Shehu Transform in Quantum Calculus and Its Applications

Abstract

In this paper, we introduce the concept of Shehu transform in q-calculus namely q-Shehu transform and establish some properties. We also give some applications of q-Shehu transform for solving some ordinary and partial differential equations with initial and boundary values problems to show its effectiveness and performance of the proposed transform.

Introduction

Researchers are actively involved in the overall transformation of theme development because it is suitable for describing and analyzing physical systems [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. Jackson [21] introduced q-calculus. Now, the q-calculus has become very important in various fields of science and technology. The concept of q-calculus can be used in fractions and control problems [22]. Some integral transformations have different q analogs. The research is carried out on the q-calculus [23,24,25]. Maitama and Zhao [1] first established the Shehu transform to solve partial differential equations in the time domain as a generalization of Sumudu and Laplace transforms. So we motivate to introduce the concept q-Shehu transform to get the advantages in q-calculus.

The Shehu transform [1] of the function $f (\varpi )$ is defined by

We introduce the concept of Shehu transform in q-calculus namely q-Shehu transform and establish some properties. We also give some applications of q-Shehu transform for solving some ordinary and partial differential equations with initial and boundary values problems to show its effectiveness and performance of the proposed transform.

Preliminaries

We use literature’s [23, 26,27,28] for our study.

The q-shifted factorials for $q\in (0,1)$ and $\kappa \in \mathbb {C}$ are defined as

$$\begin{aligned} (\kappa , q)&= 1,(\kappa , q)_{n} = \prod ^{n-1}_{k=0} (1-\kappa q^{k}), n= 1,2...,\\ (\kappa ;q)_{\infty }&= \lim _{n \rightarrow \infty } (\kappa ;q)_{n}= \prod ^{\infty }_{k=0} (1-\kappa q^{k}). \end{aligned}$$

Also,

$$\begin{aligned}{}[\kappa ]_{q} = \dfrac{1-q^{\kappa }}{1-q}, [\kappa ]_{q}! = \dfrac{(q;q)_{n}}{(1-q)^{n}},~ n\in \mathbb {N}. \end{aligned}$$

The q-derivatives $D_{q}f$ and $D_{q}^{+}$ of a function f, given by Kac and Cheung [22]

$$\begin{aligned} (D_{q}f) (\alpha ) = \dfrac{f(\alpha )-f(q\alpha )}{(1-q)\alpha } ,\,\hbox {if}\, (\alpha \ne 0) \end{aligned}$$

and $(D_{q}f) (0) = f^{\prime } ( 0)$ exists.

If f is differentiable, then $(D_{q} f) (\alpha )$ tends to $f^{\prime }(\alpha )$ as q tends to 1. For $n\in \mathbb {N}$, we have

$$\begin{aligned} D^{1}_{q} = D_{q}, ~ ~ (D_{q}^{+})^{1}= D^{+}_{q}. \end{aligned}$$

The q-derivative of the product

$$\begin{aligned} D_{q} (f.g) (\alpha ) = g(\alpha ) D_{q}f(\alpha )+f(q\alpha ) D_{q} g(\alpha ). \end{aligned}$$

The q-Jackson integral from 0 to k and from 0 to $\infty$ given by Jackson [21]

$$\begin{aligned} \int _{0}^{\kappa } f(\alpha ) d_{q}\alpha= & {} (1-q)\alpha \sum _{n=0}^{\infty } f(\alpha q^{n}) q^{n},\\ \int _{0}^{\infty } f(\alpha ) d_{q}\alpha= & {} (1-q) \sum _{n=-\infty }^{\infty } f( q^{n}) q^{n}, \end{aligned}$$

provided these sums converge absolutely.

