Martínez–Kaabar Fractal–Fractional Laplace Transformation with Applications to Integral Equations

Abstract

This paper addresses the extension of Martinez–Kaabar (MK) fractal–fractional calculus (for simplicity, in this research work, it is referred to as MK calculus) to the field of integral transformations, with applications to some solutions to integral equations. A new notion of Laplace transformation, named MK Laplace transformation, is proposed, which incorporates the MK $\alpha ,\gamma$-integral operator into classical Laplace transformation. Laplace transformation is very applicable in mathematical physics problems, especially symmetrical problems in physics, which are frequently seen in quantum mechanics. Symmetrical systems and properties can be helpful in applications of Laplace transformations, which can help in providing an effective computational tool for solving such problems. The main properties and results of this transformation are discussed. In addition, the MK Laplace transformation method is constructed and applied to the non-integer-order first- and second-kind Volterra integral equations, which exhibit a fractal effect. Finally, the MK Abel integral equation’s solution is also investigated via this technique.

1. Introduction

Integral equations (InEqs) are highly applicable in modeling various scientific and engineering phenomena. Some of the most important cases of InEqs are the first and second kind of Volterra-type InEqs. Various approaches have been developed for solving InEqs, among which the following methods stand out: successive approximation, series solution, variational iterative, and Adomian decomposition [1,2,3,4,5,6,7]. In addition, the method of Laplace transformation constitutes an efficient approach to solve the first- and second-order Volterra InEqs if the kernel of these InEqs is a difference kernel, and in such a case, the Laplace transform convolution theorem can be employed [1,2]. Symmetry and Laplace transformations play an important role in the computational process and in solving various scientific problems efficiently, where the symmetrical properties and system can simplify the computational process and offer effective solutions. Some results of Laplace transformation sometimes show the symmetrical behavior of the original function, which helps greatly in analysis. One of the essential cases of a singular InEq is called the Abel integral equation (AInEq), which has been applied in radio astronomy, electron emission, atomic scattering, radar ranging, and many other applications. In addition, AInEqs possess a difference kernel, so the method of Laplace transformation represents an effective approach to their solution [1,2,8,9].

A branch of mathematical analysis to which much relevant research has been devoted is fractional calculus. The notion of a fractional derivative is theoretically established from a non-local conception, such as Riemann–Liouville (RL) and Caputo (C) derivatives. In the past few decades, another local formulation of this concept based on incremental ratios, known as conformable derivative (CoD), has been introduced [10]. The most relevant properties of these definitions, including their interesting applications, such as fractional InEqs, have been mentioned in [11,12,13,14,15].

In recent years, as a result of the challenges associated with both non-local and local fractional derivatives, a new generalization of local definitions has been proposed. In particular, the previously discussed CoD has disadvantages in comparison with the C fractional definition for certain functions [16]. As a result, a new generalization of non-integer-order derivative has been formulated in [17], named the Abu-Shady–Kaabar derivative (ASKD), which provides a simple way to analytically solve fractional differential equations, and the results agree exactly with those produced by employing RL and C fractional derivatives.

A combination of both fractional and fractal differentiation constitutes a hybrid differentiation method, which was formulated in [18], and this approach includes the following properties in its system: fractal geometry, elastic viscosity, heterogeneity, and memory effect.

Very recently, a new generalization of fractal–fractional derivative (FrFrD) of the local type has been defined in [19], named the Martinez–Kaabar (MK) fractal–fractional derivative. As a result of the MK derivative, all obtained results of this definition were in agreement with the FrFrD’s results in the C sense with the power law, which was proposed in [18], when this definition was employed for certain elementary functions. In addition, in [19,20], this newly proposed calculus (so-called MK calculus) has been investigated, including its main mathematical analysis elements, integration, and InEqs. This included the second kind of Volterra MK InEqs, which was solved via the method of Adomian decomposition.

In comparison with the previous studies on various fractional definitions that have been employed in studying InEqs, our study indicates a high level of novelty by extending the newly proposed MK calculus to Laplace transformation. This generalized technique provides a novel mathematical tool to solving not only InEqs but also a variety of scientific, engineering, and economic problems and modelling scenarios that exhibit a fractal effect.

Motivated by all the above works, this work proposes to extend the MK calculus theory to the integral transformations, with applications in the solutions of certain InEqs. Therefore, this work is outlined as follows:

(i)

First, we define a new notion of Laplace transformation, named MK Laplace transformation in this work, which involves the MK integral operator. In relation to this new fractal–fractional transformation, we establish fundamental results, such as a sufficient condition that ensures its existence, the transformation of elementary functions, the relation of the transformation of a function and its derivative, the convolution theorem, and some qualitative properties of this transformation, among other results. The notion of inverse MK Laplace transformation is also established, and its main properties are discussed.

(ii)

Next, the MK Laplace transformation method is constructed in the context of the MK Volterra integral equations of the first and second kind.

(iii)

Finally, this new technique is employed to find the solutions to MK AInEqs and MK generalized AInEqs.

2. Preliminaries

This section includes basic concepts and results, which will be essential for the upcoming results in this research work. First of all, it is interesting to recall the definition introduced in [18]:

Definition 1. 

