Direct solutions of linear non-homogeneous difference equations
- Shu-Wen Pan^{1}Email author and
- Jia-Qiang Pan^{2}
https://doi.org/10.1186/s13662-016-0839-x
© Pan and Pan 2016
Received: 29 May 2015
Accepted: 11 April 2016
Published: 15 April 2016
Abstract
In this paper, the authors develop a direct method used to solve the initial value problems of a linear non-homogeneous time-invariant difference equation. In this method, the obtained general term of the solution sequence has an explicit formula, which includes coefficients, initial values, and right-side terms of the solved equation only. Furthermore, the authors find that when the solution sequence has a nonzero first term, it satisfies two adjoint linear recursive equations; this usually shows several new features of the solution sequence.
Keywords
MSC
1 Introduction: backward-shifting matrices of sequences
Let \(\mathbf{a}=\{a(k)\}_{k=0}^{\infty}=(a(0),a(1),a(2),\ldots,a(k),\ldots)\) be an infinite sequence. We also regard the sequence as a (\(1\times \infty\)) row vector. Two special sequences are the unit sequence \(\mathbf{e}=\{e(k)\}_{k=0}^{\infty}=\{\delta_{k,0}\}_{k=0}^{\infty}\) (\(\delta_{k,0}\) is the Kronecker delta) and the null sequence \(\mathbf{o}=\{0\}_{k=0}^{\infty}\).
For any sequence a, we always assume \(a(k)=0\) if \(k<0\). Thus, we may express all of the backward-shifting (namely right-shifting) sequences of a as \(\mathbf{a}_{(-1)}=\{a(k-1)\}_{k=0}^{\infty}=(0,a(0),a(1),a(2), \ldots)\), \(\mathbf{a}_{(-2)}=\{a(k-2)\}_{k=0}^{\infty}=(0,0,a(0),a(1),a(2),\ldots)\), \(\mathbf{a}_{(-3)}=\{a(k-3)\}_{k=0}^{\infty}=(0,0,0,a(0),a(1),a(2),\ldots)\), and so on.
We may see from (1) and (2) that, if \(\mathbf{a}\ast\mathbf{b}=\mathbf{b}\ast\mathbf{a}=\mathbf{c}\), then \(\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A}=\mathbf{C}\), where A, B, and C are the corresponding BS-matrices; and vice versa.
In the case that the first term \(a(0)\) of the sequence a is not zero, then the corresponding BS-matrix A is reversible, that is, there exists an inverse matrix \(\mathbf {A}^{-1}\) of A making \(\mathbf{A}\mathbf{A}^{-1}=\mathbf{A}^{-1}\mathbf{A}=\mathbf{E}\), where E is the \(\infty\times\infty\) unit matrix. Of course, E also is the BS-matrix of the unit sequence e. Furthermore, we may see that \(\mathbf{A}_{k}\mathbf{A}_{k}^{-1}=\mathbf {A}_{k}^{-1}\mathbf{A}_{k}=\mathbf{E}_{k}\), where \(\mathbf{A}_{k}\) and \(\mathbf{A}_{k}^{-1}\) (\(k\in \mathbb {N}_{0}\)) both are the \((k+1)\times (k+1)\) upper-left sub-matrices of the matrices A and \(\mathbf{A}^{-1}\), respectively, and \(\mathbf {E}_{k}\) (\(k\in \mathbb {N}_{0}\)) is the \((k+1)\times(k+1)\) unit matrix. The matrices \(\mathbf{A}_{k}\), \(\mathbf {A}_{k}^{-1}\) (\(k\in \mathbb {N}_{0}\)) are all upper triangular Toeplitz matrices.
If the three BS-matrices A, B, and C satisfy \(\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A}=\mathbf{C}\) and A is reversible (\(a(0)\neq0\)), then \(\mathbf{B}=\mathbf{A}^{-1}\mathbf{C}=\mathbf{C}\mathbf{A}^{-1}\), at the same time we have \(\mathbf{b}=\tilde{\mathbf{a}}\ast\mathbf{c}=\mathbf{c}\ast\tilde{\mathbf{a}}\).
In the second section of this paper, based on these relationships between sequences and their BS-matrices as mentioned above, we develop a direct method used to solve the initial value problem of a linear, time-invariant, non-homogeneous difference equation. In this method, the general term of solution sequence has an explicit formula, which includes coefficients and initial values of the solved equation only. In the third section of this paper, the authors point out that the solution sequence of the initial value problem surely satisfies two adjoint linear recursive equations, which may give the solution sequence several new features.
2 Direct solutions of linear non-homogeneous difference equations
Linear difference equations are ubiquitous in many engineering theories and mathematical branches. For example, they appear in the theory of discrete systems and control theory of discrete systems as basic models of the discrete systems [3–5], and discrete-time signal processing as basic recurrence relations of sampled signals [6]. In algebraic combinatorics, they are also one of main study topics, tying different special sequences in with their generating functions [1, 7]. For more details, the reader may refer to [8].
