On Multiple Convolutions and Time Scales

Abstract

The properties of the multiple Laplace transform and convolutions on a time scale are studied. Further, some related results are also obtained by utilizing the double Laplace transform. We also provide an example in order to illustrate the main result.

1. Introduction

A time scale ⊤ is a nonempty closed subset of the real numbers ℝ and it has the topology that it inherits from the real numbers with the standard topology. It is also well known that if the time scale is the set of real numbers, then the dynamic equation is a differential equation, whilst if the time scale is the set of integers, then the dynamic equation is reduced to a difference equation. Thus integral transform of differential and integral calculus on time scales allows us to develop a theory of differential equations. For the single Laplace transform on time scales, see [1–3]. Similarly, in [4–8], the authors defined the time scale to be any nonempty closed subset of the real numbers and provided the motivation and formulation of delta derivatives on a time scale as well as the properties of delta derivatives and integrals. In [2], Bohner and Peterson defined the Laplace transform of a time scale for function of single variable as follows.

Definition 1. For 𝑓∶⊤→ℂ, the time scale or generalized Laplace transform of 𝑓, denoted by 𝐿[𝑓(𝑡)] or 𝐹(𝑧), is given by 𝐿𝑓(𝑡)(𝑧)=^∞₀𝑒^𝜎_⊖𝑧(𝑡,0)𝑓(𝑡)Δ𝑡,for𝑧∈𝐷𝑓, (1) where 𝐷{𝑓} consists of all 𝑧∈ℂ for which the improper integral exists and 1+𝜇₁𝑧≠0 for all 𝑡∈⊤.

Definition 2. The function 𝑓∶⊤→ℝ is said to be of exponential type I if there exist constants 𝑀,𝑐>0 such that |𝑓(𝑡)|≤𝑀𝑒^𝑐𝑡. Furthermore, 𝑓 is said to be of exponential type II if there exist constants 𝑀,  𝑐>0 such that |𝑓(𝑡)|≤𝑀𝑒_𝑐(𝑡,0).

The following two theorems were proved in [3].

Theorem 3. The integral ∫^∞₀𝑒^𝜎_⊖𝑧(𝑡,0)𝑓(𝑡)Δ𝑡 converges absolutely for 𝑧∈𝐷 if 𝑓(𝑡) is of exponential type II with exponential constant 𝑐. For more details, see [3].

Theorem 4. Let 𝛼∈ℝ be regressive. Then
(I)𝐿𝑒_𝛼(𝑧)=1𝑡,0)(,𝑧−𝛼 (2) provided that lim_𝑡→∞𝑒_𝛼⊖𝑧(𝑡,0)=0,
(II)𝐿cos_𝛼(𝑧)=𝑧𝑡,0)(𝑧²+𝛼², (3) provided that lim_𝑡→∞𝑒_{𝑖𝛼⊖𝑧}(𝑡,0)=lim_𝑡→∞𝑒_{−𝑖𝛼⊖𝑧}(𝑡,0)=0,
(III)𝐿sin_𝛼(𝑧)=𝛼𝑡,0)(𝑧²+𝛼², (4) provided that lim_𝑡→∞𝑒_{𝑖𝛼⊖𝑧}(𝑡,0)=lim_𝑡→∞𝑒_{−𝑖𝛼⊖𝑧}(𝑡,0)=0.

The following theorem was also studied by Bohner and Peterson in [2].

Theorem 5. If 𝑓∶⊤→ℝ is such that 𝑓^Δ is regulated, then 𝐿𝑓^Δ(𝑧)=𝑧𝐿𝑓(𝑡)(𝑧)−𝑓(0), (5) for all 𝑧∈𝐷{𝑓} such that lim_𝑡→∞{𝑒_⊖𝑧(𝑡)𝑓(𝑡)}=0.

Throughout the study, the following notation will be useful, where ⊤₁,⊤₂ are time scales and 𝑒_𝑎⊕𝑏𝑡₁,𝑡₂=𝑒_𝑏𝑡₁𝑒_𝑏𝑡₂,𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂=𝑒_⊖𝑧1𝑡₁𝑒_2⊖𝑧𝑡₂. (6)

In [2], further it was proved that 𝑒^𝜎_⊖𝑧=𝑒_⊖𝑧11+𝜇𝑧 (7) was satisfied, where 1+𝜇₁𝑧≠0, by using the property 𝑒^𝜎_𝑧=1+𝜇𝑧𝑒_𝑧𝑧,⊖𝑧=−1+𝑧,𝑝⊖𝑞=𝑝−𝑞.1+𝜇𝑞 (8)

Later in [9], Bohner and Guseinov defined the single convolution operation on time scale of two functions 𝑓,𝑔∶⊤→ℝ by the following formula: 𝑓∗𝑔(𝑡)=𝑓^𝑡_𝑡0^⌢(𝑡,𝜎(𝑠))𝑔(𝑠)Δ𝑠. (9)

In this study, we introduce double Laplace transform on time scales and study some of the properties in solving partial differential equations. Further, we also extend the Δ-Laplace transform, which is given by Jackson in [10], to the multiple Δ-Laplace transform as follows.

