Abstract
In this paper, we introduce a general quantum Laplace transform \(\mathcal{L}_{\beta }\) and some of its properties associated with the general quantum difference operator \({D}_{\beta }f(t)= ({f(\beta (t))-f(t)} )/ ({ \beta (t)-t} )\), β is a strictly increasing continuous function. In addition, we compute the β-Laplace transform of some fundamental functions. As application we solve some β-difference equations using the β-Laplace transform. Finally, we present the inverse β-Laplace transform \(\mathcal{L}_{\beta }^{-1}\).
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1 Introduction
The Laplace transform in continuous and discrete cases has an essential role in applied mathematics and in mathematical physics, particularly in solving differential and difference equations, respectively. Recently, versions of Laplace transform in other calculi, such as q-calculus and time scale, were investigated, see [2–5]. The q-Laplace transform has a similar role in solving q-difference equations, see [1]. The general quantum difference operator \(D_{\beta }\) is defined in [12] by
where the function y is defined on an interval \(I\subseteq {\mathbb{R}}\) and β is a strictly increasing continuous general function, that is, \(\beta (t)\in {I}\) for \(t\in {I}\). The function y is said to be β-differentiable if it is classic differentiable at the fixed points of the function β. Hamza et al. (2015) [12] established the calculus based on \(D_{\beta }\) when β has only one fixed point \(s_{0}\in {I}\) that satisfies the inequality \((t-s_{0})(\beta (t)-t) \leq 0\) for all \(t\in I\), accordingly \(\lim_{k\rightarrow \infty }\beta ^{k}(t)=s_{0}\), \(\beta ^{k}(t):= \underbrace{\beta \circ \beta \circ \cdots \circ \beta }_{k\text{-times}}(t)\). Examples of this type are the Jackson q-difference operator with \(\beta (t)=qt\), \(0< q<1\), \(s_{0}=0\) and the Hahn difference operator with \(\beta (t)=qt+\omega \), \(0< q<1\), \(\omega >0\), \(s_{0}=\frac{\omega }{1-q}\). They mentioned also another type of β when it has only one fixed point \(s_{0}\in {I}\) and satisfies the inequality \((t-s_{0})(\beta (t)-t) \geq 0\) for all \(t\in I\); consequently, \(\lim_{k\rightarrow \infty }\beta ^{k}(t)=\infty \), for example, the backward Hahn difference operator with \(\beta (t)=qt+\omega \), \(q>1\), \(\omega > 0\). A study of different types of the function β according to the number of its fixed points, which can be basis for different calculi, was presented in [16]. In [13] some integral inequalities based on \(D_{\beta }\) were introduced. The homogeneous second-order linear β-difference equations and the theory of nth-order linear β-difference equations were studied in [8, 9]. In addition, some properties of the quantum exponential functions in a Banach algebra were studied in [10]. Properties of the β-Lebesgue spaces were introduced in [6]. The β-difference operator \(D_{\beta }\) and its calculus has applications in many areas in mathematics and physics such as the quantum variational calculus, the orthogonal polynomials, quantum mechanics, and scale of relativity, see [7, 14, 15].
In this paper we deduce a general quantum Laplace transform \(\mathcal{L}_{\beta }\) associated with \(D_{\beta }\), where β has only one fixed point \(s_{0}\in I\) with the inequality \((t-s_{0})(\beta (t)-t) \leq 0\) for all \(t\in I\), which will be useful in solving the β-difference equations. We organize this paper as follows: In Sect. 2, we introduce the needed preliminaries from the β-calculus. In Sect. 3, we present the β-regressive functions and define the “β-circle plus” \(\oplus _{\beta }\) and the “β-circle minus” \(\ominus _{\beta }\), and some associated relations. And then, we introduce the β-Laplace transform and some of its properties. Furthermore, we compute the β-Laplace transform of some fundamental functions. As application, we give two examples to solve some β-difference equations. Finally, we deduce the inverse β-Laplace transform \(\mathcal{L}_{\beta }^{-1}\).
2 Preliminaries
In this section, we introduce some needed preliminaries from the β-calculus, where β has only one fixed point \(s_{0}\in I\) such that \((t-s_{0})(\beta (t)-t) \leq 0\) for all \(t\in I\), \(\mathbb{X}\) is a Banach space.
