Forms of Solutions for Some Two-Dimensional Systems of Rational Partial Recursion Equations

Abstract

In this paper, we offer the closed-form expressions of systems of second-order partial difference equations. We will utilize an alternative approach to verify the results by (odd-even) dual mathematical induction. We research and enforce the specific solutions of partial difference formulas and ordinary difference formulas as a straight effect.

1. Introduction

Partial difference and differential equations are common in mathematically oriented scientific areas, such as physics and engineering. They are fundamental in the modern-day scientific understanding of mechanical engineering analysis, sound, heat, diffusion, electrodynamics, electrostatics, fluid dynamics, elasticity, general relativity, and also quantum mechanics (see for instance [1–3]). We discover both ordinary and partial difference formulas in probability, characteristics, and even other mathematical physics locations. Certainly, Laplace, as well as Lagrange, took into consideration the solution of partial difference formulas in their research of characteristics as well as probability. An instance of a partial difference formula is the following famous relationship:

Some authors research solutions for partial difference formulas.

For instance, Heins [4] considered the solution ofunder some conditions.

In [5], Carlitz researched a solution of

For more outcomes on partial difference equations, we refer the reader to [3, 6–35].

In our work, we have actually researched the solutions complying with systems of partial difference formulas:where , , , and initial , , and are real numbers.

As a straight effect, we can derive the solutions of a family of partial difference equations:where , , , and initials , and are real numbers.

In addition, we can acquire the precise solution adhering to systems of ordinary difference formulas:where , , , and initials , and are real numbers.

2. Solution Forms

We give solutions of systems (4) and (5) for values of . This system can be rewritten as

2.1. When

Theorem 1. Let be a solution of (9) withwhere and . Suppose , , , and . Then, the solutions of (9), for and , are

Proof. From system (9), we see the following:Now, we prove that (11)–(14) hold for :Moreover, we prove that (11)–(14) hold for and :Now, suppose that (11)–(14) hold for and with . So,Now, prove that (11)–(14) hold for with :Now, we prove that (11)–(14) hold for with :Similarly,From (9), we haveThere are four cases:(1)If and is even, where .(2)If and is odd, where .(3)This is as in part (1)(4)This is as in part (2)

Proposition 1. Properties for system (9) are as follows:(1)If is even and , then (2)If is odd and , then (3)If is even and , then (4)If is odd and , then (5)If is even and , then (6)If is odd and , then (7)If is even and , then (8)If is odd and , then

Proposition 2. We have the following properties for system (9):(1)If is even and , then (2)If is odd and , then (3)If is even and , then (4)If is odd and , then (5)If is even and , then (6)If is odd and , then (7)If is even and , then (8)If is odd and , then

Remark 1. As a special case of (9), we have

Corollary 1. Let be a solution of (25) with initials . Suppose and . Then,

2.2. When

Theorem 2. Let be a solution of (27) with where and . Suppose , , , and . Then,

Proof. The proof is given as in Theorem (1).

Remark 2. If we considerthen the solution of equation (29) is as follows.

Corollary 2. Let be a solution of (29) with where and . Suppose and . Then,

Proposition 3. We have the following:(1)If is even and , then (2)If is odd and , then (3)If is even and , then (4)If is odd and , then (5)If is even and , then (6)If is odd and , then (7)If is even and , then (8)If is odd and , then

Remark 3. As a special of system (27), we have

Corollary 3. Let be a solution of (31) with , , and . Then,

Remark 4. As a special case of system (31), we getThe closed-form solution of (33) iswhere and .
If n is even (or odd) and , then ().
Also, if n is even (or odd) and , then ().

2.3. When

Theorem 3. Let be a solution of (35) with where and . Suppose , , , and . Then,where .

Proof. The proof is given by piecewise double mathematical induction.

Proposition 4. We have the following:(1)If is even and , then (2)If is odd and , then (3)If is even and , then (4)If is odd and , then (5)If is even and , then (6)If is odd and , then (7)If is even and , then (8)If is odd and , then

Proposition 5. We have the following:(1)If is even and , then (2)If is odd and , then (3)If is even and , then (4)If is odd and , then (5)If is even and , then (6)If is odd and , then (7)If is even and , then (8)If is odd and , then

Remark 5. As a special case,

Corollary 4. Let be a solution of (37) with initials . Suppose and . Then,where .

Data Availability

All the data utilized for this study have been included within the article and their sources are cited accordingly.

Conflicts of Interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.

Acknowledgments

The Higher Education Commission of Pakistan partially supported A. Q. Khan’s research.