Discrete Fractional-Order Systems with Applications in Engineering and Natural SciencesView this Special Issue
A Transformation Method for Delta Partial Difference Equations on Discrete Time Scale
The aim of this study is to develop a transform method for discrete calculus. We define the double Laplace transforms in a discrete setting and discuss its existence and uniqueness with some of its important properties. The delta double Laplace transforms have been presented for integer and noninteger order partial differences. For illustration, the delta double Laplace transforms are applied to solve partial difference equation.
The origin of calculus of finite differences is found from Brook Taylor (1717), rather it was Jacob Stirling, who found the theory (1730) and introduce the delta symbol for the difference, which is in common use nowadays. The development on calculus of finite differences in the beginning of the nineteenth century by Lacroix and remarkable work of George Boole, Narlund, and Steffensen appeared later in the nineteenth century. Jordan discussed calculus of finite differences with the classical approach in . In modern era, the focus of mathematician is to correlate the continuous and the discrete, to shape in comprehensive unified mathematics, and to eliminate ambiguity. The calculus of finite differences is applicable to both continuous and discrete functions. For difference equations, Bohner and Peterson treat the dynamic equations on time scales in  and get surprisingly different results from continuous counterpart. Some results can be found in [3–19] which has helped to construct the theory of discrete fractional calculus.
Coon and Bernstein [20–22] defined the double Laplace transforms (continuous) and investigated many properties. Debnath  modified the properties and use the double Laplace transforms (continuous) to solve functional, integral, and partial differential equations. Dhunde and Waghmare  discussed convergence and absolute convergence of the double Laplace transforms (continuous) and, by application of double Laplace transforms, presented the solution of Volterra Integropartial differential equation. For applications of triple, quadruple, and -dimensional Laplace transforms (continuous), we refer the readers to [25–27]. Goodrich and Peterson  developed discrete delta Laplace transforms analogous to Laplace transforms discussed by Bohner and Peterson  in the continuous case, to solve difference and summation equations with initial data by applying the delta Laplace transforms. The delta Laplace transforms is given for newly defined Hilfer difference operator  and substantial difference operator in . Bohner et al.  generalized properties of the Laplace transforms to the delta Laplace transforms on arbitrary time scales and discussed translation theorems and transforms of periodic functions. Compatible discrete time Laplace transforms with Laplace transforms was introduced in . Savoye  highlighted the importance of discrete time problems and relationship of transforms to Laplace transforms on time scale. Fractional double Laplace transform was introduced in ; during derivation of Corollary 1, authors neglected the violation of semigroup property of Mittage-Leffler functions, and a counter example for semigroup property of Mittage-Leffler functions is given in . The qualitative analysis of delay partial difference equations is considered as discrete analog of delay partial differential equations by Zhang and Zhou . For solving partial difference equations Ozpinar and Belgacem introduced discrete homotopy perturbation Sumudu transform method in . For solving partial differential equations, double Laplace transform was applied in [37, 38].
Here, we introduce the delta double Laplace transforms similar to the one presented by Bernstein  in such a way that properties and expressions bear a resemblance to that appearing in Debnath  for the continuous calculus. The double convolution product, we consider in this article, resemble with the convolution product defined for delta calculus in [2, 10], but it differs from the one defined by Atici in . We consider the problem with constant coefficients in two independent variables and solve by applying the delta double Laplace transforms to partial difference equations with initial data.
This paper is divided into five sections. In Section 2, we shall present basic definitions and results from discrete calculus. Definition, existence, uniqueness, and series representation of the delta double Laplace transforms are given in Section 3. Some properties of the delta double Laplace transforms are proved in Section 4. In Section 5, we present the delta double Laplace transforms of partial differences.
For convenience, this section comprises of some basic definitions and results from discrete calculus for later use in the following sections. The functions we consider usually are defined on the set and , for fixed .
Falling function is defined for positive integer by
The forward jump operator is defined for by .