Definition 6. Let 0∈⊤₁,⊤₂,  sup{⊤₁,⊤₂}=∞, and 𝑡₁∈⊤₁,𝑡₂∈⊤₂. The Δ-double Laplace transform of the function 𝑓(𝑡₁,𝑡₂) (for 𝑓∈𝐶(⊤₁×⊤₂)) with respect to 𝑡₁,𝑡₂ is given by 𝐿_𝑡1𝐿_𝑡2𝑓𝑡₁,𝑡₂𝑧₁,𝑧₂=^∞₀𝑒^𝜎1𝜎2_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂𝑓𝑡₁,𝑡₂Δ₁𝑡₁Δ₂𝑡₂ (10) provided that the integral exists, further (1+𝜇₁𝑧₁)≠0, and (1+𝜇₂𝑧₂)≠0, for all (𝑡₁,𝑡₂)∈(⊤₁×⊤₂)₊.

Lemma 7. If 𝑧₁,𝑧₂∈ℂ are regressive, then 𝑒^𝜎1𝜎2_{⊖𝑧1⊖𝑧2}=𝑒_{⊖𝑧1⊖𝑧2}1+𝜇₁𝑧₁1+𝜇₂𝑧₂. (11)

Proof. By using the above analysis, we have 𝑒^𝜎1𝜎2_{⊖𝑧1⊖𝑧2}=1+𝜇₁⊖𝑧₁𝑒_⊖𝑧11+𝜇₂⊖𝑧₂𝑒_⊖𝑧2=1−𝜇₁𝑧₁1+𝜇₁𝑧₁1−𝜇₂𝑧₂1+𝜇₂𝑧₂𝑒_{⊖𝑧1𝑒2⊖𝑧}=𝑒_{⊖𝑧1⊖𝑧2}1+𝜇₁𝑧₁1+𝜇₂𝑧₂. (12)

As an example, we can easily compute 𝐿_𝑡2𝐿_𝑡1[𝑒_𝑎⊕𝑏(𝑡₁,𝑡₂,0,0)](𝑧₁,𝑧₂), where 𝑎,𝑏 are constants such that lim_{𝑡1𝑡→∞2→∞}𝑒_{𝑎⊕𝑏⊖𝑧1⊖𝑧2}(𝑡₁,𝑡₂,0,0)=0.

We have 𝐿_𝑡2𝐿_𝑡1𝑒_𝑎⊕𝑏𝑡₁,𝑡₂,0,0𝑧₁,𝑧₂=^∞₀𝑒_𝑎⊕𝑏𝑡₁,𝑡₂,0,0𝑒^𝜎1𝜎2_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂Δ₁𝑡₁Δ₂𝑡₂=^∞₀1+𝜇_𝜎1𝑡₁⊖𝑧₁𝑡₁1+𝜇_𝜎2𝑡₂⊖𝑧₂𝑡₂𝑒_𝑎⊕𝑏×𝑡₁,𝑡₂,0,0𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂Δ₁𝑡₁Δ₂𝑡₂=^∞₀𝜇1−^𝜎1𝑡₁𝑧₁1+𝜇_𝜎1⊖𝑧₁×^∞₀𝜇1−^𝜎1𝑡₂𝑧₂1+𝜇_𝜎2⊖𝑧₂𝑒_𝑏⊖𝑧2𝑡₁,𝑡₂,0,0Δ₂𝑡₂𝑒_𝑎⊖𝑧1×𝑡₁,𝑡₂Δ₁𝑡₁=^∞₀11+𝜇_𝜎1⊖𝑧₁×^∞₀11+𝜇_𝜎2⊖𝑧₂𝑒_𝑏⊖𝑧2𝑡₁,𝑡₂,0,0Δ₂𝑡₂×𝑒_𝑎⊖𝑧1𝑡₁,𝑡₂,0,0Δ₁𝑡₁=1𝑎−𝑧₁^∞₀1𝑏−𝑧₂^∞₀𝑏⊖𝑧₂𝑒_𝑏⊖𝑧2𝑡₁,𝑡₂,0,0Δ₂𝑡₂×𝑎⊖𝑧₁𝑒_𝑎⊖𝑧1𝑡₁,𝑡₂,0,0Δ₁𝑡₁=1𝑎−𝑧₁^∞₀1𝑏−𝑧₂^∞₀𝑒_𝑏⊖𝑧2𝑡₁,𝑡₂,0,0_Δ2𝑡2Δ₂𝑡₂×𝑒_𝑎⊖𝑧1𝑡₁,𝑡₂,0,0_Δ1𝑡1Δ₁𝑡₁=1𝑧₁−𝑎𝑧₂.−𝑏 (13) For 𝑡₁∈⊤₁ and 𝑡₂∈⊤₂ with 𝑡₁≥𝛼 and  𝑡₂≥𝛽, we can easily see that cosh_{(𝛼⊕𝛽)}𝑡₁,𝑡₂=𝑒_{(𝛼𝑡1⊕𝛽𝑡2)}+𝑒_{−(𝛼𝑡1⊕𝛽𝑡2)}2,sinh_{(𝛼⊕𝛽)}𝑡₁,𝑡₂=𝑒_{(𝛼𝑡1⊕𝛽𝑡2)}−𝑒_{−(𝛼𝑡1⊕𝛽𝑡2)}2,cos_{(𝛼⊕𝛽)}𝑡₁,𝑡₂=𝑒_{𝑖(𝛼𝑡1⊕𝛽𝑡2)}+𝑒_{−𝑖(𝛼𝑡1⊕𝛽𝑡2)}2,sin_{(𝛼⊕𝛽)}𝑡₁,𝑡₂=𝑒_{𝑖(𝛼𝑡1⊕𝛽𝑡2)}−𝑒_{−𝑖(𝛼𝑡1⊕𝛽𝑡2)},2𝑖 (14) and then the double Laplace transforms are given by 𝐿_𝑡2𝐿_𝑡1cosh_{(𝛼⊕𝛽)}𝑡₁,𝑡₂=1/𝑧₁−𝛼𝑧₂−𝛽+1/𝑧₁+𝛼𝑧₂+𝛽2=𝑧₁𝑧₂+𝛼𝛽𝑧²₂−𝛽²𝑧²₁−𝛼²,𝐿_𝑡2𝐿_𝑡1sinh_{(𝛼⊕𝛽)}𝑡₁,𝑡₂=1/𝑧₁−𝛼𝑧₂−𝛽−1/𝑧₁+𝛼𝑧₂+𝛽2=𝛽𝑧₁+𝛼𝑧₂𝑧²₂−𝛽²𝑧²₁−𝛼². (15)(16)

