Wolfram Mathematica application to determination of the number of solutions for certain nonlinear boundary value problems

ABSTRACT The nonlinear boundary value problem (BVP) , where or , with step-wise function , is studied. The number of nontrivial solutions for the problem is estimated. For the case, where , the exact number of solutions for the boundary value problem is given. With the help of Wolfram Mathematica, the examples show several ways to determine the number of solutions for BVP.


Introduction
The nonlinear oscillation in physics and applied mathematics has been intensively studied in many articles. Many papers, such as (Beléndez et al., 2017), (Beléndez et al., 2010), (Beléndez et al., 2016), (Elías-Zúñiga, 2013) presented analytical approximations to the periodic solutions and, in particular, to periodic solutions for oscillators described by ordinary differential equations with the odd-degree nonlinearity.
An alternative possibility of studying and solving differential equations is by using the method of Lie algebras. It was mentioned in (Shang, 2012), that Lie algebra solution of differential equations has found host of useful applications in physical systems, where wealthy symmetries exist. "In many physical or chemical systems, biological or epidemic models often lack of symmetries, which adds difficulty in finding a proper Lie algebra." An example (SIS epidemic spreading) of the application of this method was analysed in (Shang, 2012). This remark can be an impetus for the application of this method in the study of problems similar to those investigated in our article.
Motivated by these papers, the author in the current work wishes to find the exact formula for solutions (of the above-described equations) using Jacobian elliptic functions. Previous results of the author in this direction were published in a series of papers (Kirichuka, 2020), (Kirichuka, 2019), (Kirichuka & Sadyrbaev, 2018a), (Kirichuka and Sadyrbaev, Kirichuka & Sadyrbaev, 2018b).
The novelty of this research is that three ways to estimate the number of solutions to the boundary value problem are brought together.
However, insufficient attention has been paid to differential equation with even-degree nonlinearities. Although, for example, the quadratic nonlinearity has a practical application. As written in the work (Kovacic, 2020), this equation "has been used as a mathematical model of human eardrum oscillations". This fact motivated the search for an exact solution of differential equation with quadratic nonlinearity. Equations with quadratic nonlinearities were studied in (Chicone, 1987).
Solution methodology consists of three types (ways) of obtaining an estimate of the number of solutions. One of the ways that is widely used to estimate the number of solutions is the phase plane method, when we analyze the phase portrait of the equation and the monotonicity properties of solutions. The second way to determine the number of solutions is to analyze the exact graph of a solution function or graphs of systems solutions. The third method is to study the behavior of curves consisting of endpoints of trajectories on a given interval.
In our problem, we are dealing with three parameters a, b and δ and their influence on the number of solutions. There are multiple articles devoted to the study of differential equations, combined of several ones on disjoint subintervals of the main interval, for example, (Gritsans & Sadyrbaev, 2015), (Ellero & Zanolin, 2013), (Kirichuka and Sadyrbaev, 2018), (Kirichuka, 2016), (Moore & Nehari, 1959). In the paper (Kirichuka & Sadyrbaev, 2018a) an equation with cubic nonlinearity and step-wise potentials were studied together with the Dirichlet conditions.
We would like to study the same problems and compare the number of solutions. The differential equation (1) is a nonlinear equation with the quadratic or cubic nonlinearity that is switched off in a middle subinterval. We consider corresponding equations and that contain only the quadratic or cubic nonlinearity. The Equation (1) contains Equation (4) and (5) that were studied previously in (Kirichuka & Sadyrbaev, 2019), (Kirichuka, 2018), (Kirichuka, 2017), (Kirichuka, 2013), (Ogorodnikova & Sadyrbaev, 2006) and are included often in textbooks. We are not aware however of precise estimation of the number of solutions for the two-point BVP (1), (2). We study the problem (1), (2), where Equation (1) is a differential equation of the type (4) or (5) in two side subintervals I 1 and I 3 and is linear in the middle subinterval I 2 . The solutions in two side subintervals are described in terms of Jacobian elliptic functions (Gradshteyn & Ryzhik, 2000), (Milne-Thomson, 1972), (Whittaker & Watson, 1940, 1996. In the middle subinterval equation is linear x 00 ¼ À ax. The problem is to smoothly connect solutions in all subintervals. We compose a non-differential system of equations that gives the initial values of solutions for BVP (1), (2). Our results are: • the estimates of the number of solutions for the BVP (4), (2) and (5), (2) and their dependence on coefficient a; • the systems that produce solutions of the BVP (1), (2) are given for both choices of the function φðxÞ: • the estimates of the number of solutions for the BVP (1), (2) are obtained; • the examples are analyzed that show the validity of the above mentioned results and illustrate them.
