Journal of Mathematics
Volume 2019 (2019), Article ID 8256867, 14 pages
https://doi.org/10.1155/2019/8256867

Research Article

On a Theory for Analysing Second-Order Systems of Ordinary Discrete Equations

J. J. H. Bashingwa¹ and A. H. Kara²

¹Comp. B Group, University of Cape Town, South Africa
²School of Mathematics, University of The Witwatersrand Johannesburg, South Africa

Correspondence should be addressed to A. H. Kara; abdul.kara@wits.ac.za

Received 30 December 2018; Accepted 21 April 2019; Published 12 May 2019

Academic Editor: Francisco Balibrea

Copyright © 2019 J. J. H. Bashingwa and A. H. Kara. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We present geometric based methods for solving systems of discrete or difference equations and introduce a technique for finding conservation laws for such systems.

1. Introduction

Depending on a model being studied, some physical laws may be described by differential equations (DEs). Lie group theory provides us with powerful tools for obtaining analytical solutions of such equations [1]. Over the last 30 years, a considerable amount of work has been invested into applying Lie’s theory to solve and classify difference equations (see [2–6] and references therein).

The use of symmetry methods for ordinary difference equations has been introduced by Maeda [2]. He showed that the resulting linearized symmetry condition (LSC) amounts to a set of functional equations which is hard to solve in general. Hydon introduced a technique to solve LSC and obtain symmetries in closed form by repetitive differentiations [5]. For example, a full classification of second-order according to their point symmetries exists in the literature [6]. However, for systems of difference equations , most of results are based on induction methods (see [7–9] and references therein). In this paper, we present a method for solving using their underlying symmetry.

It has been proved in [5] that every second-order linear homogeneous has an eight-dimension Lie algebra isomorphic to . This is not valid for second-order system of difference equations . We shall prove this in Section 3.

2. Groundwork

Let us consider an -th order system of

We assume that for each there exists at least one such that .

Consider a point transformation

where are continuous variables. will be called one-parameter Lie group of transformations if it satisfies the following properties:

(i) is the identity map, i.e., for (ii) for every and close to 0.(iii)Each can be expanded as a Taylor series in a neighbourhood of .

Therefore, we have

where are continuous functions which we shall refer to as characteristics, , and is the “shift” operator. It is defined as follows:

We define the discrete differentiation operator as follows:

where is the identity operator.

The symmetry condition for the (1) is

whenever (1) holds.

Lie symmetries are obtained by linearizing the symmetry condition (6) about the identity. We have the following system of linearized symmetry condition (SLSC):

where the symmetry generator is given by

Definition 1. A function is invariant function under the Lie group of transformations if

where can be found by solving the characteristic equation

Theorem 2. The discrete differential operator in (5) and the generator of symmetry in (8) commute.

Proof. We prove the theorem for any generalisation is straightforward.

Corollary 3. For each invariant , is also an invariant.

Proof. We have .
Equivalently,
or .

We shall use this corollary for reductions in Section 3.

A first integral for the system (1) is a quantity such that

whenever (1) holds.

In Section 5, we shall use the condition (12) to develop a constructive technique for obtaining first integrals.

Remark 4. In this paper, we shall consider Lie point symmetry; i.e, the characteristics are given by .

We refer the reader to [1] for more information on symmetry methods for differential equations.

3. Symmetries and Reductions

3.1. Finding Characteristics

Consider a second-order system of 2 Es

We assume that or and or , so the system is of second order.

The SLCS (7) reduces to

where , and .

The functional equations (14) and (15) contain functions and with different pairs of arguments making them difficult to solve. For concreteness, if, for instance, the discrete variable stands for “state” in physics, and belong to two different states.

To overcome this, we proceed as follows.

Step 1 (elimination of and ). We differentiate (total differentiation) (14) and (15) with respect to and , respectively, keeping and fixed. Here, we take as function of and as function of .

The total derivative operators are given by

In this case, this is simplified to

So we apply the operator (17) to (14) and (18) to (15) keeping and fixed. This leads to the determining system

Step 2 (elimination of and ). We now differentiate (19) and (20) with respect to and , respectively, keeping and fixed. This means that we apply the operator on (19) and on (20). For a second-order we need at most to differentiate four times. After separating with respect to and the resulting equations, we obtain a system of DEs in and which is solvable by hand or by using a computer algebra package.

Step 3 (explicit form of constant of integration). When integrating in Step 2 to obtain the characteristics and , we have constant of integration which appears to be functions of . To obtain their explicit form, we need to substitute the results obtained in Step 2 in (19) and (20). If we do not succeed in obtaining all the constant of integration, we need further substitution in the SLSC (14) and (15).

3.2. Reductions

Consider a second-order

and its symmetry generator

The method of characteristics for partial differential equations (PDEs)

leads to three independent constants of integration . Each invariant under is function of those constant. .