Similarly, we have 𝐿_𝑡2𝐿_𝑡1cos_{(𝛼⊕𝛽)}𝑡₁,𝑡₂=𝑧₁𝑧₂−𝛼𝛽𝑧²₂+𝛽²𝑧²₁+𝛼²,𝐿_𝑡2𝐿_𝑡1sin_{(𝛼⊕𝛽)}𝑡₁,𝑡₂=𝑧₁𝑧₂+𝛼𝛽𝑧²₂+𝛽²𝑧²₁+𝛼². (17)

Proposition 8. Suppose ⊤₁,  ⊤₂ are two time scales such that 𝜇₁<𝑁₁ and 𝜇₁<𝑁₂ for some 𝑁₁,𝑁₂∈ℝ and all (𝑡₁,𝑡₂)∈⊤₁×⊤₂. Let 𝑓∶(⊤₁×⊤₂)₊→ℝ and lim_{𝑡1𝑡→∞2→∞}𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂𝑓𝜎𝑡₁,𝜎𝑡₂,0,0=0. (18) Then lim_{𝑡1𝑡→∞2→∞}𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂𝑓𝑡₁,𝑡₂,0,0=0. (19)

Proof. Since lim_{𝑡1𝑡→∞2→∞}𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂𝑓𝜎𝑡₁,𝜎𝑡₂,0,0=0 (20) and on using the following relation: 𝑒_{⊖𝑝1⊖𝑝2}𝜎𝑡₁,𝜎𝑡₂=1,0,01+𝜇₁𝑝₁1+𝜇₂𝑝₂𝑒_{⊖𝑝1⊖𝑝2}𝑡₁,𝑡₂=,0,0⊖𝑝₁⊖𝑝₂𝑝₁𝑝₂𝑒_{⊖𝑝1⊖𝑝2}𝑡₁,𝑡₂,0,0, (21) we have lim_{𝑡1𝑡→∞2→∞}⊖𝑧₁𝑡₁⊖𝑧₂𝑡₂𝑧₁𝑧₂𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂𝑓𝑡₁,𝑡₂,0,0=0,lim_{𝑡1𝑡→∞2→∞}⊖𝑧₁𝑡₁⊖𝑧₂𝑡₂𝑧₁𝑧₂1+𝜇₁𝑧₁1+𝜇₂𝑧₂𝑒_{⊖𝑧1⊖𝑧2}×𝑡₁,𝑡₂𝑓𝑡₁,𝑡₂,0,0=0,lim_{𝑡1𝑡→∞2→∞}11+𝜇₁𝑧₁1+𝜇₂𝑧₂𝑒_{⊖𝑧1⊖𝑧2}×𝑡₁,𝑡₂𝑓𝑡₁,𝑡₂,0,0=0. (22) Since 𝜇₁(𝑡₂)<𝑁₁ and 𝜇₁(𝑡₂)<𝑁₂, we get lim_{𝑡1𝑡→∞2→∞}𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂𝑓𝑡₁,𝑡₂,0,01+𝜇₁𝑧₁1+𝜇₂𝑧₂>lim_{𝑡1𝑡→∞2→∞}𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂𝑓𝑡₁,𝑡₂,0,01+𝑁₁𝑧₁1+𝑁₂𝑧₂>0. (23) Thus we obtain that 11+𝑁₁𝑧₁1+𝑁₂𝑧₂×lim_{𝑡1𝑡→∞2→∞}𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂𝑓𝑡₁,𝑡₂,0,0=0 (24) and it follows that lim_{𝑡1𝑡→∞2→∞}𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂𝑓𝑡₁,𝑡₂,0,0=0. (25)

