The Intervals of Oscillations in the Solutions

of the Legendre Differential Equations

Dimitris M. Christodoulou and James Graham-Eagle

Department of Mathematical Sciences,University of Massachusetts Lowell,

Lowell, MA 01854, USA

and Qutaibeh D. Katatbeh

Department of Mathematics and Statistics,Jordan University of Science and Technology,

Irbid, Jordan 22110

Abstract

We have previously formulated a program for deducing the intervals of oscil-lations in the solutions of ordinary second-order linear homogeneous differ-ential equations. In this work, we demonstrate how the oscillation-detectionprogram can be carried out around the regular singular points x = ±1 of theLegendre differential equations. The solutions yn(x) of the Legendre equationare predicted to be oscillatory in |x| < 1 for n ≥ 3 and nonoscillatory outsideof that interval for all values of n. In contrast, the solutions ymn (x) of theassociated Legendre equation are predicted to be oscillatory for n ≥ 3 andm ≤ n−2 only in smaller subintervals |x| < x∗ < 1, the sizes of which are de-termined by n and m. Numerical integrations confirm that such subintervalsare distinctly smaller than (−1, +1).

Keywords: oscillations, second-order linear differential equations,analytical theory, transformations, Legendre differential equations2010 MSC: 34A25, 34A30, 42C05

1. Introduction

It is well-known (e.g., [1], sec. 10.2; [2], sec. XI.1) that all ordinarysecond-order linear homogeneous differential equations can be written in the

Preprint submitted to Applied Mathematics and Computation May 1, 2014

general formy′′ + b(x)y′ + c(x)y = 0, (1)

and that this form can be transformed to the canonical form

u′′ + q(x)u = 0, (2)

where the primes denote derivatives with respect to the independent variablex, q = −(b2 + 2b′ − 4c)/4, and y(x) = u(x) exp(−1

2

∫b(x)dx). The canonical

form (2) condenses all the information about the coefficients of eq. (1) intoq(x) and Oscillation Theory focuses on this coefficient in order to derive theoscillatory properties of the solutions of eq. (1) (see the reviews in [3, 4]and references therein). In a previous paper ([5], hereafter CGK), we alsofocused on the q(x) of eq. (2). We formulated a new definition of oscillationas the appearance of successive critical points of the same kind (maxima, orminima, or inflection points) in the graph of a solution. Then we describeda program for deducing the precise intervals of oscillations in the solutionsof the general form (1), and we presented a variety of examples in which thisprocedure was successful in predicting oscillatory behavior by examining thecoefficient q(x) alone.

Some of the differential equations that were investigated by CGK weretransformed to the harmonic-oscillator form with constant coefficients in thefirst step of the program, and their oscillatory properties were then derivedeasily from the discriminant of their characteristic quadratic equations. Therest of the equations were transformed to a form with constant damping(a constant coefficient of the first-derivative term), a quantity that clearlyopposes any oscillatory tendencies in the solutions. In this second step of theprogram, a generalized Euler transformation of the independent variable xwas used:

x = c1 + c2 exp(kt), (3)

where c1, c2, and k are arbitrary constants, and a criterion for the intervalsof oscillations in the solutions was established:

q(x) >1

4(x− c1)2. (4)

Only the constant c1 appears in the criterion and corresponds to a ”horizontalshift” of the independent variable x in eq. (3).

The equations studied by CGK in the second step of the program con-tained singularities at x = 0 or no singularities at all, in which cases c1 was

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set to zero in eqs. (3) and (4). In the present work, we extend our investi-gation to the study of oscillatory properties of the Legendre and associatedLegendre equations ([6, 7]) that have regular singular points at x = ±1 (thatis, away from x = 0) . In these cases, a horizontal shift c1 6= 0 proves to bequite useful, since it can be chosen to circumvent one or the other singularityin the neighborhood of which the intervals of oscillations in the solutions arebeing sought. The other two constants in eq. (3) are chosen in ways thatallow for the investigation of specific intervals in x (in particular, 0 ≤ x < 1and 1 < x < ∞). We use the above equations in § 2 and § 3 below, wherewe analyze the Legendre equation and the associated Legendre equation,respectively. Finally, in § 4, we summarize our results.