The structure of the paper is the following. In the next section (Section 2) we describe previously obtained results on the Neumann problem for the quadratic and cubic equations. In Section 3 we obtain the systems that produce solutions of the BVP (1), (2) for both choices of the function φðxÞ: φðxÞ ¼ x or φðxÞ ¼ x 2 . The equations in those systems are obtained using the theory of Jacobian elliptic functions ( (Gradshteyn & Ryzhik, 2000), (Milne-Thomson, 1972), (Whittaker & Watson, 1940, 1996). In Section 4 we provide the main result on the number of solutions to the problem (1), (2) and we demonstrate how all the developed technique and formulas work in a specific example. In Section 5 we discuss the results and the novelty of the work.

Review of results on the number of solutions for the equations with quadratic and cubic nonlinearity
For the case, where qðtÞ ¼ b ¼ const > 0 in Equation (1). Consider the equation with quadratic nonlinearity that is given in (4). There are two critical points of Equation (4) at x 1 ¼ 0 and is a saddle as shown in Figure 1. The region bounded by homoclinic orbit is denoted G2.
Consider the Equation (5), there are three critical points of equation (5) both are saddle points. Two heteroclinic trajectories connect the two saddle points. The phase portrait of Equation (5) is depicted in Figure 2. The region bounded by two heteroclinic orbits is denoted G3.
Consider the Cauchy problem (4), It was proved in the article (Chicone, 1988), that the period of a solution to the problem (4), (2) is increasing function of x 0 . Therefore, the following statement is true. Theorem 1 Let i be a positive integer such that The Neumann problem (4), (2) has exactly 2i nontrivial solutions such that xðÀ The similar theorem and proof were provided in the article (Kirichuka & Sadyrbaev, 2019).
Consider the Cauchy problem (5), Theorem 2 Let i be a positive integer such that The Neumann problem (5), (2) has exactly 2i nontrivial solutions such that xðÀ The proof of Theorem 2 can be found in the articles (Kirichuka, 2019) and (Kirichuka & Sadyrbaev, 2018a).
Proposition 1 The number of nontrivial solutions for BVP (4), (2) and (5), (2) is the same and depends on the choice of coefficient a.

BVP with linear-quadratic equations
In the formulations below the Jacobian elliptic functions cd; sd; nd are used.
A solution of the Cauchy problem (4), xð0Þ ¼ x 0 , ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 3ð3a À 2bx 0 Þða þ 2bx 0 Þ p � � : (10) � nd ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Formulas (10) and (11) were obtained in article (Kirichuka & Sadyrbaev, 2019). Consider Equation (1), where φðxÞ ¼ x and qðtÞ is a step-wise function given by (3). Hence, we have the problems x 00 2 ¼ À a x 2 ; x 2 ðÀ 1 þ δÞ ¼ x 1 ðÀ 1 þ δÞ; x 2 ð1 À δÞ ¼ x 3 ð1 À δÞ; t 2 I 2 ; x 00 Using the change of the independent variable (t ! t À 1, t ! t þ 1) in (10), solutions of the problems are, respectively ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi and e x 3 ðt; x α Þ ¼ x 3 þ ðx α À x 3 Þcd 2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi The trajectories e x 1 ðtÞ and e x 3 ðtÞ are located in G2. In order xðtÞ to be C 2 -function both solutions e x 1 and e x 3 are to be smoothly connected by a middle function e x 2 ðtÞ: In order for the solutions e x 1 ðtÞ, e x 3 ðtÞ and e x 2 ðtÞ to connect smoothly, it is necessary for them to satisfy the following system. The following relations are to be satisfied: We solve the system (18) with respect to constants C 1 and C 2 . For this, we insert formulas (15), (16), (17) into the system (18). Then, making the certain transformations, we find constants C 1 and C 2 , equating them and find the expressions of solutions in formulas (19), (22). We get Φðx γ ; x α Þ ¼ sin ffi ffi ffi a p ðδ À 1Þ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Þk 2 cd ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi x 1 À x 3 þ ðx γ À x 1 Þcd 2 ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Ψðx γ ; x α Þ ¼ cos ffi ffi ffi a p ðδ À 1Þ: ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 2 3 b ðx 2 À x 1 Þ q ðx γ À x 2 Þk 2 cd ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi To simplify formulas (19), (22) we denote A ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi 1 6 b ðx 2 À x 1 Þ q and B ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ðδ À 1Þ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ðδ À 1Þ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi The system is obtained. We are interested in the number of solutions of boundary value problem (1), (2), where in (1) φðxÞ ¼ x. Proposition 2 For a, b and δ given a nontrivial solution ðx γ ; x α Þ of the system (23) produces a solution of the Neumann problem (1),(2), where in (1) φðxÞ ¼ x.