Theorem 9. Let 𝑓∶(⊤₁×⊤₂)₊→ℝ be regulated and let 𝐹𝑡₁,𝑡₂=^𝑡1₀𝑓𝜁,𝜂Δ𝜁Δ𝜂 (26) for (𝑡₁,𝑡₂)∈(⊤₁×⊤₂)₊. Then 𝐿_𝑡1𝐿_𝑡2𝑓𝑡₁,𝑡₂𝑧₁,𝑧₂=1𝑧₁𝑧₂𝐿_𝑡1𝐿_𝑡2𝑓𝑡₁,𝑡₂𝑧₁,𝑧₂−lim_{𝑡1𝑡→∞2→∞}𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂𝑓𝑡₁,𝑡₂,0,0, (27) for 𝑧₁,𝑧₂≠0.

Proof. By using the definition of the double Laplace transform on a time scale, we have 𝐿_𝑡1𝐿_𝑡2𝑓𝑡₁,𝑡₂𝑧₁,𝑧₂=^∞₀𝑒^𝜎1𝜎2_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂𝐹𝑡₁,𝑡₂,0,0Δ₁𝑡₁Δ₂𝑡₂, (28) and on using (21), we have 𝐿_𝑡1𝐿_𝑡2𝑓𝑡₁,𝑡₂𝑧₁,𝑧₂=1𝑧₁𝑧₂^∞₀𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂⊖𝑧₁⊖𝑧₂×𝐹𝑡₁,𝑡₂,0,0Δ₁𝑡₁Δ₂𝑡₂. (29) Now, applying integration by parts, and on using the fact that 𝐹(0,0)=0 together with the fundamental theorem of calculus, we obtain that 𝐿_𝑡1𝐿_𝑡2𝑓𝑡₁,𝑡₂𝑧₁,𝑧₂=1𝑧₁𝑧₂𝐿_𝑡1𝐿_𝑡2𝑓𝑡₁,𝑡₂𝑧₁,𝑧₂−lim_{𝑡1𝑡→∞2→∞}𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂𝑓𝑡₁,𝑡₂,0,0. (30)

Example 10. Let 𝑎∈⊤₁, 𝑏∈⊤₂, 𝑎,𝑏>0, and 𝐻_𝑎,𝑏(𝑡₁,𝑡₂)=0 for 𝑡₁∈⊤₁ and 𝑡₂∈⊤₂,𝑡₁<𝑎, and 𝑡₂<𝑏 whilst 𝐻_𝑎,𝑏(𝑡₁,𝑡₂)=1 for 𝑡₁∈⊤₁, 𝑡₂∈⊤₂ and 𝑡₁≥𝑎,  𝑡₂≥𝑏, where 𝐻_𝑎,𝑏(𝑡₁,𝑡₂)=𝐻_𝑎,(𝑡₁)⊗𝐻_𝑏(𝑡₂) and the symbol ⊗ denotes the tensor product. Then we have 𝐿_𝑡1𝐿_𝑡2𝐻_𝑎,𝑏𝑡₁,𝑡₂𝑧₁,𝑧₂=^∞₀𝑒^𝜎1𝜎2_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂𝐻_𝑎,𝑏𝑡₁,𝑡₂Δ₁𝑡₁Δ₂𝑡₂=𝑒_⊖𝑧1(𝑎)𝑒_⊖𝑧2(𝑏)𝑧₁𝑧₂. (31)

Now assume that 𝑓(𝑡₁,𝑡₂)∶⊤₁×⊤₂→ℂ such that 𝑓^Δ1 is continuous. Then the Δ-double Laplace transform 𝐿_𝑡1𝐿_𝑡2𝑓^Δ1𝑡₁,𝑡₂𝑧₁,𝑧₂=𝑧₁=𝐹𝑧₁,𝑧₂−𝐹0,𝑧₂ (32) holds for those regressive 𝑧₁,𝑧₂∈ℂ with respect to 𝑡₁ and 𝑡₂ which further satisfies lim_{𝑡1𝑡→∞2→∞}𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂𝑓𝑡₁,𝑡₂=0. (33) We next extend the result that was proved by Davis et al. in [3].

