(x)\) term, a criterion for oscillatory solutions could presumably be found by using that final canonical form of equation (2) and Sturm’s [8] comparison theorem. We analyze this procedure in Section 2 below, where we establish a program for deriving a criterion for oscillatory solutions for equation (5). Then, in Sections 3-5, we apply this methodology to some well-known and commonly used equations in applied mathematics and mathematical physics [15–17], and we predict the precise intervals of oscillations of their solutions. Finally, in Section 6, we summarize and discuss our results.
2 A program for detecting oscillatory solutions
2.1 Cauchy-Euler equation
The case of the Cauchy-Euler equation indicates that the first step in establishing a criterion for oscillatory solutions must be an attempt to transform an equation of the form (5) to another form in which all the coefficients are constant. We show in Section 2.2 that such a task cannot be accomplished by a transformation of the dependent variable \(u(x)\) because then the constant damping of the first derivative term cannot be folded into the coefficient of the final canonical form. Then we show in Section 2.3 that the task can be carried out successfully by a transformation of the independent variable x but only for some specific equations that are generalized forms of the Cauchy-Euler equation. Therefore, equation (7) is not merely a simple case that can be handled with ease; on the contrary, it is a representative of the one and only one type of differential equation that can be transformed to a damped harmonic oscillator with constant coefficients in all of its terms.
When the above step fails (for equations that do not have the symmetries of equation (7)), it is still possible to transform a given equation to a form in which only the first derivative term has a constant coefficient. Then this constant damping can be folded into the coefficient of the final canonical form where it will oppose oscillatory tendencies. We show in Section 2.4 how this step is carried out and how the criterion for oscillatory solutions then emerges. In what follows, we always begin with the canonical form (5) since all ordinary second-order linear homogeneous differential equations can be initially cast into this form. We note however that if a given equation is already in canonical form, then one may not assume that the damping has already been folded into \(q(x)\); as equation (6) shows, the given \(q(x)\) may already be contaminated by the \(-b'(x)/2\) term, which may not be acting as pure damping. It is for this reason that the above-discussed oscillation-detection program must still be carried out in its entirety for a given equation of the form (5) so that a constant-damping term will be created, and then it will be explicitly folded into the original \(q(x)\) term. This procedure will ensure that every effort has been made for pure damping to oppose the natural tendency for oscillatory behavior that the given \(q(x)\) term may possess.
2.2 The transformation of the dependent variable fails
2.3 Transformations of the independent variable
The results described indicate that differential equations of the Cauchy-Euler type should always be transformed to a form with constant coefficients before an investigation of oscillatory behavior in their solutions is carried out. At the same time, there exist differential equations that can be transformed to the Cauchy-Euler type, and the entire procedure that leads to constant coefficients must then be applied to them as well. We provide a related example in Section 3.3, where we study the Riemann-Weber [18] equations, a long-standing counterexample to the discovery of a robust criterion for oscillatory solutions by considering the \(q(x)\) term alone of a differential equation given in the canonical form (5).
2.4 Constant damping and the criterion for oscillatory solutions
Our investigation is not however limited only to the conventional definition of oscillation as a sequence of infinitely many zeros in the solution of a differential equation. In what follows, we define oscillatory behavior as the appearance of successive critical points of the same kind (maxima, or minima, or inflection points) in the graph of a solution. This definition allows us to study oscillations in solutions with a finite number of zeros or no zeros (Section 3.4) and in solutions defined in finite domains (Section 5), as well as some characteristic high-frequency oscillations that tend to occur in the vicinity of \(x=0\) (Section 3.1 and Section 4.3).
3 Confirmations of the criterion
3.1 Bessel equation and equations transformed to the Bessel type
3.2 Modified Bessel equation
3.3 Riemann-Weber equations
3.4 Wong-Willett equations with oscillatory coefficients
Wong [12] and Willett [9] have also studied several other differential equations with oscillatory coefficients. We note two equations that are extensions of equation (34):
4 Applications of the criterion in semiinfinite domains
In this section, we apply inequality (22) to equations that are known empirically to exhibit oscillatory solutions over semi-infinite intervals in the variable x.
4.1 First form of the parabolic cylinder equation
4.2 Airy equation
4.3 A complicated canonical form
5 Oscillatory solutions in finite domains
In this section, we apply inequality (22) to equations that are known empirically to exhibit oscillatory solutions over finite intervals in the variable x.
5.1 Second form of the parabolic cylinder equation
Equation (50) has four real roots, so it predicts also a small region of no oscillation around \(x=0\). This region is too narrow (its size is \(\approx1/\sqrt{-n}\) for \(n\leq-1\)) for a break in oscillation to be observed in Figure 6. The nonoscillatory part of the solution around \(x=0\) (\(|x|\leq1/6\) for \(n=-9\)) is effectively squeezed by the two larger oscillatory regions on either side of \(x=0\). Nevertheless, the finite extent of these regions raises the question of how one can define oscillatory behavior in finite domains.