2. Legendre Equation

The Legendre equation ([6, 7])

y′′ − 2x

1− x2y′ +

n(n+ 1)

1− x2y = 0, (5)

where n is a constant, is first transformed to the canonical form (2) in whichthe coefficient q(x) is

q(x) =n(n+ 1)(1− x2) + 1

(1− x2)2, (6)

and u(x) = y(x)√|1− x2|. The coefficient q(x) is an even function of x, so

we can investigate the interval x ≥ 0 and the results will be applicable forx < 0 as well.

We choose c1 = 1 and k = 1 in eq. (3), in which case the transformationbecomes

x = 1 + c2 exp(t), (7)

and the criterion (4) for oscillatory solutions becomes

q(x) >1

4(x− 1)2. (8)

By setting c2 to −1 or +1 in eq. (7), we can search for oscillatory solutionsin intervals on either side of x = 1. Specifically, for c2 = −1:

t ∈ (−∞, 0] =⇒ x ∈ [0, +1),

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and for c2 = +1:

t ∈ (−∞, +∞) =⇒ x ∈ (+1, +∞).

In both cases, eq. (8) is the applicable criterion for oscillatory solutions, sothese two intervals in x can be investigated simultaneously. The pole atx = +1 is eliminated from the analysis when eqs. (6) and (8) are combined,in which case we find that the interval over which the solutions may oscillateis given by the solution of the algebraic inequality

Nx2 + 2x− (N + 2) < 0 =⇒ x ∈ (−1− 2/N, +1), (9)

whereN ≡ (2n+ 1)2. (10)

The endpoint x− = −1− 2/N lies on the negative side of the x−axis, there-fore oscillatory solutions are predicted in the entire interval x ∈ [0, +1).Furthermore, all solutions in x ∈ (+1, +∞) are predicted to be nonoscilla-tory.

Using the even symmetry of the canonical form (2) with q(x) given byeq. (6), we conclude that the solutions of the Legendre equation can oscillateonly in the interval

|x| < 1. (11)

On the other hand, we know from the results obtained by CGK that low-orderpolynomial solutions will be nonoscillatory and that the lowest-order solu-tions that may exhibit at least two critical points of the same kind (maxima,or minima, or inflection points) for certain choices of boundary conditionshave n = 3. The borderline case with n = 3 is discussed in the NodalAnalysis below.

Numerical Results.—Figs. 1 and 2 show two low-order numerical solutionsof eq. (5) in (−1, +1). In Fig. 1, n = 3 and the boundary conditions arey(0) = 1 and y′(0) = 0. In Fig. 2, n = 4 and the boundary conditions arey(0) = 0 and y′(0) = 1. In both cases, the solutions are oscillatory accordingto our definition since they exhibit two extrema of the same kind. Additionalnumerical integrations confirm that, for larger values of n, the solutions ex-hibit increasingly more oscillations in (−1, +1), while the solutions outsideof that interval are nonoscillatory for all values of n.

Nodal Analysis.—The above results can be better understood by countingthe number of nodes N (n) in x ∈ [−1, +1] for the two types of boundary

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Figure 1: The numerical solution of eq. (5) with n = 3 and boundary conditions y(0) = 1and y′(0) = 0.

conditions used. For y(0) = 1, the solutions are even and the number ofnodes is

N (n) =

{n, if n is even

n+ 1, if n is odd, (12)

while for y(0) = 0, the solutions are odd and then the number of nodes is

N (n) =

{n+ 1, if n is evenn, if n is odd

, (13)

where the node at x = 0 has been counted for the odd solutions of eq. (13).Nodes at x = ±1 do not occur for any n, not even in solutions that remainfinite at these endpoints.

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Figure 2: The numerical solution of eq. (5) with n = 4 and boundary conditions y(0) = 0and y′(0) = 1.