Proposition 11. Let (𝑎,𝑏)∈(⊤₁×⊤₂)₊, 𝑎,𝑏>0. Assume that 𝑓(𝑡₁,𝑡₂) is one of the following functions: 𝑒_𝑎⊕𝑏𝑡₁,𝑡₂,0,0,sinh_{(𝛼⊕𝛽)}𝑡₁+𝑡₂,cosh_{(𝛼⊕𝛽)}𝑡₁+𝑡₂,cos_{(𝛼⊕𝛽)}𝑡₁+𝑡₂,𝑜𝑟sin_{(𝛼⊕𝛽)}𝑡₁+𝑡₂. (34) If 𝑧₁,𝑧₂,𝛼,and𝛽 are regressive and satisfy lim_{𝑡1𝑡→∞2→∞}𝑒_{𝛼⊖𝑧1𝛽⊖𝑧2}𝑡₁,𝑡₂𝑓𝑡₁,𝑡₂,𝑎,𝑏=lim_{𝑡1𝑡→∞2→∞}𝑒_{𝑖𝛼⊖𝑧1𝑖𝛽⊖𝑧2}𝑡₁,𝑡₂𝑓𝑡₁,𝑡₂,𝑎,𝑏=lim_{𝑡1𝑡→∞2→∞}𝑒_{−𝑖𝛼⊖𝑧1−𝑖𝛽⊖𝑧2}𝑡₁,𝑡₂𝑓𝑡₁,𝑡₂,𝑎,𝑏, (35) then 𝐿_𝑡1𝐿_𝑡2𝐻_𝑎,𝑏𝑡₁,𝑡₂𝑓𝑡₁,𝑡₂,𝑎,𝑏𝑧₁,𝑧₂=𝑒_{⊖𝑧1⊖𝑧2}(𝑎,𝑏,0,0)𝐿_𝑡1𝐿_𝑡2𝑓𝑡₁,𝑡₂,0,0. (36)

Proof. Let us study the case 𝑓(𝑡₁,𝑡₂,𝑎,𝑏)=𝑒_𝛼⊕𝛽(𝑡₁,𝑡₂,𝑎,𝑏). By using (21), we get 𝑒_𝛼⊕𝛽𝑡₁,𝑡₂,𝑎,𝑏𝑒_{⊖𝑧1⊖𝑧2}𝜎𝑡₁,𝜎𝑡₂=𝑒,0,0_𝛼⊕𝛽𝑡₁,𝑡₂,𝑎,𝑏𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂,0,01+𝜇₁𝑧₁1+𝜇₂𝑧₂ (37) and by using Theorem 4 and (21) the right hand side of equation (37) is =𝑒_{⊖𝑧1⊖𝑧2}(𝑎,𝑏,0,0)𝛼−𝑧₁𝛽−𝑧₂𝛼−𝑧₁𝛽−𝑧₂1+𝜇₁𝑧₁1+𝜇₂𝑧₂×𝑒_{𝛼⊖𝑧1𝛽⊖𝑧2}𝑡₁,𝑡₂=𝑒,𝑎,𝑏_{⊖𝑧1⊖𝑧2}(𝑎,𝑏,0,0)𝛼−𝑧₁𝛽−𝑧₂𝛼⊖𝑧₁𝛽⊖𝑧₂𝑒_{𝛼⊖𝑧1𝛽⊖𝑧2}𝑡₁,𝑡₂,𝑎,𝑏. (38) Then 𝐿_𝑡1𝐿_𝑡2𝐻_𝑎,𝑏𝑡₁,𝑡₂𝑓𝑡₁,𝑡₂,𝑎,𝑏𝑧₁,𝑧₂=^∞₀𝑒_{⊖𝑧1⊖𝑧2}𝜎𝑡₁,𝜎𝑡₂,0,0𝐻_𝑎,𝑏𝑡₁,𝑡₂𝑒_𝛼⊕𝛽×𝑡₁,𝑡₂,𝑎,𝑏Δ₁𝑡₁Δ₂𝑡₂=𝑒_{⊖𝑧1⊖𝑧2}(𝑎,𝑏,0,0)𝛼−𝑧₁𝛽−𝑧₂^∞₀𝐻_𝑎,𝑏𝑡₁,𝑡₂×𝛼⊖𝑧₁𝛽⊖𝑧₂𝑡₁,𝑡₂×𝑒_{𝛼⊖𝑧1𝛽⊖𝑧2}𝑡₁,𝑡₂,𝑎,𝑏×Δ₁𝑡₁Δ₂𝑡₂=𝑒_{⊖𝑧1⊖𝑧2}(𝑎,𝑏,0,0)𝛼−𝑧₁𝛽−𝑧₂^∞_𝑏^∞_𝑎𝛼⊖𝑧₁𝛽⊖𝑧₂×𝑒_{𝛼⊖𝑧1𝛽⊖𝑧2}𝑡₁,𝑡₂,𝑎,𝑏×Δ₁𝑡₁Δ₂𝑡₂=𝑒_{⊖𝑧1⊖𝑧2}(𝑎,𝑏,0,0)𝛼−𝑧₁𝛽−𝑧₂=𝑒_{⊖𝑧1⊖𝑧2}(𝑎,𝑏,0,0)𝐿_𝑡1𝐿_𝑡2𝑒_𝛼⊕𝛽𝑡₁,𝑡₂,0,0. (39)