A q-analogue of integration by parts formulae is given by the following relation:

Gasper and Rahamen [27], Kac and Cheung [22] have given the following relation:

$$\begin{aligned} E_{q}^{\rho }&= \sum _{n=0}^{\infty } q^{\dfrac{n(n-1)}{2}} \dfrac{\rho ^{n}}{[n]_{q}!}= (-(1-q)z;q)_{\infty }, \end{aligned}$$

(1)

$$\begin{aligned} e^{\rho }_{q}&= \sum _{n=0}^{\infty } \dfrac{\rho ^{n}}{[n]_{q}!} = \dfrac{1}{((1-q)\rho ;q)_{\infty }},~ ~|z| < \dfrac{1}{1-q}. \end{aligned}$$

(2)

The above Eqs. (1) and (2) satisfy the following equations:

$$\begin{aligned} D_{q}e^{\rho }_{q}= & {} e^{\rho }_{q}, ~ D_{q} E^{\rho }_{q} = E^{q\rho }_{q},\,\hbox { and } \\ e^{\rho }_{q}E^{-\rho }_{q}= & {} E^{-\rho }_{q} e^{\rho }_{q} =1. \end{aligned}$$

Jackson [21] has introduced the following concept and many researchers [22, 26, 28] have given important results on it,

If satisfies the following conditions

$$\begin{aligned} \varGamma _{q}(\vartheta +1)= & {} [\vartheta ]_{q} \varGamma (\vartheta ) , \varGamma _{q} (1)= 1,\, \hbox {and}\\ \lim _{q\rightarrow 1^{-}} \varGamma _{q} (\vartheta )= & {} \varGamma (\vartheta ),~~ Re (\vartheta ) >0. \end{aligned}$$

The function $\varGamma _{q}$ has the following q -integral representations

The q-integral representation $\varGamma _{q}$ is defined in [23, 29] as follows:

For all $\gamma ,\vartheta > 0$, we have

where,

$$\begin{aligned} K_{q} (\vartheta ) = \dfrac{(-q,-1;q)_{\infty }}{(-q^{\vartheta },-q^{1-\vartheta };q)_{\infty }}. \end{aligned}$$

If $\dfrac{\log (1-q)}{\log (q)}\in \mathbb {Z},$ we obtain

Main results

Definition 3.1

$\varLambda$ be a function defined by $\mathbb {R}_{q_{1},+},$ we defined the q-Shehu transform of a function $\varLambda$ as

Property 3.2

(Linearity property) Let the functions $M \varLambda (\varpi )$ and $N \rho (\varpi )$ be in set A, then $[M \varLambda (\varpi ) + N \rho (\varpi )] \in A$, where M and N are non-zero arbitrary constants, and

$$\begin{aligned} S_{q} [M \varLambda (\varpi ) + N \rho (\varpi )] = M S_{q} [\varLambda (\varpi )] + N S_{q} [\rho (\varpi )]. \end{aligned}$$

Proof

Property 3.3

(Change of the scale property) Let the function $\varLambda (M(\varpi ))$ be in a set A, where M is an arbitrary constant, then

$$\begin{aligned} S_{q} [\varLambda (M \varpi )] = \dfrac{\varrho }{M} R \Big (\dfrac{\tau }{M}, \varrho \Big ). \end{aligned}$$

Proof

We obtain

Substituting $\eta = M \varpi \Rightarrow \varpi = \dfrac{\eta }{M}$ and $d_{q} \varpi = \dfrac{d_{q} \eta }{M}.$

Therefore,

Property 3.4

q-Shehu transform shoes the following:

(1) If $\varLambda (\varpi ) = 1$ be in set A, then

$$\begin{aligned} S_{q} (1) = \dfrac{\varrho }{(1-q) \tau }. \end{aligned}$$

Proof

Using the concept (3), we obtain

(2) If $\varLambda (\varpi ) = \varpi$ in a set A, then

$$\begin{aligned} S_{q} (\varpi ) = \dfrac{ \varrho ^{2} }{(1-q) \tau ^{2} }. \end{aligned}$$

Proof

Using the concept (3), we obtain

(3) If $\varLambda (\varpi )$ = $\dfrac{\varpi ^{n}}{n!}$ ,   n= 0, 1, 2.. in a set A, then

$$\begin{aligned} S_{q} \Big [ \dfrac{\varpi ^{n}}{n!} \Big ] = \dfrac{1}{(1-q)} \Big (\dfrac{\varrho }{\tau } \Big )^{n+1}. \end{aligned}$$

Proof

Using the concept (3), we obtain

(4) If $\varLambda (\varpi ) =e_{q}^{(M \varpi )}$ in set A, then

$$\begin{aligned} S_{q} \left[ e_{q}^ {(M \varpi )}\right] = \dfrac{1}{(1-q)} \dfrac{\varrho }{(\tau -M \varrho )}. \end{aligned}$$