The node counts clarify why the lowest-order oscillatory solutions occurfor n = 3: The even solution has N (3) = 4 and develops two minima, so itis oscillatory. On the other hand, the odd solution has N (3) = 3 and twoextrema of the same kind do not occur, so it is nonoscillatory. For n ≥ 4,both types of solutions have enough nodes to guarantee oscillatory behavior,and the number of nodes and oscillations increases with increasing n.

3. Associated Legendre Equation

The associated Legendre equation ([6, 7])

y′′ − 2x

1− x2y′ + [

n(n+ 1)

1− x2− m2

(1− x2)2]y = 0, (14)

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where n and m are constants, is first transformed to the canonical form (2)in which the coefficient q(x) is

q(x) =n(n+ 1)(1− x2) + (1−m2)

(1− x2)2, (15)

and u(x) = y(x)√|1− x2|. The coefficient q(x) is an even function of x, so

we can investigate the interval x ≥ 0 and the results will be applicable forx < 0 as well.

We choose again c1 = 1 and k = 1 in eq. (3), in which case the transforma-tion (7) is applicable and the criterion for oscillatory solutions in x ∈ [0, +∞)is given by inequality (8). Substituting eq. (15) into eq. (8), we find that theinterval over which the solutions may oscillate is given by the solution of thealgebraic inequality

Nx2 + 2x− (N + 2M) < 0 =⇒ x ∈ (x−, x+), (16)

where

x± =1

N(−1±

√N2 + 2MN + 1), (17)

N is given by eq. (10), and

M ≡ 1− 2m2. (18)

The endpoint x− lies on the negative side of the x−axis, therefore oscillatorysolutions are predicted in the interval x ∈ [0, x+). Furthermore, all solutionsin x ∈ (x+, +∞) are predicted to be nonoscillatory.

Using the even symmetry of the canonical form (2) with q(x) given byeq. (15), we conclude that the solutions of the associated Legendre equationcan oscillate only in the interval |x| < x∗, where from eq. (17):

x∗ ≡ x+ =1

N(−1 +

√N2 + 2MN + 1). (19)

It is easy to show that x∗ < 1 for all values of m 6= 0. For m = 0, thenM = 1 and eq. (19) reduces to x∗ = 1, as was also found in § 2 above forthe Legendre equation. Therefore, in all cases of interest (in which m 6= 0and |m| ≤ |n|), the interval (−x∗, +x∗) is a distinct subinterval of (−1, +1)and the oscillations of the solutions of the associated Legendre equation willhave to cease before the solution curves approach the endpoints x = ±1.

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Figure 3: The numerical solution of eq. (14) with n = 5, m = 2, and boundary conditionsy(0) = 1 and y′(0) = 0. The oscillations terminate at x∗ ≈ ±0.93, as predicted by eq. (19).

Indeed, this behavior is observed in the numerical solutions of various caseswith m ≥ 1.

Numerical Results.—Figs. 3 and 4 show two numerical solutions of eq. (14)that oscillate in a distinct subinterval of (−1, +1). In Fig. 3, n = 5, m = 2,and the boundary conditions are y(0) = 1 and y′(0) = 0. In Fig. 4, n = 12,m = 8, and the boundary conditions are y(0) = 0 and y′(0) = 1. Ad-ditional numerical integrations show that, for larger values of n and a fixedvalue of m, the solutions exhibit increasingly more oscillations in (−x∗, +x∗),while the solutions outside of that interval are nonoscillatory for all valuesof n. Furthermore, for a fixed n and increasing m, the oscillation interval(−x∗, +x∗) shrinks and the solutions exhibit progressively fewer nodes. Fi-

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Figure 4: The numerical solution of eq. (14) with n = 12, m = 8, and boundary conditionsy(0) = 0 and y′(0) = 1. The oscillations terminate at x∗ ≈ ±0.77, as predicted by eq. (19).

nally, for m ≥ n−2, some solutions do appear that are nonoscillatory over theentire interval (−1, +1). In the borderline case withm = n−2, such nonoscil-latory solutions depend critically on the adopted boundary conditions (seeFigs. 5 and 6 below), while for m > n − 2, all solutions are nonoscillatory(see the Nodal Analysis below). Two examples of the borderline case withm = n − 2 = 10 are shown in Figs. 5 and 6. The odd solution in Fig. 5 isnonoscillatory, while the even solution in Fig. 6 exhibits two minima. Evenin cases such as that of Fig. 5, however, the nonoscillatory solutions aredistinctly nonmonotonic within the intervals specified by eq. (19).