Theorem 12. Let 𝑓(𝑡₁,𝑡₂)∶⊤₁×⊤₂→ℂ and suppose that 𝑓^Δ𝑖 and 𝑓^Δ𝑗 are continuous for all 𝑖=1,2,…𝑛 and 𝑗=1,2,…,𝑚, respectively; then the Δ-double Laplace transforms for 𝑓^Δ𝑖 and 𝑓^Δ𝑗 are given by 𝐿_𝑡1𝐿_𝑡2𝑓^Δ1𝑛𝑡₁,𝑡₂𝑧₁,𝑧₂=𝑧^𝑛₁𝐹𝑧₁,𝑧₂−𝑧₁^𝑛−1𝐹0,𝑧₂−⋯−𝐹^{−Δ1𝑛−1}0,𝑧₂,𝐿_𝑡1𝐿_𝑡2𝑓^Δ2𝑚𝑡₁,𝑡₂𝑧₁,𝑧₂=𝑧^𝑚₂𝐹𝑧₁,𝑧₂−𝑧₂^𝑚−1𝐹𝑧₁,0−⋯−𝐹^{−Δ2𝑚−1}𝑧₁,0, (40)(41) for all 𝑧₁,𝑧₂∈ℂ with lim_𝑡1→∞𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂𝑓𝑡₁,𝑡₂=lim_𝑡1→∞𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂𝑓^Δ1𝑛𝑡₁,𝑡₂⋯=lim_𝑡1→∞𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂𝑓^Δ1𝑛−1𝑡₁,𝑡₂=0,lim_𝑡2→∞𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂𝑓𝑡₁,𝑡₂=lim_𝑡2→∞𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂𝑓^Δ2𝑚𝑡₁,𝑡₂⋯=lim_𝑡2→∞𝑒_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂𝑓^Δ2𝑚−1𝑡₁,𝑡₂=0, (42) respectively.

Proof. Let 𝑛=1. On using the definition of the Δ-single Laplace transform and integrating parts of the function 𝑓^Δ11(𝑡₁,𝑡₂), we have 𝐿_𝑡1𝑓^Δ11𝑡₁,𝑡₂𝑧₁=^∞₀𝑒^𝜎1_⊖𝑧1𝑓^Δ11𝑡₁,𝑡₂Δ𝑡₁=𝑓𝑡₁,𝑡₂^∞₀+𝑧₁^∞₀𝑒^𝜎1_⊖𝑧1𝑓𝑡₁,𝑡₂Δ𝑡₁=−𝑓0,𝑡₂+𝑧₁𝐹𝑧₁,𝑡₂. (43) Then the single Laplace transform with respect to 𝑡₁ is given by 𝐿_𝑡1𝑓^Δ11𝑡₁,𝑡₂𝑧₁,𝑧₂=𝑧₁𝐹𝑧₁,𝑡₂−𝑓0,𝑡₂. (44) Similarly, Laplace transform for (44) with respect to 𝑡₂ is given by 𝐿_𝑡2𝐿_𝑡1𝑓^Δ11𝑡₁,𝑡₂𝑧₁=𝑧₁𝐹𝑧₁,𝑧₂−𝐹0,𝑧₂. (45) Then (45) is called the Δ-double Laplace transform for the function 𝑓^Δ11(𝑡₁,𝑡₂). Similarly, if 𝑚=1 in (41), the Δ-double Laplace transform for the function 𝑓^Δ21(𝑡₁,𝑡₂) is given by 𝐿_𝑡2𝐿_𝑡1𝑓^Δ21𝑡₁,𝑡₂𝑧₁,𝑧₂=𝑧₂𝐹𝑧₁,𝑧₂−𝐹𝑧₁,0. (46) Now, in order to obtain the Δ-double Laplace transform for the function 𝑓^Δ12𝑡₁,𝑡₂=𝜕²𝑓𝑡₁,𝑡₂Δ₁𝑡²₁ (47) with respect to 𝑡₁,𝑡₂, we proceed as follows: first of all by taking the Δ-single Laplace transform with respect to the variable 𝑡₁ and on using the fact that lim_𝑡1→∞𝑒_⊖𝑧1𝑡₁𝑓^Δ11𝑡₁,𝑡₂=0,lim_𝑡1→∞𝑒_⊖𝑧1𝑡₁𝑓𝑡₁,𝑡₂=0, (48) we have 𝐿_𝑡1𝑓^Δ12𝑡₁,𝑡₂𝑧₁=𝑓^Δ11𝑡₁,𝑡₂^∞₀+𝑧₁^∞₀𝑒^𝜎1_⊖𝑧1𝑓^Δ11𝑡₁,𝑡₂Δ𝑡₁=−𝑓^Δ110,𝑡₂+𝑧₁^∞₀𝑒^𝜎1_⊖𝑧1𝑓^Δ11𝑡₁,𝑡₂Δ𝑡₁. (49) Thus on using (44), we obtain that 𝐿_𝑡1𝑓^Δ12𝑡₁,𝑡₂𝑧₁=𝑧²₁𝐹𝑧₁,𝑡₂−𝑧₁𝐹0,𝑡₂−𝑓^Δ10,𝑡₂. (50) In a similar way, the single Laplace transform with respect to 𝑡₂ is given by 𝐿_𝑡2𝐿_𝑡1𝑓^Δ12𝑡₁,𝑡₂𝑧₁,𝑧₂=𝑧²₁𝐹𝑧₁,𝑧₂−𝑧₁𝐹0,𝑧₂−𝐹^Δ10,𝑧₂. (51) Further, the Δ-double Laplace transform for the function 𝑓^Δ22(𝑡₁,𝑡₂)=𝜕²𝑓(𝑡₁,𝑡₂)/Δ₂𝑡²₂ with respect to 𝑡₁,𝑡₂ is given by 𝐿_𝑡2𝐿_𝑡1𝑓^Δ22𝑡₁,𝑡₂𝑧₁,𝑧₂=𝑧²₂𝐹𝑧₁,𝑧₂−𝑧₂𝐹𝑧₁,0−𝐹^Δ1𝑧₁,0. (52) Finally, note that we can generalize the proof for 𝑛,𝑚.