Proof

Using the concept (3), we obtain

(5) If $\varLambda (\varpi ) = \varpi e_{q}^{(M \varpi )}$ in set A, then

$$\begin{aligned} S_{q} \Big [ \varpi e_{q}^{(M \varpi )} \Big ] = \dfrac{1}{1-q} \dfrac{\varrho ^{2}}{(\tau -M \varrho )^{2}}. \end{aligned}$$

Proof

Using the concept (3), we obtain

(6) If $\varLambda (\varpi ) = \sin _{q}{(M \varpi )}$ in set A, then

$$\begin{aligned} S_{q} \left[ sin_{q} (M \varpi )\right]= & {} \dfrac{M \varrho ^{2}}{\tau ^{2}- \tau ^{2} q+ M^{2} \varrho ^{2} - M^{2} \varrho ^{2} q }. \end{aligned}$$

Proof

Using the concept (3), we obtain

(7) If $\varLambda (\varpi ) = \cos _{q} {(M \varpi )}$ in set A, then

$$\begin{aligned} S_{q} [cos_{q} (M \varpi )] = \dfrac{\varrho \tau }{\tau ^{2}- \tau ^{2} q + M^{2} \varrho ^{2} - q M^{2} \varrho ^{2} } . \end{aligned}$$

Proof

Using the concept (3), we obtain

Theorem 3.5

If the Shehu transform of a function $\varLambda (\varpi )$ exists, then

$$\begin{aligned} S_{q} \left[ \varLambda (\varpi -M) H ( \varpi -M ) \right] = e_{q}^{-\dfrac{\tau M }{\varrho }} S_{q } [\varLambda (\varpi -M) ], \end{aligned}$$

where $H ( \varpi )$ is Heaviside unit step function defined by $H ( \varpi -M )$ = 1, when $\eta > M$ and $H ( \varpi -M )$ = 0 when $\eta < M.$

Proof

We have by definition,

By putting $A = \varpi - M$

Theorem 3.6

If the Shehu transform of the $\varLambda (\varpi )$ exists where $\varLambda (\varpi )$ is a periodic function of periods $\varLambda$ (That is $\varLambda (\varpi + A) = \varLambda (\varpi ), \forall \varpi ),$ then

Proof

Setting $\varpi = A + M$ in the second integral, we have

Theorem 3.7

(q - Shehu Convolution product) The Convolution of two functions $\varLambda (\varpi )$ and $\rho (\varpi )$ is denoted by $( \varLambda * \rho ) (\varpi )$ and defined as

Convolution theorem

Statement. Let $\varLambda _{1} (\varpi )$ and $\varLambda _{2} (\varpi )$ be a positive scalar functions of $\varpi$ and let $\rho _{1} (\varpi )$ and $\rho _{2} (\varpi )$ be their q- Shehu transforms, then

$$\begin{aligned} S_{q} \{ \varLambda _{1} (\varpi ) * \varLambda _{2} (\varpi ) \} = \rho _{1} (\varpi ) . \rho _{2} (\varpi ), \end{aligned}$$

where $\varLambda _{1} (\varpi ) * \varLambda _{2} (\varpi )$ =

Proof

We have

Let $\varpi - N= M$ $\Rightarrow d_{q} \varpi = d_{q} M.$

Applications

Application 4.1

We take the first order ODE

$$\begin{aligned} \dfrac{d_{q} \varLambda (\varpi )}{d_{q} \varpi } + \varLambda (\varpi ) = 0, \end{aligned}$$

(4)

with $\varLambda (0) = 1.$

Applying the concept of q-Shehu transform to the Eq. (4), we obtain

$$\begin{aligned} \dfrac{\tau }{\varrho } S_{q} (\tau , \varrho ) - S_{q} \{\varLambda (0)\} + S_{q} (\tau , \varrho ) =0. \end{aligned}$$

By applying initial condition, we get

$$\begin{aligned} S_{q} (\tau , \varrho ) = \dfrac{\varrho ^{2}}{ (1-q) (\tau + \varrho ) \tau }. \end{aligned}$$

(5)