For second-order systems, two invariants suffice to do reduction of the systems.

Let

be the invariants functions under . We choose them in a way that the Jacobian is nonzero.

That is, (24) can be inverted as follows:

By Corollary 3, and are also invariant functions. Therefore, the solution of (21) satisfies

(27) is a first-order SEs which can be solved by further reductions or by using computer algebra software (maple, Mathematica,…) for linear systems. Note that there exist some first-order systems which cannot be solved analytically.

The general solution is

for some constant .

So the second-order system (21) is equivalent to the first-order system obtained by substituting (28) in (26):

(29) also admits the symmetries generated by . The best way to integrate any first-order analytic E is to use its canonical coordinates [10],

which satisfy

The obvious choice of canonical coordinates is (see [10])

3.3. Applications

3.3.1. Example 1

Consider the most general homogeneous second-order linear system of difference equations

where are arbitrary functions.

One can readily verify that the determining system (19) and (20) amounts to

Therefore

where are constants.

The characteristics in (35) must satisfy the SLSC (14) and (15). Hence, we have

and

So, (35) is simplified to

The first generator of symmetry for a second-order homogeneous linear system (36) is the scaling symmetry given by

The system (36), which governs the remaining generators of the Lie point symmetry for the system (33), is of second order in and . Its general solution is

where are constants.

So, the most large Lie algebra of symmetry generators which can be obtained from a homogeneous second-order system of 2 difference equations has dimension five.

For clarification let us consider

and . The system (33) becomes

The system which governs the remaining generators of the Lie point symmetry in this case is given by

The general solutions for this system will be

Therefore we have 5 generators of the Lie point symmetry spanned by

3.3.2. Example 2

Consider the system

(45) is a special case of systems investigated in [11], where the author looked at the stability of the systems.

We choose the ansatz .

The determining system (19) and (20) amounts to

Differentiating twice (46) with respect to and twice (47) with respect to keeping and fixed we obtain, after separating with respect to and , the following system of Des:

whose most general solutions are

To obtain the nature of functions we substitute (49) in (46) and (47). After separating with respect to and we get the following SEs:

whose solutions are

The remaining unknown functions and are determined by substituting (51) and (49) into the SLSC (14) and (15). This leads to the SEs

The general solutions to (52) are given by

where are arbitrary constants. It follows that the characteristics are given by

Therefore, we have six generators of Lie point symmetry

Each generator in (55) can be used to reduce the order of (45).

Let us consider . By the characteristic method for Partial Differential Equations, the invariants are given by following equation:

We get

where are constants.

If we choose , we have

and if we choose , we have

From (58) and (59)

we deduce

Let us now consider the generator . The resulting invariants are

where

Note also the relationship between them

From (61), we deduce the following relation:

One can readily check that the general solution to (64) is given by

where and are defined in (62).

From (61) we obtain

which is a first-order system after substitution of by the results given in (65). Its solutions can be obtained by using the following canonical coordinates:

This leads to the the following linear system with variable coefficients:

where and .

The latter is a linear first-order system with variable coefficients. Its general solution is

The general solution of (45) is obtained by substituting (69) into (67).

4. Conservation Laws

In Section 2, we have defined a first integral associated with a second-oreder SEs. It is given by (12)

Let

By differentiating (70) with respect to and we obtain

and

The substitution of (72) in (73) leads to the following second-order system of functional equations:

As for SLSC, we differentiate repeatedly to obtain a system of DEs for and . Given the solutions of (74) we easily construct . For consistency of our solutions, we must check the integrability conditions

and

The first integral is then given by

The constant of integration which is a function depending on is determined by substituting (77) in (70).

4.1. Applications

Let us consider the second-order SEs

By choosing the ansatz and one can readily check that the determining system (74) is simplified to

where and denote the right-hand side of (78).

Differentiating (79) with respect to and leads to

Thus, we have from (72)

Substituting (80) in (79) and separating with respect to and we obtain the system

The solutions to (82) will provide us with the explicit form of

The first integral is then given by

for some constants .

For clarification, let us consider , that is,

The solutions to (82) will be

where are constants. We have twelve solutions for and . That is,

(1)If

 then

(2)If

 then

(3)If

 then

(4)If

 then

(5)If

 then

(6)If

 then

(7)If

 then

(8)If

 then

(9)If

 then

(10)If

 then

(11)If

 then

(12)If

 then

Therefore, we obtain twelve conservation laws for the system (84). They are given by

5. Conclusion and Discussions

We have presented a method for obtaining nontrivial symmetries and how to use them for solving a second-order SEs. Each symmetry can be used to reduce the order. However, different symmetries lead to different reductions (see (60) and (64)), but the same solution. We also proposed a technique to construct first integral associated to second-order systems of difference equations.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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