A Group Theory Approach towards Some Rational Difference Equations

Abstract

A full Lie point symmetry analysis of rational difference equations is performed. Nontrivial symmetries are derived, and exact solutions using these symmetries are obtained.

1. Introduction

Over a century ago, symmetries became a centre of interest of several authors after the work of Sophus Lie [1] on differential equations. He studied the continuous group of transformations that leaves the differential equations invariant. This concept of symmetries is strongly related to the existence of conservation laws, and the relationship between them has attracted great interest among researchers following the work of Noether [2]. The extension of this idea to difference equations is now well documented (see [3] and references herein). In [3], Hydon developed a symmetry-based algorithm enabling one to derive solutions of difference equations without making any special lucky guesses. Hydon emphasized on second-order difference equations, although his algorithm is valid for any order. When it comes to higher-order equations, computations can be cumbersome and extra ansatz may be needed to ease the calculations.

We aim to extend the work by Elsayed [4] where the author investigated the dynamics and solutions ofwhere the initial conditions , and are arbitrary nonzero real numbers. For related work, see [5, 6]. One can notice that equation (1) is just a special case of a more general formwhere and are arbitrary sequences. We will use a symmetry-based method to solve (2). Equivalently, we studyinstead, where and are arbitrary sequences. This means we can only compare with . Furthermore, we use the same technique to obtain exact solutions ofwhere , and again we studyinstead. Note that solutions of (4) are found in [7]; however, their method is completely different from ours.

2. Definitions and Notation

The definitions are taken from Hydon [3], and most of the notation follows from the same book.

Definition 1. A parameterized set of point transformations,where and are continuous variables, is a one-parameter local Lie group of transformations if the following conditions are satisfied:(1) is the identity map if when (2) for every a and b sufficiently close to 0(3)Each can be represented as a Taylor series (in a neighborhood of that is determined by x), and thereforeConsider the pth-order difference equationfor some function . Assume the point transformations are of the formwith the corresponding infinitesimal symmetry generatorwhere S is the shift operator, i.e., . The symmetry condition is defined aswhenever (8) is true. Substituting the Lie point symmetries (9) into the symmetry condition (11) leads to the linearized symmetry condition:whenever (8) holds.

Definition 2. is invariant under the Lie group of transformations (9) if and only if .
We define the functionsand we adopt the standard conventionsWe refer the reader to [3] for a deeper understanding of the concept of symmetry analysis of difference equations.

3. Main Results

3.1. On the Difference Equation (3)

Consider the sixth-order difference equations of the form (3), that is,

We impose the symmetry condition (12) and simplify the resulting equation to get

To solve for Q, we first differentiate (16) with respect to (keeping fixed and viewing as a function of , and ). This leads, after simplification, towhere denotes the derivative with respect to the independent variable. We then differentiate (17) with respect to to get

The solutions of (18) are given byfor some functions and of n. To obtain more information on and , we substitute (19) in (16) and we split the resulting equation to get the following:

These equations ((20a)–(20e)) reduce to

The expression of in (21) is merely obtained by solving the corresponding characteristic equation for r. Assuming that , are the solutions of this characteristic equation, then is a linear combination of the ’s. In other words, the solutions of (21) arefor some arbitrary constants , where . Thus, we obtain four characteristics given by

The four corresponding symmetry generators , and are given by

Here, using , we introduce the canonical coordinate [8]

In view of (21), we introduce the variable

It is easy to check that

Therefore, is invariant under . It is advantageous to usethat is, . Here, we choose to use the plus sign and we show, using (3), that

Therefore,

It is worthwhile to mention that equation (30) gives the solution of (29) for all n. By reversing all the change of variables, we havewhere is given in (30), , and .

Note. Equation (31) gives the solution of (3) in a unified manner.
We can simplify (31) further by splitting it into six categories. We haveSimilarly, after a straightforward but lengthy computation, we get

Note. We can obtain (33) using (28) without considering the absolute value function.
Now, using (30) and (33), we obtain the solutions of (3) as follows:whenever the denominators do not vanish, i.e., and .
The implication is that solutions of (2) are given bywhenever the denominators do not vanish, i.e., and .

