Integrable Abel equations and Vein’s Abel equation

Integrable Abel equations and Vein’s Abel equation

Stefan C. Mancas stefan.mancas@erau.edu Fakultät für Informatik und Mathematik
Hochschule München - Munich University of Applied Sciences, Germany
   Haret C. Rosu hcr@ipicyt.edu.mx IPICyT, Instituto Potosino de Investigacion Cientifica y Tecnologica,
Camino a la presa San José 2055, Col. Lomas 4a Sección, 78216 San Luis Potosí, S.L.P., Mexico
online at Math. Meth. Appl. Sci. – 7/28/2015
Abstract

We first reformulate and expand with several novel findings some of the basic results in the integrability of Abel equations. Next, these results are applied to Vein’s Abel equation whose solutions are expressed in terms of the third order hyperbolic functions and a phase space analysis of the corresponding nonlinear oscillator is also provided.

Keywords: Abel equation; Appell invariant; normal form; canonical form; third order hyperbolic function.

Math. Meth. Appl. Sci. 39 (2016) 1376-1387

I Introduction

The first order Abel nonlinear equation of the second kind has the form

(1)

where , , , , and are all some functions of . The case has been introduced more than two hundred years ago by Abel a1 (). Many solvable equations of this type are collected in polzai () and other ones can be found in more recent works ct1 (); ct2 (); mak1 (); mak2 (); pz (); sh (); mr (). The transformation converts this equation in the form

(2)

which is Abel’s equation of the first kind. We see that Abel’s original equation of the second kind is actually a homogeneous case of that of the first kind.

One can also write (2) in the canonical form

(3)

where only the case leads to the simple identifications: , , , and .

The integrability features of Abel’s equation in the canonical form are extremely important because of its connection with nonlinear second order differential equations that phenomenologically describe a wide class of nonlinear oscillators which in this way can be treated analytically. In particular, Vein ve () introduced an Abel equation with and rational forms of , , and whose solutions are expressed in terms of third order hyperbolic functions. However, to the best of our knowledge, the paper of Vein went unnoticed for many years and only recently Yamaleev yama1 (); yama2 () provided a generalization in the framework of third order multicomplex algebra. Needless to say, the corresponding Vein’s nonlinear oscillator has not been studied in the literature. This was the main motivation for writing this paper, which is organized as follows. In section II, we present a lemma that encodes the connection of Abel equation with nonlinear oscillator equations and some related simple results. In section III, we reformulate several integrability results for Abel’s equation with the purpose to apply them to Vein’s Abel equation, which is the subject of section IV. The dynamical systems analysis of Vein’s oscillator is developed in section V, and we end up with the conclusions.

Ii Connection with second order nonlinear ODEs

The known importance of Abel’s equation in its canonical form (3) stems from the fact that its integrability leads to closed form solutions to a general nonlinear ODE of the form

(4)

where the variables and are some parametric solutions that depend on a generalized coordinate . This can be expressed by the following lemma mr ()

Lemma 1. Solutions to a general second-order ODE of type (4) may be obtained via the solutions to Abel’s equation (3) and vice versa using the following relationship

(5)


Proof. To show the equivalence, one just needs the chain rule

(6)

which turns (4) into the Abel equation of the second kind in canonical form

(7)

Via the inverse transformation

(8)

of the dependent variable, equation (7) becomes (3). Moreover, the linear term in (3) can always be eliminated via the transformation

(9)

which gives

(10)

where

(11)

The case where is seen from (2) to be the one actually considered by Abel, for in this case the reduced Abel equation

(12)

can be always put in the form

(13)

where and are reduced functions and are given by the expressions

(14)

Using the lemma, the reduced Abel equation corresponds to a linear ODE without higher-order dissipative terms

(15)

On the other hand, in the case of , the Abel equation of the first-kind (3) becomes a Riccati equation, while the second-kind homogeneous Abel equation (7) is reduced to the Riccati equation

(16)

which is equivalent to

(17)