By applying inverse on Eq. (5), we get

$$\begin{aligned} \varLambda (\varpi )= & {} S_{q}^{-1} \Big [ \dfrac{\varrho ^{2}}{ (1-q) (\tau + \varrho ) \tau } \Big ]. \\= & {} S_{q}^{-1} \Big [ \dfrac{\varrho }{(1-q)\tau }\Big ] - S_{q}^{-1} \Big [ \dfrac{\varrho }{(1-q) (\tau +\varrho )} \Big ] \\= & {} 1-e_{q}^{\varpi }. \end{aligned}$$

Application 4.2

Consider the following second order ODE

$$\begin{aligned} \dfrac{d^{2}_{q} \varLambda (\varpi )}{d \varpi ^{2}}+ \dfrac{d_{q} \varLambda (\varpi )}{d_{q} \varpi } = 1 \end{aligned}$$

(6)

with $\varLambda (0) = 0,~ \dfrac{d_{q} \varLambda (\varpi (0))}{d_{q} \varpi }= 0$.

Applying the concept of q-Shehu transform to the Eq. (6), we obtain

$$\begin{aligned}&\dfrac{\tau ^{2}}{\varrho ^{2}} S_{q} (\tau , \varrho ) - \dfrac{\tau }{\varrho } S_{q} \{ \varLambda (0)\} - S_{q} \{ \varLambda ^{/} (0)\} + \dfrac{\tau }{\varrho } S_{q} (\tau , \varrho ) - S_{q} \{ \varLambda (0)\} = \dfrac{1}{(1-q)} \dfrac{\varrho }{\tau } \\&\Rightarrow \Big [ \dfrac{\tau ^{2}+ \varrho \tau }{\varrho ^{2}} \Big ] S_{q}(\tau , \varrho ) = \dfrac{1}{(1-q)} \dfrac{\varrho }{\tau }. \end{aligned}$$

By applying initial conditions, we get

$$\begin{aligned}&\Rightarrow R(\tau , \varrho ) = \dfrac{\varrho ^{3}}{(1-q) (\tau ^{2} + \varrho \tau ) \tau } \\&\quad \Rightarrow R(\tau , \varrho ) = \dfrac{\varrho ^{3}}{(1-q) (\tau ^{3} + \varrho \tau ^{2}) }. \end{aligned}$$

Taking the inverse q-Shehu transform, we get

$$\begin{aligned} \varLambda (\varpi )= & {} S_{q}^{-1} \Big [ \dfrac{\varrho ^{3}}{(1-q) \tau ^{2} (\tau + \varrho ) } \Big ] \\= & {} S_{q}^{-1} \left\{ \dfrac{\varrho ^{2}}{(1-q) \tau } \Big [ \dfrac{1}{\tau } - \dfrac{1}{\tau + \varrho } \Big ] \right\} \\= & {} S_{q}^{-1} \Big [ \dfrac{\varrho ^{2}}{\tau ^{2}(1-q)} \Big ] - S_{q}^{-1} \Big [ \dfrac{\varrho ^{2}}{\tau (\tau + \varrho ) (1-q)} \Big ] \\= & {} \varpi - S_{q}^{-1} \Big [ \dfrac{\varrho }{(1-q)\tau } - \dfrac{\varrho }{(1-q) (\tau + \varrho )} \Big ] \\= & {} \varpi -1 -e_{q}^{-\varpi }. \end{aligned}$$

Application 4.3

Consider the following second non-homogeneous order ODE

$$\begin{aligned} \dfrac{d^{2}_{q} \varLambda (\varpi )}{d \varpi ^{2}} -3 \dfrac{d_{q} \varLambda (\varpi ) }{d_{q} \varpi } + 2 \varLambda (\varpi ) = e_{q}(3 \varpi ) \end{aligned}$$