Theorem 2.1
([12])
Assume that \(f:{I}\rightarrow \mathbb{X}\) and \(g:{I}\rightarrow \mathbb{R}\) are β-differentiable functions on I. Then:
-
(i)
The product \(fg:I\rightarrow \mathbb{X}\) is β-differentiable at \(t\in{I}\) and
$$ \begin{aligned} {D}_{\beta }(fg) (t) &=\bigl({D}_{\beta }f(t) \bigr)g(t)+f\bigl(\beta (t)\bigr){D}_{ \beta }g(t) \\ &=\bigl({D}_{\beta }f(t)\bigr)g\bigl(\beta (t)\bigr)+f(t){D}_{\beta }g(t), \end{aligned} $$ -
(ii)
\(f/g\) is β-differentiable at \(t\in{I}\) and
$$ {D}_{\beta } ({f}/{g} ) (t)= \frac{({D}_{\beta }f(t))g(t)-f(t){D}_{\beta }g(t)}{g(t)g(\beta (t))}, $$provided that \(g(t)g(\beta (t))\neq {0}\).
Lemma 2.2
([12])
The following statements are true:
-
(i)
The sequence of functions \(\{\beta ^{k}(t)\}_{k=0}^{\infty }\) converges uniformly to the constant function \(\hat{\beta }(t):=s_{0}\) on every compact interval \(J\subseteq I \) containing \(s_{0}\).
-
(ii)
The series \(\sum_{k=0}^{\infty } \vert \beta ^{k}(t)-\beta ^{k+1}(t) \vert \) is uniformly convergent to \(\vert t-s_{0} \vert \) on every compact interval \(J \subseteq I \) containing \(s_{0}\).
Theorem 2.3
([12])
If \(f:I\rightarrow \mathbb{X}\) is continuous at \(s_{0}\), then
-
(i)
the sequence \(\{f(\beta ^{k}(t))\}_{k=0}^{\infty }\) converges uniformly to \(f(s_{0})\),
-
(ii)
the series \(\sum_{k=0}^{\infty } \Vert (\beta ^{k}(t)-\beta ^{k+1}(t) )f( \beta ^{k}(t)) \Vert \) is uniformly convergent
on every compact interval \(J\subseteq I\) containing \(s_{0}\).
Definition 2.4
([12])
Let \(f:{I}\rightarrow {\mathbb{X}}\) and \(a,b\in {I}\). The β-integral of f from a to b is defined by
where
provided that the series converges at \(x=a\) and \(x=b\). f is called β-integrable on I if the series converges at a and b for all \(a,b\in {I}\). Clearly, if f is continuous at \(s_{0}\in {I}\), then f is β-integrable on I.
Theorem 2.5
([12])
Assume that f, g are β-differentiable functions on I and \(D_{\beta }f\), \(D_{\beta }g \) are both continuous at \(s_{0}\). Then
Here, at least one of the functions f and g is a real-valued function.
Definition 2.6
([11])
The β-exponential functions \(e_{p,\beta }(t)\) and \(E_{p,\beta }(t)\) are defined by
and
where \(p:I \rightarrow \mathbb{C}\) is a continuous function at \(s_{0}\). Clearly, both products in (2.1) and (2.2) are convergent to a non-zero number for every \(t\in I\), since \(\sum_{k=0}^{\infty } \vert p(\beta ^{k}(t)) (\beta ^{k}(t)-\beta ^{k+1}(t) ) \vert \) is uniformly convergent.
Theorem 2.7
([11])
The β-exponential functions \(e_{p,\beta }(t)\) and \(E_{p,\beta }(t)\) are the unique solutions of the β-initial value problems
respectively.