This corresponds to the simple fact that an inverse power of a Riccati solution also satisfies a Riccati equation with redistributed coefficients. Now, we eliminate the linear part to obtain the reduced Riccati equation

(18)

which corresponds to a nonlinear ODE with higher-order dissipative terms

(19)

where now the coefficients are

(20)

Iii Some Abel integrability cases

iii.1 Abel’s equation of constant coefficients

Denote the coefficients of (3), where are constants and so that . It is obvious that the roots of the equation are themselves solutions of (3). More generally, the general solution of (3) is obtained via factorization of the denominator in the right-hand side of

(21)

which leads to the following cases:

  • If , so integration of (21) has the form

    (22)
  • If , then the integration of (21) is of the form

    (23)
  • If , so (21) becomes

    (24)
  • If and , the integration of (21) leads to

    (25)

iii.2 Integrability based on the normal form of Abel’s equation

If the following transformations as given in Kamke’s book kam ()

(26)

are applied to equation (3), then one obtains Abel’s equation in normal form

(27)

where the invariant is given by

(28)

Thus, we conclude that if , then (27) is integrable since it is separable. If one chooses relations between the functions such that the invariant is null and letting , then

(29)

is a Riccati equation, which is always integrable because it is obtained from Abel’s equation in normal form, which is integrable. Thus, the solution to (27) is

(30)

and explicitly, the solution to (3) with null invariant is

(31)

iii.3 Integrability of Abel equation with non-constant coefficients and

In this subsection, we will consider the original Abel equation (2) of the first-kind

(32)

which corresponds to (4) without the cubic nonlinearity.

Let us use the transformation

(33)

which is (9) with that allows us to put (32) into a differential form

(34)

where

(35)

Thus, the original equation (1) is considerably reduced only by having .

For equation (34), if there exists an integrating factor davis ()

(36)

where is a constant, then

(37)

is integrable, provided that

(38)

which leads to . Hence, the necessary condition for the integration of (32) is

(39)

Using , we find that there exists the potential , which satisfies

(40)

Therefore, the solution to (34) is

(41)

or in terms of the integrating factor

(42)

By substituting the condition (39), the invariant (28) has the form

(43)

iii.4 Canonical form of Abel’s equation and the integrating factor

According to the book of Kamke kam (), for equations of the type (32) for which there is no condition (39), one should change the variables according to

(44)

which lead to the canonical form

(45)

where

(46)

is the Appell invariant. Then, the integrating factor can be used to formulate the following interesting result:

Lemma 2. Any Abel equation in the canonical form (45) is integrable as long as the invariant is constant, with , and solution given by (41).

Proof. In (34), let , where , which leads to

(47)

that is separable, with solution

(48)

which after simplification leads back to (41).

Iv Vein’s Abel equation

The following Abel equation:

(49)

where , is known from a paper of Vein to be integrable ve (). We first notice that if we define the following determinants,

then they allow us to write the coefficients of Vein’s Equation (49) as , , and . Thus, (49) can be written in the compact form

(50)

Next, using the transformation given by (26) with the coefficients from (49), we have

(51)

and

(52)

Using these expressions, the invariant becomes

(53)

Therefore, the normal form of Abel’s Equation (45) is

(54)

Using (26) and integrating (54), we obtain

(55)

But because , we find

(56)

From the last two equations, one might infer that

(57)

and consequently using (26) would get

(58)

Unfortunately, because for (57), the argument of is satisfied, but the argument of is not satisfied, then we conclude that (58) is not the solution of (49).

To see how the implicit solutions (55), (57) can be untangled and written explicitly, we will proceed as Vein did. Firstly, we show that there are actually three solutions of the type (58), which are the solutions of (49) and can be put in the form ve ():

(59)

with the -functions defined cyclically as

where is an arbitrary constant as it does not appear in the differential equation. The functions are the following triad:

(60)

also known as the third-order hyperbolic functions, and are plotted in Fig. 1.

Figure 1: The third-order hyperbolic functions , of Equation (IV).