Result on the number of solutions to the BVP (1), (2)
Analysis of some examples have shown that the following assertions hold. We have considered several examples concerning the problems (1), (2), where 0 < δ < 1: One might expect that for δ ! 1 the equations (1) "tend" to the limiting equations (4) and (5). Numerical experiments show that this is not the case.
We have observed for the case of quadratic nonlinearity that if ffi ffi ffi a p is in the interval iπ 2 ; ðiþ1Þπ 2 � � and i is sufficiently large, the number of nontrivial solutions of the Neumann problem (1), (2) is less than 2i provided that δ is close to unity. The detailed analysis of the respective situation is given when considering Example 4. The problem (1), (2), where in (1) φðxÞ ¼ x has no nontrivial solutions for δ close to zero (the equation is then almost linear).
Similarly, we have observed for the case of cubic nonlinearity that if ffi ffi ffi a p is in the interval iπ 2 ; ðiþ1Þπ 2 � � and i is sufficiently large, the number of nontrivial solutions of the Neumann problem (1), (2) is greater than 2i provided that δ is close to unity. The evidence of this is in Example 4. The problem (1), (2), where in (1) φðxÞ ¼ x 2 has no nontrivial solutions for δ close to zero (the equation is then almost linear). Remark 1 At δ ¼ 0 (the equation is linear) the functions Φðx γ ; x α Þ and Ψðx γ ; x α Þ in (21), (22) are respectively À ffi ffi ffi a p ðx γ À x α Þ cos ffi ffi ffi a p and À ffi ffi ffi a p ðx γ þ x α Þ sin ffi ffi ffi a p . The system (23) for δ ¼ 0 takes the form where ffi ffi ffi a p � iπ 2 , i is a positive integer. Then the system (35) has only the trivial solutions x γ ¼ x α ¼ 0 and the BVP has no solutions for δ sufficiently small.
Remark 2 We note the following properties of the functions Φðx γ ; x α Þ and Ψðx γ ; x α Þ. The function Φ satisfies These relations mean that if a point ðx γ ; x α Þ solves the system (23) then symmetrical with respect to the bisectix point ðx α ; x γ Þ is also a solution.
In examples 4 and 4 we consider BVP, where equations contain only quadratic and cubic nonlinearities.
In Figures 5 and 6 we see the behaviors of curves of endpoints (at t ¼ 1Þ for equation (36). The curve of values ðxð1; x γ Þ; x 0 ð1; x γ ÞÞ for equation (36) is a spiral around the origin. Any point of intersection of these curves with the axis x 0 ¼ 0 corresponds to a solution of the BVP (36), (2).
Example 2 Consider equation (1), φðxÞ ¼ x 2 with a ¼ 50, qðtÞ ¼ b ¼ 25: Consider differential equation (38), where the initial conditions are xðÀ , then the number of solutions satisfying the boundary conditions (2) is four and for initial conditions xðÀ 1Þ ¼ x α , x 0 ðÀ 1Þ ¼ 0, À ffi ffi ffi 2 p < x α < 0 there are also four solutions to the problem, totally eight solutions. Therefore, the Theorem 2 is fulfilled. This is the case for i ¼ 4 (namely 4 π 2 < ffi ffi ffi ffi ffi 50 p < 5 π 2 ) in the inequality (9). On the other hand, the number of solutions to the problem (38), (2) can be determined using the formula (25) and the replacement t ! t þ 1. We get equation where k ¼ ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi The graph of f ð2; 50; 25; x α Þ is depicted in Figure 7. There are eight zeros of (39) and, respectively, eight initial values x α , which have solutions to the problem (38), (2). Therefore, Proposition 1 is fulfilled.
In Figures 8 and 9 we see the behaviours of curves of end-points (at t ¼ 1Þ for equation (38). The curve of values ðxð1; x α Þ; x 0 ð1; x α ÞÞ for equation (38) is a spiral around the origin. Any point of intersection of these curves with the axis x 0 ¼ 0 corresponds to a solution of the BVP (38), (2).