Definition 2.8
([11])
The β-trigonometric functions are defined by
Definition 2.9
([11])
The β-hyperbolic functions are defined by
Theorem 2.10
([11])
Let \(p:I\rightarrow \mathbb{C}\) be a continuous function at \(s_{0}\). Then the following properties hold:
-
(i)
\(e_{p,\beta }(\beta (t))= [1+(\beta (t)-t)p(t) ]e_{p,\beta }(t)\), \(t \in I\),
-
(ii)
\(D_{\beta } (\frac{1}{e_{p,\beta }(t)} )= \frac{-p(t)}{e_{p,\beta }(\beta (t))}\),
-
(iii)
\(\frac{1}{e_{p,\beta }(t)}\) is the unique solution of the first-order β-difference equation
$$ D_{\beta }y(t) =\frac{-p(t)e_{p,\beta }(t)}{e_{p,\beta }(\beta (t))}y(t), \quad y(s_{0})=1. $$
Theorem 2.11
([11])
Assume that \(p,q:I\rightarrow \mathbb{C}\) are continuous functions at \(s_{0}\in I\). The following properties are true:
-
(i)
\(\frac{1}{e_{p,\beta }(t)}=e_{-p/[1+(\beta (t)-t)p]}(t)\),
-
(ii)
\(e_{p,\beta }(t)e_{q,\beta }(t)=e_{p+q+(\beta (t)-t)pq}(t)\),
-
(iii)
\(e_{p,\beta }(t)/e_{q,\beta }(t)=e_{(p-q)/[1+(\beta (t)-t)q]}(t)\).
3 Main results
In this section, we present the β-regressive functions and define the “β-circle plus” \(\oplus _{\beta }\) and the “β-circle minus” \(\ominus _{\beta }\). We introduce the β-Laplace transform and some of its main properties. Furthermore, we compute the β-Laplace transform of the β-exponential and the β-trigonometric functions. As application, we give two examples to solve some β-difference equations. Finally, we deduce the inverse β-Laplace transform \(\mathcal{L}_{\beta }^{-1}\).
3.1 β-Regressive functions
Definition 3.1
A function \(p:I\rightarrow \mathbb{C}\) is said to be β-regressive on I if \(1+ (\beta (t)-t )p(t)\neq 0\) for all \(t\in I\).
We denote the set of all β-regressive functions \(p:I\rightarrow \mathbb{C}\) and continuous at \(s_{0}\) by \(\mathcal{R}_{\beta }\), and the set of all β-regressive constants \(z\in \mathbb{C}\) by \(\mathcal{R}_{\beta }^{c}\).
Definition 3.2
Let \(p,q\in \mathcal{R}_{\beta }\). Then we define \(p\oplus _{\beta }q\), \(\ominus _{\beta }p\), and \(p\ominus _{\beta } q\) by
-
(i)
\((p\oplus _{\beta }q)(t)=p(t)+q(t)+(\beta (t)-t)p(t)q(t)\), \(t\in I\),
-
(ii)
\((\ominus _{\beta }p)(t)=\frac{-p(t)}{1+(\beta (t)-t)p(t)}\), \(t\in I\),
-
(iii)
\((p\ominus _{\beta }q)(t)= (p\oplus _{\beta }(\ominus _{\beta }q) )(t)\), \(t\in I \).
From the definition we conclude that \(p\ominus _{\beta }p=0\), \(\ominus _{\beta }(\ominus _{\beta }p)=p\), \(\ominus _{\beta }(p \ominus _{\beta }q)=q \ominus _{\beta } p\), \(\ominus _{\beta }(p\oplus _{\beta }q)=(\ominus _{\beta }p)\oplus _{ \beta }(\ominus _{\beta }q)\), and \((\mathcal{R}_{\beta },\oplus _{\beta })\) form an abelian group.
Note that at \(t=s_{0}\), \(\oplus _{\beta }\) and \(\ominus _{\beta }\) reduce to the classic addition and subtraction operations.