They are independent because their Wronskian , and they satisfy the following relationships

They also fulfill the following relationships

(61)

which are cyclic. That is, the fourth-order derivatives have the same property as the first ones, the fifth-order derivatives as the second ones, and so forth. In particular, they are independent solutions of the differential equation , but also of and of any , with .

Let

(62)

where is a constant. Then, fulfills

(63)

with . Suppose is defined implicitly as a multi-valued function of as follows

(64)

and denote the inverse relation by the explicit formula

(65)

Then satisfies Abel’s equation (49).

Indeed, from (64), differentiating with respect to and by the usage of (IV), one obtains

The quotient can be eliminated by means of (64), which leads to

(66)

Differentiating again, this time with respect to we have,

Eliminating the -quotients by means of (64) and (66), one can find after some reduction the following equation

Substituting and dividing by lead to Abel’s equation (49). Thus, the solution can be written explicitly as

(67)

with an arbitrary constant.

The other two solutions can be obtained by cyclically replacing the ’s in the function

(68)

V Vein’s nonlinear oscillator

To see what kind of oscillator corresponds to Vein’s Abel equation, we will proceed as follows. First, we will eliminate the linear term from (49) using

(69)

to obtain

(70)

where

(71)

Then using lemma 1 with , we obtain

(72)

which represents a nonlinear oscillator with rational friction and nonlinearity that are plotted in Figures 2 and 3, respectively, for the following values (i) ; (ii) ; (iii) ; (iv) .

Figure 2: The friction function of Vein’s oscillator at the fixed points , and .
Figure 3: The nonlinearity function of Vein’s oscillator at the fixed points , and .

Writing equation (72) as a dynamical system

(73)

and because (73) cannot be put in the form of (40), because , we will be using instead the standard methods of phase-plane analysis and use the linear approximation at the equilibrium points of (73) to classify them instead of solving the equation by finding the potential .

Also, notice that the system is non-Hamiltonian because there is no potential such that

(74)

The Jacobian matrix of (73) is

(75)

We have three equilibrium points of the system (73) from which one is real , while the other two are a pair of complex conjugates, , with and . The particular case which Vein considered, will reduce the three fixed points to the case of one real fixed point . The characteristic polynomial of the Jacobian matrix is

(76)

while the discriminant is . By evaluating the Jacobian at the real fixed point , where is either real or complex, we obtain

(77)

from which

(78)

1. Real fixed point: In this case we have

(79)

then, the real fixed point living on the axis is a stable spiral. The eigenvalues of the Jacobian matrix are complex conjugate pairs . By calculating the eigenvectors, the linearized solution around the fixed point is

(80)

where and are arbitrary constants.

2. Complex conjugated fixed points: In this case the coefficients of the characteristic polynomial are complex, and their values are shown in Table 1 and given in (81).

(81)

Because both coefficients and are not real nothing can be said about these complex fixed points.


The phase-plane portrait of the system (73) is shown in Fig. 4. The real fixed point is identified by the red dot, and the isoclines by the dotted curves.

Fixed Points Type
stable spiral
no conclusion
Table 1: Classification of equilibrium points of (73).
Figure 4: Phase-plane portrait at the fixed points for (top left); (top right); (bottom left); (bottom right).

Vi Conclusion

This paper recalls several integrability properties of Abel equations together with some simple consequences, followed by a discussion of Vein’s Abel equation related to third-order hyperbolic functions. The corresponding nonlinear oscillator system is introduced here, and its dynamical systems analysis is provided. Finally, it is worth noticing that the nonlinear oscillators corresponding to Abel equations are in fact a large class of generalized Liénard equations of the form , where , and are arbitrary functions and , where are constants hl () and, as such, have many applications in physics, biology, and engineering m10 (); nay95 ().

Vii Acknowledgment

The first author wishes to acknowledge support from Hochschule München while on leave from Embry-Riddle Aeronautical University in Daytona Beach, Florida. We also wish to thank the referees for their suggestions that led to significant improvements of this work.

References

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