Nodal Analysis.—The above results can be better understood by countingthe number of nodes N (n,m) in x ∈ [−1, +1] for the two types of boundary

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Figure 5: The numerical solution of eq. (14) with n = 12, m = n− 2 = 10, and boundaryconditions y(0) = 0 and y′(0) = 1. The solution is nonoscillatory as two extrema of thesame kind do not appear in the interval (−0.60, +0.60) that is determined from eq. (19).

conditions used. For y(0) = 1, the solutions are even and for 1 ≤ m ≤ n, thenumber of nodes is

N (n,m) =

{n−m+ 2, if n−m is evenn−m+ 1, if n−m is odd

, (20)

while for y(0) = 0, the solutions are odd and for 1 ≤ m ≤ n, the number ofnodes then is

N (n,m) =

{n−m+ 1, if n−m is evenn−m+ 2, if n−m is odd

, (21)

where the node at x = 0 has been counted for the odd solutions of eq. (21).The term +2 in both cases represents two nodes that occur at x = ±1. For

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Figure 6: As in Fig. 5 with m = n−2 = 10, but for different boundary conditions y(0) = 1and y′(0) = 0. The solution is oscillatory in the interval (−0.60, +0.60) that is determinedfrom eq. (19).

m 6= 0, such nodes always occur for even n −m in even solutions (eq. [20])and for odd n−m in odd solutions (eq. [21]).

The node counts clarify the borderline case with m = n − 2. In thiscase, n − m = 2 (even) and the even solutions all have 4 nodes (eq. [20]),just enough for two extrema of the same kind to develop; so they are oscil-latory. In contrast, the odd solutions have 3 nodes (eq. [21]) and they areall nonoscillatory. Finally, for m > n − 2, the solutions have a very smallnumber of nodes (1 ≤ N (n,m) ≤ 3) and then oscillations cannot occur.

Alternative Analysis of Oscillations.—The results shown in Figs. 3-6 havebeen verified by an independent analysis in x ∈ (−1, +1) as follows: For

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|x| < 1, the substitutionx = sin(t), (22)

transforms the associated Legendre equation (14) to the form

y − tan(t)y + [n(n+ 1)−m2sec2(t)]y = 0, (|t| < π/2), (23)

where dots denote derivatives of y(t) with respect to t. The canonical form (2)of this equation has u(t) = y(t)

√cos(t) and

q(t) =1

4[N + (2M − 1)sec2(t)], (24)

where N and M are given by eqs. (10) and (18), respectively. Since theseequations are strictly applicable in |t| < π/2, then the criterion (4) for oscil-latory solutions can be used with c1 = 0, in which case it reads

q(t) >1

4t2. (25)

Substituting eq. (24) into eq. (25), we find that oscillatory solutions can occurin a subinterval of |t| < t∗ < π/2 in which

N + (2M − 1)sec2(t)− 1

t2> 0, (26)

or, using eq. (22), in a subinterval of |x| < x∗ < 1 in which

N +2M − 1

1− x2− 1

[sin−1(x)]2> 0. (27)

We have solved this inequality numerically for various values of N and M ,and the resulting values of x∗ for all cases, including those depicted in Figs. 3-6, are virtually identical to those calculated analytically from eq. (19). Theagreement between results in the above two analyses justifies the use of a hori-zontal shift (c1 = 1) in the former analysis. Using horizontal shifts to removesingularities simplifies the calculations and leads to quadratic inequalitiesthat can be solved analytically.