Theorem 3.3
Let \(p,q \in \mathcal{R}_{\beta }\), \(t\in I\). Then the following statements are true:
- (\(i_{1}\)):
-
\(e_{\ominus _{\beta }p,\beta }(t)=\frac{1}{e_{p,\beta }(t)}=\prod_{k=0}^{ \infty }[1-p(\beta ^{k}(t))(\beta ^{k}(t)-\beta ^{k+1}(t))]=E_{-p, \beta }(t)\),
- (\(i_{2}\)):
-
\(e_{\ominus _{\beta }p,\beta }(t)\) is the unique solution of the first-order β-difference equation
$$ D_{\beta }y(t)=(\ominus _{\beta }p) (t)y(t),\quad y(s_{0})=1,$$(3.1) - (\(i_{3}\)):
-
$$ \begin{aligned} e_{\ominus _{\beta }p,\beta }\bigl(\beta (t)\bigr)&= \bigl[1+\bigl( \beta (t)-t\bigr) ( \ominus _{\beta }p) (t) \bigr]e_{\ominus _{\beta }p,\beta }(t) = \frac{e_{\ominus _{\beta }p,\beta }(t)}{1+(\beta (t)-t)p(t)} \\ &=-\frac{(\ominus _{\beta }p)(t)}{p(t)}e_{\ominus _{\beta }p,\beta }(t)=- \frac{(\ominus _{\beta }p)(t)}{p(t)e_{p,\beta }(t)}, \end{aligned} $$
- (\(i_{4}\)):
-
\(D_{\beta } (e_{\ominus _{\beta }p,\beta }(t) ) = \frac{(\ominus _{\beta }p)(t)}{e_{p,\beta }(t)}= (\ominus _{\beta }p)(t)e_{ \ominus _{\beta }p,\beta }(t)= -p(t) [e_{\ominus _{\beta }p,\beta }( \beta (t)) ]\),
- (\(i_{5}\)):
-
\(e_{p,\beta }(t)e_{q,\beta }(t)=e_{p\oplus _{\beta }q,\beta }(t)\),
- (\(i_{6}\)):
-
\(\frac{e_{p,\beta }(t)}{e_{q,\beta }(t)}=e_{p\ominus _{\beta }q,\beta }(t)\).
Proof
- (\(i_{1}\)):
-
Using Definition 2.6 and Theorem 2.11\((i)\), we have
$$\begin{aligned} e_{\ominus _{\beta }p,\beta }(t)&=e_{ \frac{-p(t)}{[1-(\beta (t)-t)p(t)]},\beta }(t)= \frac{1}{e_{p,\beta }(t)} \\ &=\prod_{k=0}^{\infty }\bigl[1-p\bigl(\beta ^{k} (t)\bigr) \bigl(\beta ^{k}(t)-\beta ^{k+1}(t) \bigr)\bigr]=E_{-p, \beta }(t). \end{aligned}$$ - (\(i_{2}\)):
-
Since \((\ominus _{\beta }p)(t)=\frac{-p(t)}{1+(\beta (t)-t)p(t)}= \frac{-p(t)e_{p,\beta }(t)}{e_{p,\beta }(\beta (t))}\). Then equation (3.1) can be written as
$$ D_{\beta }y(t) =\frac{-p(t)e_{p,\beta }(t)}{e_{p,\beta }(\beta (t))}y(t), \quad y(s_{0})=1. $$By \((i_{1})\) and Theorem 2.10\((\mathit{iii})\), we get the desired result.
- (\(i_{3}\)):
-
Using \((i_{1})\), \((i_{2})\), we have
$$\begin{aligned} e_{\ominus _{\beta }p,\beta }\bigl(\beta (t)\bigr)&= e_{\ominus _{\beta }p,\beta }(t)+\bigl( \beta (t)-t \bigr) \bigl(D_{\beta }e_{\ominus _{\beta }p,\beta }(t) \bigr) \\ &= e_{\ominus _{\beta }p,\beta }(t)+\bigl(\beta (t)-t\bigr) (\ominus _{\beta }p) (t)e_{ \ominus _{\beta }p,\beta }(t) \\ &= \bigl[1+\bigl(\beta (t)-t\bigr) (\ominus _{\beta }p) (t) \bigr]e_{\ominus _{\beta }p, \beta }(t) \\ &= \biggl[1-\frac{(\beta (t)-t)p(t)}{1+(\beta (t)-t)p(t)} \biggr]e_{ \ominus _{\beta }p,\beta }(t) \\ &= \biggl[\frac{1}{1+(\beta (t)-t)p(t)} \biggr]e_{\ominus _{\beta }p,\beta }(t) \\ &=-\frac{(\ominus _{\beta }p)(t)}{p(t)}e_{\ominus _{\beta }p,\beta }(t) \\ &=-\frac{(\ominus _{\beta }p)(t)}{p(t)e_{p,\beta }(t)}. \end{aligned}$$ - (\(i_{4}\)):
-
From \((i_{1})\) and Theorem 2.10, we get
$$\begin{aligned} D_{\beta } \bigl(e_{\ominus _{\beta }p,\beta }(t) \bigr)&=D_{\beta } \biggl( \frac{1}{e_{p,\beta }(t)} \biggr)= -\frac{p(t)}{e_{p,\beta }(\beta (t))} \\ &= \frac{1}{e_{p,\beta }(t)} \frac{-p(t)}{ [1+(\beta (t)-t)p(t) ]} \\ &=\frac{1}{ e_{p,\beta }(t)}(\ominus _{\beta }p) (t) \\ &= (\ominus _{\beta }p) (t)e_{\ominus _{\beta }p,\beta }(t). \end{aligned}$$On the other hand, from \((i_{2})\), \((i_{3})\)
$$ -p(t) \bigl[e_{\ominus _{\beta }p,\beta }\bigl(\beta (t)\bigr) \bigr]=(\ominus _{ \beta }p) (t)e_{\ominus _{\beta }p,\beta }(t)=D_{\beta } \bigl(e_{\ominus _{ \beta }p,\beta }(t) \bigr). $$ - (\(i_{5}\)):
-
From Theorem 2.11\((\mathit{ii})\) and Definition 3.2, we get the desired result.