The set of regressive functions is defined for by .
The circle plus sum of is given by .
The additive inverse of is given by for .
Definition 1. (see ). Assume and . Then, the delta exponential function is given byBy the empty product convention, for any function .
Example 1. If is a constant such that (that is ), then the delta exponential function for a constant is given byFor a particular choice of , that is, the initial point of the domain of definition,
Definition 2. (see ). Assume and are in ; then, the delta definite integral is defined byNote that the value of integral , depending on the set . Also, by the empty sum convention,The delta indefinite integral is defined by
Definition 3. (see ). Assume . Then, the delta Laplace transform of based at is defined byfor all complex numbers such that this improper integral converges.
Note that throughout this article, we take the delta Laplace transform at the initial point of the set , unless stated otherwise.
The following concepts are also discussed in [10, 16].
Definition 4. (see ). A function is of exponential order if there exist a constant and the following inequality:If is of exponential order, then converges absolutely for , which ensures the existence of the Laplace transform. Even though the converse in not true, we restrict ourselves to only exponential order functions. For , the following are useful expressions for the delta Laplace transform of based at :for all complex numbers such that this infinite series converges.
Example 2. If , then for , we have
Definition 5. (see ). Assume . The convolution product is defined byNote that by the empty sum convention .
Lemma 1. (convolution theorem, see ). Assume . If both and exist, then the delta Laplace transform of the convolution product is given by
Lemma 2. (see ). Assume two functions . Let such that , and we have the summation by parts formula:
Definition 6. The generalized falling function is defined in term of gamma function bygiven that the expression in the above equation is justifiable. It is convenient to take , whenever is natural number and is a zero or negative integer.
Definition 7. The discrete Taylor monomial based at is defined byand the order Taylor monomial is defined by
Lemma 3. (see ). The following hold for delta Laplace of Taylor monomial:(i), for , (ii), for , (iii), for , .
In the next definition, we consider only delta difference with increment 1, and do not bother the different operators that we will not be using here. One can find the details of Definition 8 in [1, 39].
Definition 8. Assume , a function of two independent variables. Then, the partial difference of with respect to , regarding as a constant is given byThe partial difference of with respect to , regarding as a constant, is given byPartial difference equation is an equation containing partial differences.
Note that . Followed by the rule for integer order difference operator , we adopt the symbol for partial differences as follows: and .
3. The Delta Double Laplace Transforms
In this section, we give abstract definition of the delta double Laplace transform. For convenience, we simplify definition to series representation followed by Goodrich and Peterson  simplification of the delta Laplace transform. Also, condition for existence, uniqueness, and linearity of the delta double Laplace transform has been revealed.
Definition 9. Assume . Then, the delta double Laplace transform of based at () is the successive application of the delta Laplace transform on and in any orderwhere and are the delta Laplace transforms (single) based at with respect to and , respectively, and is the delta double Laplace transform based at . The delta double Laplace transform of a function of two variables and is defined in - plane provided the following double sum converges:for all complex numbers and .
One can easily verify by using Lemma 4 that . Later, in Theorem 2, we will prove that the double infinite series is absolutely convergent. It is well known that absolutely convergent series behave nicely and change in the order of summation allowed. Therefore, we can operate in either way .
Lemma 4. Assume . Then,for all complex numbers and such that the infinite series converges.
Proof. By using the definition of the delta double Laplace transform, consider the following:Now, by the definition of delta integral from discrete calculus, we obtainIn preceding steps, we use the definition of delta exponential function and the fact that and , since and are regressive functions. In the following step, we use and to reindex the sums as follows:
Theorem 1. Assume functions , , and such that the delta double Laplace transforms exist, then the following holds:(i)(ii)(iii)
Proof. Under the assumption stated above and by Lemma 4,(i)For and , we have(ii)The proof is similar to part (i).(iii)For and , we have
Example 3. (i)If for , then ,(ii)If for , then .(iii)By Lemma 4(iv)By using Theorem 1 part (iii), we obtainBy using Lemma 3,If we choose either or , then as a special case of the aboveCoon and Bernstein [20, 21] defined the double Laplace transforms and discussed convergence and existence for the continuous case. We discuss discrete analogue of the double Laplace transforms.