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4. Summary

We have presented an analysis of the oscillatory properties of the solutionsof the Legendre and associated Legendre differential equations ([6, 7]) thatshow two regular singular points in their coefficients away from x = 0 (at x =±1). The analysis makes use of a program that was described in [5] by CGKfor investigating the x-intervals in which the solutions have an oscillatorycharacter. According to this procedure, an attempt must first be made totransform a given equation to a form with constant coefficients. Unlike in thecase of the Chebyshev equation that also has two poles at x = ±1 but can berecast to a form with constant coefficients (CGK), the Legendre equationscannot be simplified in ths manner and their analysis must proceed to thesecond step of the program, where a transformation of x is applied in orderto cast these equations into forms with constant damping (i.e., a constantcoefficient of the first derivative term). As was described in §§ 2 and 3 above,such a transformation is capable of eliminating from the analysis one or theother singularity, but not both of them at the same time. This leaves onepole in the coefficient of the canonical form of the final equation (eq. [6]and eq. [15]) and its presence complicates the search for oscillatory solutions.Because the coefficients (6) and (15) of the canonical form are even functionsof x, it turns out that we can investigate only the interval x ≥ 0 (which canbe made free of the singularity at x = +1), and then use symmetry to derivethe corresponding results for x < 0.

The results of our analysis are as follows:

(a) The solutions yn(x) of the Legendre equation are predicted to be os-cillatory in |x| < 1 for n ≥ 3 (according to the definition of oscillationgiven by CGK and adopted here as well) and nonoscillatory outsideof that interval for all values of n. Numerical integrations of manycases with a variety of boundary conditions confirm these predictions.Two examples of oscillatory solutions in (−1, +1) are shown in Figs. 1and 2.

(b) The solutions ymn (x) of the associated Legendre equation are predictedto be oscillatory for n ≥ 3 only and m ≤ n− 2 in smaller subintervalsof |x| < x∗ < 1, where x∗ is given by eq. (19). (Note however thatin the borderline case with m = n − 2, only the even solutions areoscillatory.) Numerical integrations, as well as a separate analysis of thealternative form (23) of the associated Legendre equation, confirm that

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such subintervals are predicted correctly and that they are distinctlysmaller than the interval (−1, +1) of the m = 0 case. Three examplesof oscillatory solutions are shown in Figs. 3, 4, and 6 and an exampleof a nonoscillatory solution is shown in Fig. 5.

In the example of Fig. 5 with m = n − 2 = 10, two extrema of thesame kind do not occur in the predicted interval |x| < 0.60 because of therelatively large value of m. In general, in the borderline case with m = n−2,some solutions are oscillatory and other solutions are not, depending onthe adopted boundary conditions (as in Figs. 6 and 5, respectively); butfor m > n − 2, all solutions are nonoscillatory because they exhibit just asmall number of nodes (1 to 3). Even these types of solutions are howevercharacteristically nonmonotonic as they attempt to oscillate, but the smallnumber of nodes does not permit for a full ”cycle” to develop in the intervals|x| < x∗ with x∗ specified by eq. (19).

References

[1] Whittaker, E. T., & Watson, G. N. 1920, A Course of Modern Analysis,3rd Ed., Cambridge Univ. Press, Cambridge

[2] Hartman, P. 1964, Ordinary Differential Equations, Wiley, New York

[3] Wong, J. S. W. 1968, Funkcialaj. Ekvacioj, 11, 207-234

[4] Agarwal, R. P., Grace, S. R., & O’Regan, D. 2002, Oscillation Theory forSecond Order Linear, Half-Linear, Superlinear and Sublinear DynamicEquations, Kluwer, Dordrecht

[5] Christodoulou, D. M., Graham-Eagle, J., & Katatbeh, Q. D. 2014, Appl.Math. Comp., submitted (CGK)

[6] Abramowitz, M., & Stegun, I. A., eds., 1972, Handbook of Mathemati-cal Functions with Formulas, Graphs, and Mathematical Tables, Dover,New York

[7] Bell, W. W. 2004, Special Functions for Scientists and Engineers, Dover,New York

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