(7)

with $\varLambda (0) = 1, ~ ~ \dfrac{d_{q} \varLambda (0) }{d_{q} \varpi } = 0.$

Applying the concept of q-Shehu transform to the Eq. (7), we obtain

$$\begin{aligned}&\dfrac{\tau ^{2}}{\varrho ^{2}} \overline{\varLambda } - \dfrac{\tau }{\varrho } S_{q} \{\varLambda (0)\} - S_{q} \{ \varLambda ^{'}(0)\} -3 \Big ( \dfrac{\tau }{\varrho } \overline{\varLambda }(\tau , \varrho ) - S_{q} \{\varLambda (0)\} \Big ) + 2 \overline{\varLambda } (\tau , \varrho ) \\&\quad = \dfrac{\varrho }{(1-q) (\tau -3\varrho ) }. \\&\quad \Rightarrow \Big [ \dfrac{\tau ^{2}}{\varrho ^{2}}-3 \dfrac{\tau }{\varrho } + 2 \Big ] \overline{\varLambda }(\tau , \varrho ) - \dfrac{1}{1-q} + \dfrac{3 \varrho }{ (1-q) \tau } = \dfrac{\varrho }{(1-q) (\tau -3\varrho ) } \\&\quad \Rightarrow \Big [ (\dfrac{\varrho }{\tau }-2) (\dfrac{\varrho }{\tau }-1) \Big ] \overline{\varLambda } (\tau , \varrho ) = \dfrac{\varrho }{(1-q) (\tau -3\varrho ) } -\dfrac{3 \varrho }{ (1-q) \tau } + \dfrac{1}{1-q} \\&\quad \Rightarrow \overline{\varLambda } (\tau , \varrho ) = \dfrac{ \varrho ^{3}}{(1-q) (\tau -3\varrho ) (\tau -2\varrho )(\tau -\varrho )}- \dfrac{3 \varrho ^3 }{ (1-q)\tau (\tau -2\varrho )(\tau -\varrho ) }\\&\qquad + \dfrac{ \varrho ^2 }{(1-q)(\tau -2\varrho )(\tau -\varrho )} \\&\quad \Rightarrow \overline{\varLambda } (\tau , \varrho ) = \dfrac{\varrho ^{2}}{(1-q) (\tau -3 \varrho ) } \Big [ \dfrac{1}{\tau -2\varrho } - \dfrac{1}{\tau -\varrho } \Big ] - \dfrac{3\varrho ^{2} }{(1-q) \tau } \Big [ \dfrac{1}{\tau -2 \varrho } - \dfrac{1}{\tau -\varrho } \Big ] \\&\qquad + \dfrac{\varrho }{1-q} \Big [ \dfrac{1}{\tau -2\varrho } - \dfrac{1}{\tau -\varrho }\Big ] \\&\quad \Rightarrow \overline{\varLambda } (\tau , \varrho ) = \dfrac{\varrho ^{2}}{(1-q)~ (\tau -3\varrho )~ (\tau -2\varrho )} - \dfrac{\varrho ^{2}}{(1-q)~ (\tau -3\varrho )~ (\tau -\varrho )} - \dfrac{3\varrho ^{2}}{(1-q)~ \tau ~ (\tau -2\varrho )} \\&\qquad + \dfrac{3\varrho ^{2}}{(1-q)~ \tau ~ (\tau -\varrho )} + \dfrac{\varrho }{(1-q)~ (\tau -2\varrho )} - \dfrac{\varrho }{(1-q)~ (\tau -\varrho )} \\&\quad \Rightarrow \overline{\varLambda } (\tau , \varrho ) = \dfrac{\varrho }{(1-q) (\tau -3\varrho )} - \dfrac{\varrho }{(1-q) (\tau -2\varrho )} - \dfrac{\varrho }{2~(1-q) (\tau -3\varrho )} + \dfrac{\varrho }{2~(1-q) (\tau -\varrho )} \\&\qquad - \dfrac{3\varrho }{2~(1-q) (\tau -2\varrho )} + \dfrac{3\varrho }{2~(1-q) \tau } - \dfrac{3\varrho }{(1-q) \tau } + \dfrac{3\varrho }{(1-q) (\tau -\varrho )} \\&\qquad + \dfrac{\varrho }{(1-q) (\tau -2\varrho ) } - \dfrac{\varrho }{(1-q) (\tau -\varrho ) }. \end{aligned}$$