We have defined in Section 2.4 oscillatory behavior as the appearance of successive critical points of the same kind (maxima, or minima, or inflection points as in Figure 2) in the graph of a solution of a differential equation. In addition, for oscillation in a finite interval \([x_{1}, x_{2}]\), we expect to see at least one full ‘cycle’ in the open interval \((x_{1}, x_{2})\), that is, at least two critical points of the same kind. As a result of this definition, low-order polynomial solutions of degree \(n\leq2\) are not oscillatory in any interval, but higher-order polynomial solutions will be called oscillatory in \([x_{1}, x_{2}]\) if two or more critical points of the same kind do appear in \((x_{1}, x_{2})\). Some borderline cases with just two such critical points are described in Sections 5.2, 5.4, and 5.5.
5.2 Hermite equation
Equation (53) also predicts a region of no oscillation around \(x=0\), but this region is too narrow for a break in oscillation to be observed in Figure 7. We call this solution (a polynomial of degree \(\lambda= 4\)) oscillatory because, according to the definition given at the end of Section 5.1, a full ‘cycle’ (two minima) can be seen in the open interval \((-3, 3)\). In fact, numerical integrations using the same boundary conditions show that the lowest-order Hermite solution that exhibits such an oscillation has \(\lambda= 3\); for \(\lambda\leq2\), the predicted region for oscillation, \(|x| < \sqrt{2\lambda+ 1}\), does not host two critical points of the same kind, and we call such solutions nonoscillatory. We note however that the \(\lambda= 3\) oscillatory case is a borderline case; for different choices of boundary conditions (e.g., for \(y(0)=0\) and \(y'(0)=1\)), numerical integrations produce solutions that are nonoscillatory. This demonstrates the heavy influence of the adopted boundary conditions to the low-order polynomial solutions.
5.3 CDOS equation
5.4 Chebyshev equation
5.5 Laguerre equation
6 Summary and discussion
6.1 Summary
In this paper, we have presented a new methodology for predicting the intervals of oscillations in the solutions of ordinary second-order linear homogeneous differential equations by examining the behavior of their coefficients. We have defined oscillatory behavior as the appearance of successive critical points of the same kind (maxima, minima, or inflection points) in the graph of a solution (Section 2.4). According to this definition, low-order polynomial solutions can be oscillatory even in finite intervals (see Section 5.1). Because of the subtleties involved in investigating oscillations in finite and infinite intervals, an entire program must be carried out with the goal of transforming a given equation to its simplest possible form. The starting point of the program is the canonical form (5) of any given equation in the form (2).
In the first step, an attempt must be made to transform the canonical form into a damped harmonic oscillator with constant coefficients in the form (1). Then the criterion for oscillatory solutions can be established, quite easily, from the discriminant of the characteristic quadratic equation, as is well known. It turns out that the Cauchy-Euler equation (Section 2.1), the Riemann-Weber equations (Section 3.3), the CDOS equation (Section 5.3), and the Chebyshev equation (Section 5.4) can all be transformed to equations with constant coefficients during this step of the program.
In the event that the above step is not viable, a different type of transformation may still be applied to the independent variable (Section 2.3; see also Section 2.2 for a failure to transform the dependent variable). In this second step, the canonical form (5) must be cast to a form in which the coefficient of the first derivative is constant. This constant represents pure damping that is capable of opposing any natural oscillatory tendency that the solutions may possess (Section 2.4). Finally, when this form is transformed to its canonical form, the constant damping is folded into the coefficient of this final form, where it will certainly oppose oscillation. Then, an application of Sturm’s [8] comparison theorem produces a criterion (equations (21) and (22)) that can detect the intervals of oscillatory behavior in this step of the program. In Sections 3-5, we have presented several examples of equations from applied mathematics and mathematical physics in which our oscillation-detection program can be carried out successfully.
6.2 Discussion
Two more interesting cases that make use of special forms of \(b(x)\) can be delineated from inequality (70):
If \(q(x)\) contains two or more regular singular points, then the oscillatory properties of the solutions can still be investigated across each singularity by applying a different \(x=f(t)\) transformation in each interval that contains one such singularity.
This is also the case for elementary periodic functions such as \(\sin(kx)\) and \(\cos(kx)\). For \(n=0\), the solution (44) of equation (43) reduces to \(u(x)=\sin(kx)\), and the criterion (46) for oscillations reduces to \(|x| > 1/(2|k|)\). Thus, the oscillations set in outside of a finite interval of width \(\pm P/(4\pi)\), where \(P=2\pi/|k|\) is the fundamental period. This interval is however too narrow for a break in oscillation to be observed in the graphs of \(\sin(kx)\) and \(\cos(kx)\) (cf. Section 5.1).
When Laplace’s equation is separated in Cartesian or spherical coordinates, the resulting inertial terms \(by'\) are 0 and of the form \((2/x)y'\), respectively. It is interesting that the criterion (70) reduces to \(c(x) > 1/(4x^{2})\) in both of these cases. This inequality indicates that the lowest-level resistance to oscillation is present in these coordinate systems unlike in the case of cylindrical coordinates.
Declarations
Acknowledgements
During this research project, DMC and JG-E were supported by the University of Massachusetts Lowell, whereas QDK was on a sabbatical visit and was fully supported by the Jordan University of Science and Technology.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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