3.2. The Case Where and Are Two Periodic Sequences

Let , , , and . The solution in this case is given by the following equations:where and .

3.3. The Case Where and Are Constants

Here, , , , and . Equation (2) becomes .

3.3.1.

The solution given in (35a)–(35f) simplifies towhere and .(i)If , then equations in (37a)–(37c) are exactly the ones obtained in Theorem 2.1 in [4] for and their restriction (the initial conditions are arbitrary nonzero positive real numbers) is a special case of our restriction (the initial conditions are arbitrary nonzero real numbers and ).(ii)If , then equations in (37a)–(37c) are exactly the ones obtained in Theorem 3.1 in [4] for and their restriction (the initial conditions are arbitrary nonzero real numbers and ) coincides with our restriction (the initial conditions are arbitrary nonzero real numbers and ).

3.3.2. The Case Where

In this case, the solution given in (35a)–(35f) simplifies towhere and .

Note. If , then the solution given in Section 3.3.2 simplifies to(i)When setting in Section 3.3.2, we get the result obtained in Theorem 4.1 in [4] for and their restriction coincides with our restriction (the initial conditions are arbitrary nonzero real numbers, and ).(ii)When setting in Section 3.3.2, we get the result obtained in Theorem 5.1 in [4] for and their restriction coincides with our restriction (the initial conditions are arbitrary nonzero real numbers, and ).

4. On the Difference Equation (5)

Consider the fifth-order difference equations of the form (5), i.e.,

Here, the procedure for finding the characteristics of (5) is similar as above and is as follows:(i)Impose the symmetry condition (12) to (5)(ii)Differentiate with respect to (keeping fixed) and viewing as a function of , and (iii)Differentiate with respect to twice (keeping fixed)(iv)Use the method of separation

After performing this series of operations, we obtain the following characteristics:where . Thus, we obtain two characteristics with corresponding generators given by

From the characteristic equationswe obtain the invariants , and . We readily notice thatand we choose , i.e.,

By shifting (49) thrice, we getwhose solution is given by

The constants , can be obtained from the following equations:

Thanks to (49), we can express in terms of as follows:where is given in (51) with . The constants and must satisfy

Equations in (53a) and (53b) give the solutions of (5) in a unified manner.

For the sake of clarification, we split solutions (53a) to realize the solutions in the existing literature. Using (53a) and (53b), we have

Using the same approach, we show that

4.1. The Case of

Equation (5) becomesand we said earlier that the solution of (50), in this case, is (51), i.e.,

We haveand using (52a) in (58), we get

Using the same approach, we show that

Let and , , , and . Using (60) in (55), we obtain the solution of (56) as follows:whenever the denominators do not vanish.

4.2. The Case of

In this case, as we found earlier, the solution of (50) is given by (51), i.e.,

Using this, we getand using (52d) in (63), we find

Using the same approach, we show that

Using (65) in (55), we obtain the solution of (5) as follows:where . Equations in Section 4.2 give the exact solution of (5) for any real values of λ and μ provided that the denominators do not vanish.

Recall that we acted the shift operator on (4) to get (5). Hence, the solutions of (4) are obtained, using Section 4.2, as follows:for any real values of λ and μ as long as the denominators do not vanish.(i)When and , equations in Section 4.2 yield the results obtained by Yasin Yazlik in Theorem 5 in [9] for where , and e are positive real numbers.(ii)When and , equations in Section 4.2 yield the results obtained by Yasin Yazlik in Theorem 9 in [9] for where , and e are positive real numbers with and .

Note. There should not be a minus sign right after the expression of in Theorem 9 in [9].(i)When and , equations in Section 4.2 yield the results obtained by Yasin Yazlik in Theorem 7 in [9] for where , and e are nonzero real numbers with , , and .(ii)When and , equations in Section 4.2 yield the results obtained by Yasin Yazlik in Theorem 11 in [9] for where , and e are nonzero real numbers with , , and .

Note. There should be a minus sign right after the expression of in Theorem 11 in [9].

5. Conclusion

In this paper, we have obtained nontrivial symmetries of some rational ordinary difference equations and their exact solutions were obtained. Most importantly, we note that (21) gives a clear idea, without making any lucky guesses, for the most convenient choice of the invariant of (3).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.