Taking the inverse q-Shehu transform, then

$$\begin{aligned}&\Rightarrow \overline{\varLambda } (\varpi ) = S_{q}^{-1} \Big [ \dfrac{\varrho }{2 (1-q) (\tau -3\varrho )} \Big ] - S_{q}^{-1} \Big [ \dfrac{3\varrho }{2 (1-q) (\tau -2\varrho )} \Big ] + S_{q}^{-1} \Big [ \dfrac{5\varrho }{2 (1-q) (\tau -\varrho )} \Big ] \\&\qquad - S_{q}^{-1} \Big [ \dfrac{5\varrho }{2 (1-q) (\tau -\varrho )} \Big ] - S_{q}^{-1} \Big [ \dfrac{3\varrho }{2~ (1-q) ~\tau } \Big ] \\&\quad \Rightarrow \overline{\varLambda } (\varpi ) = \dfrac{1}{2} e_{q}^{\varpi } - \dfrac{3}{2} e_{q}^{2\varpi } + \dfrac{5}{2} e_{q}^{\varpi } - \dfrac{3}{2}. \end{aligned}$$

Application 4.4

A semi-infinite solid $\eta > 0$ is initially at temperature zero. At time $\varpi > 0,$ a constant temperature $R_{0} > 0$ is applied and maintained at the face $\eta =0$. Find the temperature at any point of the solid at any time $\varpi > 0$.

Here the temperature $\varrho (\eta , \varpi )$ at any point of the solid at any time $\varpi > 0$ is governed by one dimensional heat equation

$$\begin{aligned} \dfrac{\partial _{q} \varrho }{ \partial _{q} \varpi } = C^{2} \dfrac{\partial ^{2}_{q} \varrho }{\partial _{q} \eta ^{2} }, (\eta>0, \varpi > 0), \end{aligned}$$

(8)

with the initial and boundary conditions

$$\begin{aligned} \varrho (0, \varpi ) = R_{0},~ \varrho (\eta , 0) = 0. \end{aligned}$$

Applying the concept of q-Shehu transform to the Eq. (8), we obtain

$$\begin{aligned}&\dfrac{\tau }{\varrho } \overline{\varrho } (\eta , \varpi ) - S_{q} \{ \varrho ( \eta , 0)\} = C^{2} \dfrac{d^{2}_{q} \overline{\varrho } }{d_{q} \eta ^{2}} \\&\Rightarrow \dfrac{d^{2}_{q} \overline{\varrho } }{d_{q} \eta ^{2}} - \dfrac{\tau }{\varrho C^{2} } \overline{\varrho } =0 . \end{aligned}$$

The solution is

$$\begin{aligned} \overline{\varrho }= A~ e^{\sqrt{ \dfrac{\tau }{\varrho C^{2}}}\eta }_{q} + B~ e^{-\sqrt{ \dfrac{\tau }{\varrho C^{2}}}\eta }_{q}. \end{aligned}$$

(9)

Since $\varrho$ is finite, when $\eta \rightarrow \infty$.

$$\begin{aligned}&\therefore \overline{\varrho } \,\hbox {is also finite, when }\,\eta \rightarrow \infty .\\&\quad \therefore \,\hbox {A = 0, otherwise}\,\overline{\varrho } \rightarrow \infty \, \hbox {as} \,\eta \rightarrow \infty . \end{aligned}$$

Taking the q-Shehu transform of the both sides, then we get

$$\begin{aligned} \varrho (0, \varpi ) = R_{0}. \end{aligned}$$

Therefore, $\overline{\varrho }$ = $R_{0} \dfrac{\varrho }{(1-q) \tau }.$

$\therefore$ From (9), we have $\overline{\varrho } = B = R_{0} \dfrac{\varrho }{(1-q) \tau }.$

Hence $\overline{\varrho }$= $R_{0} \dfrac{\varrho }{(1-q) \tau }~ e^{-\sqrt{ \dfrac{\tau }{\varrho C^{2}}}\eta }_{q}$.