In this section, by means of the BS-matrices of sequences we give a direct computational method used to solve the initial value problem of a linear, time-invariant, non-homogeneous difference equation.
Theorem 2.1
Proof
We can rewrite equation (4) as a form of sequence convolution: \(\mathbf{a}\ast\mathbf{b}=\mathbf{c}\), where the general terms of the sequences b and c are shown in (6) and (7). Therefore, denoting the BS-matrices of a, b, and c by A, B and C, respectively, we have \(\mathbf{A}\mathbf{B}=\mathbf {C}\). For \(b(0)=1\), B is reversible and thus \(\mathbf{A}=\mathbf{C}\mathbf{B}^{-1}\). Hence, \(\mathbf{a}=\mathbf{c}\ast\tilde{\mathbf{b}}\), that is, equation (8) holds. □
Next, let us see several simple but enlightening examples, in which p is 1 or 2 only.
Example 2.2
Example 2.3
Example 2.4
Example 2.5
Remark 1
As we know, traditional methods used for solving linear non-homogeneous difference equations face several difficulties in practical applications. Here, the traditional methods include the classical method (the discrete analogue of the technique of solving linear differential equations, namely the solution of a non-homogeneous equation is the sum of the general solution of corresponding homogeneous equation and a particular solution of the non-homogeneous equation) [1], and the generating function method [1] or similar Z-transform method [5, 6] (usually, the former is used in mathematics, and the latter in engineering theories). Summarily, these methods are all unable to give explicit expressions of the general term of solution sequences by using coefficients and the non-homogeneous right-side terms. This is because they always need to find all complex roots of a polynomial (the characteristic polynomial of the equation). In general and higher order cases, that is very difficult. Besides, finding particular solutions of the non-homogeneous equations as in the classical method, or expanding rational fraction functions as a sum of simple fractions, or finding power series expansions or finding inverse Z-transforms as in the generating function method and Z-transform method, are also very difficult in general cases.
However, by using the direct method developed in this paper, we can directly solve the initial value problems of linear non-homogeneous difference equations (recursive relation). We can explicitly express the general term of the solution sequence by using the coefficients, initial values, and non-homogeneous right-side terms of the solved equation only. This is a distinct difference of the direct method.
Remark 2
3 Adjoint linear recursive equations
Let a be the solution sequence of the initial value problem of linear non-homogeneous difference equation (4). We may find that the sequence a satisfies an adjoint linear recursive relation, as shown in the following theorem.
Theorem 3.1
Proof
For the initial value problem (4) and (5), we have \(\mathbf{A}\mathbf{B}=\mathbf{C}\), where A, B, and C are BS-matrices of three sequences a, b, and c. The general terms of sequences b and c are shown in (6) and (7). Because \(b(0)=1\) and \(c(0)=a_{0}\neq0\), the matrices B and C both are reversible. Hence, we have \(\mathbf{C}^{-1}\mathbf{A}=\mathbf{B}^{-1}\), that is, \(\tilde{\mathbf{c}}\ast\mathbf{a}=\tilde{\mathbf{b}}\). □
Example 3.2
Example 3.3
We may give another adjoint linear recursive equation in a similar way, as follows.
Theorem 3.4
Proof
For the initial value problem (4) and (5), we have \(\mathbf{A}\mathbf{B}=\mathbf{B}\mathbf{A}=\mathbf{C}\), where A, B and C are the BS-matrices of the three sequences a, b, and c. The general terms of sequences b and c are shown in (6) and (7). Because \(c(0)=a_{0}\neq0\), the matrix C is reversible. Hence, we have \(\mathbf{C}^{-1}\mathbf{B}\mathbf{A}=\mathbf{E}\), that is, \(\tilde{\mathbf{c}}\ast\mathbf{b}\ast\mathbf{a}=\mathbf{e}\). □
Example 3.5
Example 3.6
4 Conclusions
In this paper, the authors develop a direct method used to solve the initial value problems of a linear non-homogeneous time-invariant difference equation. In this method, the obtained general term of the solution sequence has an explicit formula, which includes coefficients, initial values, and right-side terms of the solved equation only. Furthermore, when the solution sequence has a nonzero first term, it satisfies two adjoint linear recursive equations; this usually shows several new features of the solution sequence.
Declarations
Acknowledgements
This research is partially funded by China Scholarship Council (201408330553), National Natural Science Foundation of China (61573314), Natural Science Foundation of Zhejiang Province, China (LY14F030002 LY14F030003), Hangzhou Returned Oversea Student Innovation Project, Hangzhou science and Technology Projects (20140432B08, 20142013A64), and Teacher Science Research Foundation of Zhejiang University City College (J-14023). The authors would like to thank Professor Jie Pan and The University of Western Australia for research facilities and the referees for their helpful comments and suggestions.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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