Next, we define the double convolution of two time scale functions as follows.

Definition 13. The double convolution of the function 𝑓,  𝑔∶𝐓₁×𝐓₂→ℝ is given by 𝑓∗∗𝑔𝑡₁,𝑡₂=^𝑡1₀𝑓𝜁,𝜂𝑔𝑡₁,𝑡₂,𝜎₁(𝜁),𝜎₂𝜂Δ𝜁Δ𝜂. (53)

In the next theorem we prove the properties of the double convolution on a time scale as follows.

Theorem 14. Let 𝑓,𝑔,ℎ∶𝐓₁×𝐓₂→ℝ be integrable functions on a time scale. Then the following properties are satisfied: 𝑓∗∗𝑔𝑡₁,𝑡₂=𝑔∗∗𝑓𝑡₁,𝑡₂,𝑓∗∗𝑔∗∗ℎ=𝑓∗∗𝑔∗∗ℎ. (54)

Proof. We can easily show that convolution is commutative by using the definition of the double convolution 𝑓∗∗𝑔𝑡₁,𝑡₂=^𝑡1₀𝑓𝜁,𝜂𝑔𝑡₁,𝑡₂,𝜎₁(𝜁),𝜎₂𝜂=Δ𝜁Δ𝜂_𝑡10𝑓𝑡₁,𝑡₂,𝜎₁(𝜁),𝜎₂𝜂𝑔𝜁,𝜂Δ𝜁Δ𝜂. (55) For the proof of the associative law, we also use the definition as follows: =𝑓∗∗𝑔∗∗ℎ_𝑡10𝑓∗∗𝑔𝑡₁,𝑡₂,𝜎₁(𝜁),𝜎₂𝜂ℎ=𝜁,𝜂Δ𝜁Δ𝜂_{𝑡10𝑡1𝜎2𝜂𝑡1𝜎1(𝜁)}𝑓𝑡₁,𝑡₂,𝜎₁(𝛼),𝜎₂𝛽×𝑔𝛼,𝛽,𝜎₁(𝜁),𝜎₂𝜂=×ℎ𝜁,𝜂Δ𝜁Δ𝜂Δ𝛼Δ𝛽_𝑡10𝑓𝑡₁,𝑡₂,𝜎₁(𝛼),𝜎₂𝛽𝑡𝑓∗∗𝑔𝛼,𝛽Δ𝛼Δ𝛽×𝑓∗∗𝑔∗∗ℎ₁,𝑡₂. (56)

Theorem 15 (convolution theorem). Let 𝑓,  𝑔∶𝐓₁×𝐓₂→ℂ be regulated functions. Then the double Laplace transform of the double convolution is given by 𝐿_𝑡2𝐿_𝑡1𝑓∗∗𝑔𝑡₁,𝑡₂;𝑧₁,𝑧₂=𝐹𝑧₁,𝑧₂𝐺𝑧₁,𝑧₂. (57)