Taking the inverse q-Shehu transform, then

$$\begin{aligned} \overline{\varrho } (\eta ,\varpi )= & {} S_{q}^{-1} \Big [ R_{0} \dfrac{\varrho }{(1-q) \tau }~ e^{-\sqrt{ \dfrac{\tau }{\varrho C^{2}}}\eta }_{q} \Big ] \\= & {} S_{q}^{-1} \Big [ \dfrac{R_{0}}{(1-q) \dfrac{\tau }{\varrho } \tau }~ e^{-\sqrt{ \dfrac{\tau /\varrho }{ C^{2}}}\eta }_{q} \Big ] \\= & {} R_{0}~ erfc_{q} \Big ( \dfrac{\eta }{2C \sqrt{\varOmega }} \Big ). \end{aligned}$$

Application 4.5

Find the solution of the equation

$$\begin{aligned} \dfrac{\partial _{q} \varrho }{ \partial _{q} \varpi } = K \dfrac{\partial _{q}^{2} \varrho }{ \partial _{q}^{2} \eta ^{2} } \end{aligned}$$

(10)

which tends to zero as $\eta \rightarrow \infty$ and which satisfies the conditions $\varrho = f(\varpi )$ when $\eta =0,$ $\varpi > 0$ and $\varrho = 0$ when $\eta > 0$, $\varpi = 0$.

Applying the concept of q-Shehu transform to the Eq. (10), we obtain

$$\begin{aligned} \dfrac{\tau }{\varrho } \overline{\varrho } (\eta , \varpi ) -S_{q} \{ \varrho (\eta ,0)\}= & {} K \dfrac{d^{2}_{q} \overline{\varrho } }{d_{q} \eta ^{2}}\Rightarrow \dfrac{d^{2}_{q} \overline{\varrho } }{d_{q} \eta ^{2}} - \dfrac{\tau }{K \varrho } \overline{\varrho } = 0 . \end{aligned}$$

The solution is

$$\begin{aligned} \overline{\varrho } = A~ e_{q}^{\sqrt{\dfrac{\tau }{K \varrho }}\eta } + B~ e_{q}^{-\sqrt{\dfrac{\tau }{K \varrho }}\eta } . \end{aligned}$$

Since $\varrho \rightarrow 0$  as  $\eta \rightarrow \infty$.

$$\begin{aligned} \therefore \overline{ \varrho } \rightarrow 0 ~ as ~ \eta \rightarrow \infty . \end{aligned}$$

From which it follows that $A = 0.$

Therefore,

$$\begin{aligned} \overline{\varrho } = B~ e_{q}^{-\sqrt{\dfrac{\tau }{K \varrho }}\eta }. \end{aligned}$$

(11)

Again when $\eta = 0,$

$\therefore$ From (11), we have

$$\begin{aligned} \overline{\varLambda }(\rho ) = B. \end{aligned}$$

Hence, $\overline{\varrho } = \overline{\varLambda }(\rho )~ e_{q}^{-\sqrt{\dfrac{\tau }{K \varrho }}\eta }.$

If we take $\varLambda (\varpi ) = M$ (constant),

where, constant may be real or complex.

then $\overline{\varLambda }(\rho ) = \dfrac{M}{(1-q)} \dfrac{\varrho }{\tau }.$

Taking the inverse q-Shehu transform, we get

$$\begin{aligned} \varrho (\eta , \varpi )= & {} S_{q}^{-1} \Big [ \dfrac{M}{(1-q)} . \dfrac{\varrho }{\tau }~ e_{q}^{-\sqrt{\dfrac{\tau /\varrho }{K^{2} }}\eta } \Big ]\\= & {} \hbox {M} \,erfc_{q} \Big ( \dfrac{\eta }{2 \sqrt{K} \varpi } \Big ). \end{aligned}$$

Application 4.6

An infinite long string having one end $\eta = 0$ is initially at rest on the $\eta$-axis. The end $\eta = 0$ undergoes a periodic transverse displacement given by $A_{0} \sin {\rho \varpi }, ~ \varpi > 0,$ find the displacement of any point of string at $\varpi > 0$.

Here the displacement of any point of any point of the string is governed by the equation

$$\begin{aligned} \dfrac{\partial _{q}^{2} \varrho }{ \partial _{q} \varpi ^{2} } = K^{2} \dfrac{\partial _{q}^{2} \varrho }{ \partial _{q} \eta ^{2}}, \end{aligned}$$

(12)

with the boundary and initial conditions

$$\begin{aligned} \varrho (0, \varpi )= & {} A_{0} \sin {\rho \varpi }, ~ \varpi> 0, \\ \varrho (\eta , 0)= & {} 0,~ \varrho _{\varpi } (\eta , 0) = 0, ~ \eta > 0 \end{aligned}$$

and the displacement is finite.