- (\(i_{6}\)):
-
From (\(i_{1}\)), (\(i_{5}\)), we get the result.
□
Lemma 3.4
Let \(z,x\in {R}_{\beta }^{c}\) such that \(z=x+iy\), where \(z\in \mathbb{C}\), \(x,y\in \mathbb{R}\). Then \(\vert e_{\ominus _{\beta }z,\beta }(t) \vert \leq e_{\ominus _{\beta }x, \beta }(t)\).
Proof
Using Theorem 2.11\((\mathit{ii})\), we get
So,
Then
Since \(\frac{1}{e_{z,\beta }(t)}=e_{\ominus _{\beta }z,\beta }(t)\). Therefore,
□
3.2 The β-Laplace transform
In this section, let \(\sup {I}=\infty \), \(s_{0}\in I\). We assume that \(z,\ominus _{\beta }z\in \mathcal{R}_{\beta }^{c}\) and hence \(e_{\ominus _{\beta }z,\beta }\) is well defined. Furthermore, we denote by \(V ([s_{0},\infty ),\mathbb{C} )\) the set of β-integrable functions over each compact subinterval of \([s_{0},\infty )\).
Definition 3.5
Let \(\sup {I}=\infty \), \(s_{0}\in I\) and \(f(t)\) be continuous at \(s_{0}\) on \([s_{0},\infty )\). We define the improper β-integral by
provided this limit exists, and we say that the improper β-integral converges in this case. If this limit does not exist, then we say that the improper β-integral diverges.
Definition 3.6
A function \(f\in V ([s_{0},\infty ),\mathbb{C} )\) is said to be of exponential order \(\lambda >0\), \(\lambda \in \mathbb{R}\) if there exists a constant \(M>0\) such that \(\vert f(t) \vert \leq M e_{\lambda ,\beta }(t)\) for all \(t\in [s_{0},\infty )\).
Definition 3.7
Suppose \(f\in V ([s_{0},\infty ),\mathbb{C} )\). Then the Laplace transform of f is defined by
for all \(z\in \mathcal{R}_{\beta }^{c}\) for which the β-integral (3.3) exists.
Note that in the usual differential case, \(\ominus _{\beta }z=-z\), \(\beta (t)=t\), \(e_{\ominus _{\beta }z,\beta }(\beta (t))=e^{-zt}\), and (3.3) becomes the usual Laplace transform
Moreover, in the case of \(\beta (t)=qt\), \(q\in (0,1)\), then \(s_{0}=0\), \(e_{\ominus _{\beta }z,\beta }(\beta (t))=e_{\ominus _{q}z,q}(qt)\), and we obtain the q-Laplace transform of the form
see [4].
Theorem 3.8
Let \(f\in V ([s_{0},\infty ),\mathbb{C} )\) be of exponential order λ, \(z\in \mathcal{R}_{\beta }^{c}\) such that \(z=x+iy\), \(x,y\in \mathbb{R}\). Then the integral in the β-Laplace transform (3.3) converges absolutely for \(\vert z \vert >\lambda \), provided that \(\lim_{t\rightarrow \infty }e_{\lambda \ominus _{\beta }z,\beta }(t)=0\).