Definition 10. A function is of exponential order with respect to and , respectively, if there exist a constant and such that, for each and , the inequality holds, where for and .
Theorem 2. If a function is of exponential order , then the delta double Laplace transform converges absolutely for and provided , .
Proof. Assume is of exponential order . Then, there exists a constant and such that, for each and , . Thus, for and , we consider the following:Since and , therefore and . Hence, the delta double Laplace transform of converges absolutely.
Theorem 2 ensures the existence of the delta double Laplace transform. In general, the converse does not hold. We should consider functions of some exponential order , to ensure the delta double Laplace transform of which does converge somewhere in the complex plane outside the both closed balls of radius , centered at , that is, we can choose for and .
Theorem 3. Suppose . If the delta double Laplace transform of converges for and , where , and let , , then the delta double Laplace transform of converges for , , and , converges for and .
Proof. Since and the delta double Laplace transform of converges for and , where . We have that, for and ,Theorem 3 exposed the linearity property of the delta double Laplace transform, and Theorem 4 revealed the uniqueness.
Theorem 4. Suppose and , . If , provided , , and , then for all .
Proof. By hypothesis, we havefor and . This implies thatfor and . Since, by Theorem 2, the double infinite series is absolute convergent, therefore comparison of both sides implies thatFor each fix and for all , this implies thatFor each fix , we obtain
4. Basic Properties of the Delta Double Laplace Transform
In this section, following Bohner et al. , we prove some properties of the delta double Laplace transform. We also define double convolution product of discrete functions followed by Goodrich and Peterson  convolution product (single) of discrete functions. We present, the delta double Laplace transform of double convolution product for later use to solve difference equations.
Theorem 5. Assume and exists. If , thenwhere is the Heaviside unit step function defined by
Proof. We have, by Lemma 4,Reindex by and ,In the last step, we use Lemma 4 with the fact and .
Theorem 5 gives different results from its continuous counterpart stated in . We state the useful shifting Theorem 6 for discrete setting.
Theorem 6. Assume and exist. If , then(i).(ii), where is the Heaviside unit step function defined by
Theorem 7. Assume is periodic with and exist; then,
Proof. Under the assumption, we have, by Lemma 4,In the last step, we used and to reindex second double summation. In second double summation, periodicity of implies that
Definition 11. Assume . The double convolution product is defined byNote, by empty sum convention, .
Lemma 5. Assume . The double convolution product is commutative:
Proof. By Definition 11 and the change of variables and , we have
Theorem 8. (convolution theorem). Assume . If both and exist, then the delta double Laplace transform of double convolution product is
Proof. Under given assumption, we have, by Lemma 4 and the fact ,In the last step, we used Definition 11; next, making the change of variables and gives us thatIn the previous steps, we interchanged the order of first pairs and second pairs of summation and change variables and .
Proof. By double convolution theorem, we haveSince ,The last step is followed from single convolution Lemma 1.
5. The Delta Double Laplace Transforms of Partial Differences
In this section, we examine the action of the delta double Laplace transforms on first order partial differences. The results developed for first order partial differences are further used to establish properties of the delta double Laplace transforms of generalized order partial difference, similar to that appeared in  for fractional order partial derivatives. We usually consider functions , of exponential order with respect to and , respectively, ensuring that delta Laplace and the delta double Laplace transforms of and its partial differences does exist.
Lemma 6. Assume , such that the delta Laplace transforms exist for constants and . Then,
Proof. By definition of the delta Laplace transforms on ,Apply summation by parts (Lemma 2) on