In this case, coefficient a is large enough but the number of nontrivial solutions is less than 2i.  ðxð1; x γ Þ; x 0 ð1; x γ ÞÞ for equation (40) leave the region G2 and therefore the number of solutions has decreased.
In Figures 14 and 15 we see the behaviors of curves of end-points (at t ¼ 1Þ for equation (40). The curve of values ðxð1; x γ Þ; x 0 ð1; x γ ÞÞ for equation (40) is more complicated spiral-like curve than that for quadratic equation (36) (see Figures 5 and 6). Any point of intersection of these curves with the axis x 0 ¼ 0 corresponds to a solution of the BVP (40), (2). Figures 14 and 15 show that there are fewer intersections points with the axis x 0 ¼ 0 than in the corresponding quadratic equation (36).
In this case, coefficient a is large enough and the number of nontrivial solutions is greater than 2i. For a ¼ 50, i ¼ 4 the number of solutions for Equation (41) must be 8, but there are 12 for Equation (41).           In Figures 23 and 24 we see the behaviors of curves of end-points (at t ¼ 1Þ for equation (41), where we can see how the additional solutions arise. The curve of values ðxð1; x α Þ; x 0 ð1; x α ÞÞ for equation (41) is more complicated spiral-like curve than for cubic equation (38) (see Figures 8  and 9). Any point of intersection of these curves with the axis x 0 ¼ 0 corresponds to a solution of the BVP (41), (2).
These solutions are depicted in Figure 26. In this case coefficient a is small enough and the number of nontrivial solutions is 2i. For a ¼ 4, i ¼ 1 the number of solutions must be two.
This estimate is in agreement with Theorem 2. Next, let us consider the case δ < 1.
These solutions are depicted in Figure 28. In this case coefficient a is small enough and the number of nontrivial solutions is 2i. For a ¼ 4, i ¼ 1 the number of solutions must be two.

Concluding discussion
In this paper, we investigated the BVP x 00 ¼ À ax þ qðtÞxφðxÞ, where φðxÞ ¼ x or φðxÞ ¼ x 2 , x 0 ðÀ 1Þ ¼ x 0 ð1Þ ¼ 0 with step-wise function qðtÞ given in (3). The systems that produce the solutions of the BVP (1), (2) are given for both cases of the function φðxÞ: φðxÞ ¼ x or φðxÞ ¼ x 2 . Using the possibilities the instruments of Wolfram Mathematica, the trajectories of those systems are constructed. Therefore, it is possible to determine the number of solutions to the problem and the initial values of solutions. This can be observed in Example 4 and Example 4. These examples show two ways to determine the number of BVP solutions. One of them uses the above-mentioned system, the second one uses behavior of curves of endpoints.
Example 1 and Example 2 consider BVP, where equations contain only quadratic and cubic nonlinearities. This example shows that the number of solutions to BVP can be estimated in three ways and the results obtained are the same. One of them is using results of Theorem 1 or Theorem 2 accordingly to inequality (7) or (9). The second way is to use the graph of exact solution (10) or (24) obtained in the author's works (Kirichuka & Sadyrbaev, 2019), (Kirichuka, 2019). The third way is using the behavior of curves of end points.
In Example 5 and Example 6 the estimates of the number of solutions for the BVP (1), (2) are obtained for small enough coefficient a and it was shown that the number of solutions is the same as in Theorem 1 or Theorem 2.
Despite the fact that the equation of quadratic nonlinearity looks simpler, finding a solution is more difficult. This can be explained by the fact that for cubic nonlinearity the solution trajectory in the phase plane is symmetric in all four quadrants, but for quadratic nonlinearity this is not the case.
Further research in the indicated direction can be conducted taking into account the following. More polynomial right hand sides f ðxÞ can be studied. The period annuli surrounding critical points appear often in theoretical research and in applications. The trajectories that escape regions like G2 go away and can tend to infinity. The reason is the step-wise character of the coefficient qðtÞ: Therefore the study of such resonant behaviour is possible. Evidently, this can be of practical value. Adding the damping terms of the form f ðxÞx 0 2 in the equation allows to consider more general cases. Certain transformations of dependent variables can reduce problems with damping to equations of the form studied here. The functions f ðxÞ can be considered which are not polynomials, but the equations have similar properties to what was studied in this paper.