Proof
Using Definition 3.6, Lemma 3.4, we get
Then \(\mathcal{L}_{\beta }\{f(t)\}\) converges absolutely. □
Example 3.9
Find the β-Laplace transform of \(f(t)\equiv 1\).
Sol. Using Theorem 3.3\((i_{3})\), \((i_{4})\), we have
provided that \(\lim_{t\rightarrow \infty }e_{\ominus _{\beta }z,\beta }(t)=0\).
Theorem 3.10
For \(z,\lambda \in \mathcal{R}_{\beta }^{c}\),
provided that \(\lim_{t\rightarrow \infty }e_{\lambda \ominus _{\beta }z,\beta }(t)=0\).
Proof
We find
provided that \(\lim_{t\rightarrow \infty }e_{\lambda \ominus _{\beta }z,\beta }(t)=0\). □
Corollary 3.11
Let \(\lambda ,\mu ,z\in \mathcal{R}_{\beta }^{c}\). Then
provided that \(\lim_{t\rightarrow \infty }e_{(\lambda +\mu )\ominus _{\beta }z,\beta }(t)=0\).
Proof
Using Theorem 3.3\((i_{5})\), and since
Therefore, we have
□
Theorem 3.12
(Linearity)
Let \(f,g\in V ([s_{0},\infty ),\mathbb{C} )\), and \(c_{1}\), \(c_{2}\) be constants. Then
Proof
□
Example 3.13
Find the β-Laplace transform of the following functions:
Sol. By Definitions 2.8, 2.9 and since
we have
and
Theorem 3.14
(β-Laplace transform of the β-derivative function)
Let \(f\in V ([s_{0},\infty ),\mathbb{C} )\) be a function of exponential order λ. Then
provided that \(\lim_{t\rightarrow \infty }f(t)e_{\ominus _{\beta }z,\beta }(t)=0\).
Proof
Using Theorems 2.5, 3.3\((i_{4})\), we have
□
Corollary 3.15
Let \(f\in V ([s_{0},\infty ),\mathbb{C} )\) be a function of exponential order λ. Then, for any \(n\in \mathbb{N}\), we have
Proof
As a consequence of Theorem 3.14 and using induction, we get
\(\mathcal{L}_{\beta }\{D_{\beta }^{3}f(t)\}= z^{3} \mathcal{L}_{\beta } \{f(t)\}-z^{2} f(s_{0})-zD_{\beta }f(s_{0})-D_{\beta }^{2}f(s_{0})\).
Assume that the corollary is true for \(k\in \mathbb{N}\)
Then
Hence, the corollary holds for any \(n\in \mathbb{N}\). □
Example 3.16
Using the β-Laplace transform, find the solution of the β-initial value problem
Sol. By taking the β-Laplace transform and using equation (3.4), we have
so that
and hence
Theorem 3.17
(β-Laplace transform of the β-integral function)
Let \(f\in V ([s_{0},\infty ),\mathbb{C} )\) be a function of exponential order λ. Then
where
provided that \(\lim_{t\rightarrow \infty }F(t)e_{\ominus _{\beta }z,\beta }(t)=0\).
Proof
Using Theorem 2.5, we have
provided \(\lim_{t\rightarrow \infty }F(t)e_{\ominus _{\beta }z,\beta }(t)=0\) holds. □
Corollary 3.18
Assume \(f\in V ([s_{0},\infty ),\mathbb{C} )\) and \(\mathcal{L}_{\beta }\{f(t)\}=F(z)\). Then
Proof
Using Theorem 3.3\((i_{5})\) and since \(\ominus _{\beta }(z\oplus _{\beta }\lambda )=(\ominus _{\beta }\lambda ) \oplus _{\beta }(\ominus _{\beta }z)\), we have
Then
□
Definition 3.19
Let \(\lambda \in \mathcal{R}_{\beta }^{c}\). We define the functions \(\psi _{k}:I\rightarrow \mathbb{C}\) for each \(k\in \mathbb{N}_{0}\) recursively by taking \(\psi _{0}(t):=1\), and
Theorem 3.20
Let \(\lambda \in \mathcal{R}_{\beta }^{c}\) and \(n\in \mathbb{N}_{0}=\{0,1,2,\ldots \}\) be given. Then
provided that
Proof
Using induction, for \(n=0\), we have
For any \(n\in \mathbb{N}\),
Suppose \(\mathcal{L}_{\beta }\{\psi _{n-1}(t)e_{\lambda ,\beta }(t)\}= \frac{1}{(z-\lambda )^{n}}\) for some \(n \geq 1\). Then, by using Theorems 2.5, 3.3, we get
Thus the desired result is satisfied for all \(n\in \mathbb{N}\). □
In the following theorem, we deduce the inverse β-Laplace transform \(\mathcal{L}_{\beta }^{-1}\).