Applying the concept of q-Shehu transform to the Eq. (12), we obtain

$$\begin{aligned} \dfrac{\tau ^{2}}{\varrho ^{2}} \overline{\varrho } (\eta , \varpi ) - \tau S_{q} \{ \varrho (\eta , 0)\} - S_{q} \{ \varrho _{\varpi } (\eta , 0) \}= & {} K^{2} \dfrac{d^{2}_{q} \overline{ \varrho } }{ d_{q} \eta ^{2}}.\Rightarrow \dfrac{d^{2}_{q} \overline{ \varrho } }{ d_{q} \eta ^{2}} - \dfrac{\tau ^{2}}{ \varrho ^{2} K^{2} } \overline{\varrho } = 0.\qquad \end{aligned}$$

(13)

Also $\overline{\varrho } (0, \varpi ) = A_{0} { S_{q}} \{ \sin {\rho \varpi } \} = A_{0} \dfrac{\rho \varrho ^{2}}{\tau ^{2}- \tau ^{2} q+ \rho ^{2} \varrho ^{2} - \rho ^{2} \varrho ^{2} q }$ and $\overline{\varrho } (\eta , \varpi )$ is finite for $\eta > 0$.

Now solution of (13) is given by

$$\begin{aligned} \overline{\varrho } = A~ e_{q}^{\dfrac{\tau }{K \varrho }\eta } + B~ e_{q}^{-\dfrac{\tau }{K \varrho }\eta }. \end{aligned}$$

Since $\overline{\varrho } (\eta , \varpi )$ is finite $\therefore$ A= 0, otherwise $\overline{\varrho } (\eta , \varpi )$ becomes infinite when $\eta \rightarrow \infty .$

$$\begin{aligned} \therefore \overline{\varrho } (\eta , \varpi ) = B~ e_{q}^{-\dfrac{\tau }{K \varrho }\eta }. \end{aligned}$$

Now $\overline{\varrho } (0, \varpi ) = B = A_{0} \dfrac{\rho \varrho ^{2}}{\tau ^{2}- \tau ^{2} q+ \rho ^{2} \varrho ^{2} - \rho ^{2} \varrho ^{2} q }$.

Hence,

Taking the inverse q-Shehu transform, then

$$\begin{aligned} \varrho (\eta , \varpi )= & {} sin_{q} (\rho \varpi -\eta /K ) \,\hbox {when} \, \rho ~\varpi > \eta /K\\= & {} 0\,\hbox {when} \, \rho ~\varpi < \eta /K. \end{aligned}$$

Discussion

Maitama and Zhao [1] have introduced a new integral transform named Shehu transform to generalize Sumudu and Laplace transform for solving differential equations in the time domain. Quantum calculus is a calculus without limits. So we have applied quantum calculus in Shehu transform to explore the quantum concept in the application of Shehu transform. This is the nobility of the proposed transform.

The q-Shehu transform may be applied to solve heat and transport equations, Volterra integral equations of the first kind, Bessel’s functions, and cryptography in the quantum calculus.

The proposed method is an analytic method that gives the solution of ordinary and partial differential equation with initial and boundary conditions when we compare with the model introduced by Hadid et al. [13], which optimizes the bounded interval using the fraction entropy while Zhang et al. [30] established the existence of solution type solutions for a class of fractional Choquard equations. In Zhang et al. [30], the technique was based on constrained minimization of arguments, whereas in our proposed method as Knill [31] has given advantages of quantum calculus that in the calculus, the differential form and geometric objects are treated in the same way and also it allows to do calculus on continuous functions which do not need to be smooth. So by the features mentioned above, the proposed method is more effective and high performs for solving real-life problems in differential equations.

Conclusion

We introduced the concept of Shehu transform in q-calculus namely q-Shehu transform and established some properties. We applied q-Shehu transform for solving some ordinary and partial differential equations with initial and boundary values problems to show its effectiveness and performance of the proposed transform.

Open Problems

(1) As Alfaqeih and Misirli [3] have introduced the concept of double Shehu transform and its properties with applications. Is this concept is applicable in q-calculus?

(2) Does the above study also applicable for the other transform or fractional operator?