Proof. By using the definition of the double Laplace transform on a time scale, we obtain that 𝐿_𝑡2𝐿_𝑡1𝑓∗∗𝑔𝑡₁,𝑡₂;𝑧₁,𝑧₂=^∞₀𝑒^𝜎1𝜎2_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂,0,0𝑓∗∗𝑔𝑡₁,𝑡₂Δ₁𝑡₁Δ₂𝑡₂=^∞₀^𝑡1₀𝑓𝜁,𝜂𝑔𝑡₁,𝑡₂,𝜎₁(𝜁),𝜎₂𝜂×Δ𝜁Δ𝜂𝑒^𝜎1𝜎2_{⊖𝑧1⊖𝑧2}𝑡₁,𝑡₂,0,0Δ₁𝑡₁Δ₂𝑡₂=^∞₀𝑓𝜁,𝜂^∞_𝜎1(𝜁)^∞_𝜎2𝜂𝑔𝑡₁,𝑡₂,𝜎₁(𝜁),𝜎₂𝜂^𝜎1𝜎2_{⊖𝑧1⊖𝑧2}×𝑡₁,𝑡₂,0,0Δ₁𝑡₁Δ₂𝑡₂=Δ𝜁Δ𝜂_∞0𝑓𝜁,𝜂𝐿_𝑡2𝐿_𝑡1𝐻_{𝜎1(𝜁),𝜎2(𝜂)}𝑡₁,𝑡₂×𝑔𝑡₁,𝑡₂,𝜎₁(𝜁),𝜎₂𝜂Δ𝜁Δ𝜂, (58) where 𝐻_{𝜎1(𝜁),𝜎2(𝜂)}(𝑡₁,𝑡₂)=𝐻_𝜎1(𝜁),(𝑡₁)⊗𝐻_𝜎2(𝜁),(𝑡₂) and the symbol ⊗ denotes the tensor product. Thus we have 𝐿_𝑡2𝐿_𝑡1𝑓∗∗𝑔𝑡₁,𝑡₂;𝑧₁,𝑧₂=^∞₀𝑓𝜁,𝜂𝐺𝑧₁,𝑧₂^𝜎1𝜎2_{⊖𝑧1⊖𝑧2}𝑧𝜁,𝜂,0,0Δ𝜁Δ𝜂=𝐺₁,𝑧₂^∞₀𝑓𝜁,𝜂^𝜎1𝜎2_{⊖𝑧1⊖𝑧2}𝑧𝜁,𝜂,0,0Δ𝜁Δ𝜂=𝐹₁,𝑧₂𝐺𝑧₁,𝑧₂. (59)

Finally, we give the solution to the wave equation in one dimension as follows. Consider the wave equation in the form of 𝑢^Δ1Δ1−𝑢^Δ2Δ2=0, (60) under the conditions 𝑢(𝑥,0)=𝑒₁(𝑥),𝑢(0,𝑡)=𝑒₁(𝑡),𝑢^Δ2(0,𝑡)=𝑒₁(𝑡),𝑢^Δ1(𝑥,0)=𝑒₁(𝑥). (61) Then by formally taking the double Laplace transform, we get 𝐿_𝑡2𝐿_𝑡1𝑢(𝑡,𝑥);𝑧₁,𝑧₂=𝑧₁+1𝑧₂−1𝑧²₁−𝑧²₂−𝑧₂+1𝑧₁−1𝑧²₁−𝑧²₂. (62) By using partial fractions, we have 𝐿_𝑡2𝐿_𝑡1𝑢(𝑡,𝑥);𝑧₁,𝑧₂=12𝑧₂−1𝑧₁−𝑧₂+12𝑧₂−1𝑧₁+𝑧₂+12𝑧₂𝑧₂−1𝑧₁−𝑧₂−12𝑧₂𝑧₂−1𝑧₁+𝑧₂−12𝑧₁−1𝑧₁−𝑧₂+12𝑧₁−1𝑧₁+𝑧₂−12𝑧₁𝑧₁−1𝑧₁−𝑧₂−12𝑧₁𝑧₁−1𝑧₁+𝑧₂. (63) Thus the solution to the above equation is given by 𝑢(𝑡,𝑥)=𝑒_1⊕1(𝑡,𝑥,0,0). (64) In particular, if we consider the case 𝐓₁×𝐓₂=ℝ×ℝ, 𝜎(𝑡)=𝑡,𝜎(𝑥)=𝑥, for all (𝑡,𝑥)∈𝐓₁×𝐓₂, then 𝑒_𝑎⊕𝑏(𝑡,𝑥,0,0)=𝑒^{𝑎𝑥+𝑏𝑡} for any constant 𝑎,𝑏∈ℝ and so the solution is given by 𝑢(𝑡,𝑥)=𝑒^𝑡+𝑥. (65)

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of paper.

Acknowledgment

The authors are grateful to the referees for the useful comments regarding presentation in the early version of the paper. The authors also gratefully acknowledge that this project was supported by King Saud University, Deanship of Scientific Research College of Science Research Center.