Theorem 3.21
For \(z\in \mathcal{R}_{\beta }^{c}\) and \(\lambda \neq 0\),
such that
Proof
Let \(\lambda \neq 0\) be given. By the partial fraction
then taking the inverse β-Laplace transform and applying Theorem 3.10 and Theorem 3.20, we obtain
□
Corollary 3.22
Let \(\lambda \neq 0\), \(z\in \mathcal{R}_{\beta }^{c}\). The following relations hold:
-
(1)
\(\mathcal{L}_{\beta }^{-1} \{\frac{z}{(z^{2}+\lambda ^{2})^{2}} \}=\frac{\sin _{\lambda ,\beta }(t)}{2\lambda }\int _{s_{0}}^{t} \frac{1}{1+ \lambda ^{2} (\beta (\tau )-\tau )^{2}}\,d_{\beta } \tau -\frac{\cos _{\lambda ,\beta }(t)}{2}\int _{s_{0}}^{t} \frac{ (\beta (\tau )-\tau )}{1+\lambda ^{2} (\beta (\tau )-\tau )^{2}}\,d_{\beta }\tau \).
-
(2)
\(\mathcal{L}_{\beta }^{-1} \{\frac{z^{2}}{(z^{2}+\lambda ^{2})^{2}} \} =\frac{\sin _{\lambda ,\beta }(t)}{2\lambda } + \frac{\cos _{\lambda ,\beta }(t)}{2}\int _{s_{0}}^{t} \frac{1}{1+\lambda ^{2} (\beta (\tau )-\tau )^{2}}\,d_{\beta } \tau +\frac{\lambda \sin _{\lambda ,\beta }(t)}{2}\int _{s_{0}}^{t} \frac{ (\beta (\tau )-\tau )}{1 + \lambda ^{2} (\beta (\tau )-\tau )^{2}}\,d_{\beta }\tau \).
-
(3)
\(\mathcal{L}_{\beta }^{-1} \{\frac{z^{3}}{(z^{2}+\lambda ^{2})^{2}} \}= \cos _{\lambda ,\beta }(t)- \frac{\lambda \sin _{\lambda ,\beta }(t)}{2}\int _{s_{0}}^{t} \frac{1}{1+\lambda ^{2} (\beta (\tau )-\tau )^{2}}\,d_{\beta } \tau +\frac{\lambda ^{2}\cos _{\lambda ,\beta }(t)}{2}\int _{s_{0}}^{t} \frac{ (\beta (\tau )-\tau )}{1+ \lambda ^{2} (\beta (\tau )-\tau )^{2}}\,d_{\beta }\tau \).
Example 3.23
Using the β-Laplace transform, find the solution of the β-initial value problem
Sol. By applying the β-Laplace transform of equation (3.5), we get
and then
Therefore,
Since
then
Hence,
4 Conclusion
In this paper, a general quantum Laplace transform \(\mathcal{L}_{\beta }\) associated with the general quantum difference operator \(D_{\beta }\) and some of its properties were introduced. Moreover, the β-Laplace transform of some fundamental functions was computed. Finally, the inverse β-Laplace transform \(\mathcal{L}_{\beta }^{-1}\) was presented.
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Shehata, E.M., Faried, N. & El Zafarani, R.M. A general quantum Laplace transform. Adv Differ Equ 2020, 613 (2020). https://doi.org/10.1186/s13662-020-03070-5
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DOI: https://doi.org/10.1186/s13662-020-03070-5