Suppose that $f\left(t\right)\in C^n\left([k,\infty )\right)$ is differentiable on $[k,\infty )$, where $k\ge 0$; if $f$ is a fractal differentiable on $[k,\infty )$ of order $\gamma$, then the FrFrD of $f$ of order $\alpha$ in the context of C with the power lawis written as

$$\begin{gathered} {}_k{}^{FrFrD}D_t^{\alpha ,\gamma }f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(n-\alpha \right)}}{{\displaystyle \int }}_k^t{\left(t-\tau \right)}^{n-\alpha -1}{\displaystyle \frac{df\left(\tau \right)}{d{\tau }^{\gamma }}}d\tau , \\ n-1<\alpha ,\gamma \le n, n\in N, \end{gathered}$$

(1)

where

$${\displaystyle \frac{df\left(\tau \right)}{d{\tau }^{\gamma }}}=\underset{t\to \tau }{\mathit{lim}}{\displaystyle \frac{f\left(t\right)-f\left(\tau \right)}{t^{\gamma }-{\tau }^{\gamma }}} .$$

(2)

In the above definition, Γ(⋅) is the Gamma function, the order of FrFrD is controlled by $\alpha$ and $\gamma$, and the integration term with the power-law kernel indicates the non-locality and memory-effect properties of fractal systems. In addition, Equation (2) can be updated by including the two-scale fractal derivative for the case of multi-fractal systems, where each of the two scales exhibits different fractal effects, indicating that various processes happen at various scales, as in the cases of turbulence and porous media.

Remark 1. 

(i)

${\displaystyle \frac{df\left(\tau \right)}{d{\tau }^{\gamma }}}$ in Equation (2) is the fractal derivative of order $\gamma$, with $\gamma >0$, which was proposed in [21,22].

(ii)

If we assume that $k=0$ and $n=1$, then Equation (1) can be expressed as

$${}_0{}^{FrFrD}D_t^{\alpha ,\gamma }f\left(t\right)={\displaystyle \frac{1}{\Gamma \left(1-\alpha \right)}}{{\displaystyle \int }}_0^t{\left(t-\tau \right)}^{-\alpha }{\displaystyle \frac{df\left(\tau \right)}{d{\tau }^{\gamma }}}d\tau .$$

(3)

It is also interesting to recall, in relation to the FrFrD of order α defined in Equation (3), that the following two results [18] are obtained:

Theorem 1. 

Suppose that $0<\alpha ,\gamma \le 1$, and $\delta >-1$. Then, we obtain

$${}_0{}^{FrFrD}D_t^{\alpha ,\gamma }\left(t^{\delta }\right)={\displaystyle \frac{\delta \Gamma \left(\delta -\gamma +1\right)}{\gamma \Gamma \left(\delta -\alpha -\gamma +2\right)}}{t}^{\delta -\alpha -\gamma +1}.$$

(4)

Remark 2. 

If $f\left(t\right)=w$ for every real constant$w$, then ${}_0{}^{FrFrD}D_t^{\alpha ,\gamma }\left(w\right)=0$.

Theorem 2. 

Assume that$0<\alpha ,\gamma \le 1$, a function $f\left(t\right)$ is supposed to be analytic at the origin with McLaurin expansion, written as follows:

$$f\left(t\right)={\displaystyle \sum }_{j=0}^{\infty }{a}_j{t}^j,$$

(5)

for   $t\in [0,k)$ with $k\in R^{+}$. Then, we obtain

$${}_0{}^{FrFrD}D_t^{\alpha ,\gamma }f\left(t\right)={\displaystyle \sum }_{j=0}^{\infty }{a}_j{}_0{}^{FrFrD}D_t^{\alpha ,\gamma }\left(t^j\right).$$

(6)

Remark 3. 

From Theorem 1, Equation (6) can be expressed as

$${}_0{}^{FrFrD}D_t^{\alpha ,\gamma }f\left(t\right)={\displaystyle \sum }_{j=1}^{\infty }{a}_j{\displaystyle \frac{j\Gamma \left(j-\gamma +1\right)}{\gamma \Gamma \left(j-\alpha -\gamma +2\right)}}{t}^{j-\alpha -\gamma +1}.$$

(7)

From Definition 1 and the above Theorems 1 and 2, the MK derivative of order $\alpha$ is defined [19] as follows:

Definition 2. 

A function: $f:[0,\infty )\to R$, the MK derivative of order $0<\alpha \le 1$, of $f$ at $t>0$ is written as

$${}^{MK}D^{\alpha ,\gamma }f\left(t\right)=\underset{\epsilon \to 0}{lim}{\displaystyle \frac{f\left(t+\epsilon M\left(\alpha ,\gamma ,\delta \right)t^{2-\alpha -\gamma }\right)-f\left(t\right)}{\epsilon }},$$

(8)

where   $M\left(\alpha ,\gamma ,\delta \right)={\displaystyle \frac{\Gamma \left(\delta -\gamma +1\right)}{\gamma \Gamma \left(\delta -\alpha -\gamma +2\right)}}$ with $0<\gamma \le 1$ and $\mathsf{\delta }>-1$.

If $f$ is MK $\alpha ,\gamma$-differentiable in some $\left(0,m\right)$, $m>0$, and $\underset{t\to 0^{+}}{lim}{}^{MK }D^{\alpha ,\gamma }f\left(t\right)$ exists, then it is written as

$${}^{MK }D^{\alpha ,\gamma }f\left(0\right)=\underset{t\to 0^{+}}{\lim }{}^{MK }D^{\alpha ,\gamma }f\left(t\right).$$

(9)

Remark 4. 

Note that if $\alpha =\gamma =1$, then Equation (8) is the usual limit-based definition of the derivative.

Theorem 3 

([19]). Suppose that $0<\alpha ,\gamma \le 1,$ and $f$ is a MK $\alpha ,\gamma$-differentiable at a point $t>0$. If, further, $f$ is a differentiable function, then

$${}^{MK}D^{\alpha ,\gamma }f\left(t\right)=M\left(\alpha ,\gamma ,\delta \right)t^{2-\alpha -\gamma }{\displaystyle \frac{df\left(t\right)}{dt}},$$

(10)

where $M\left(\alpha ,\gamma ,\delta \right)={\displaystyle \frac{\Gamma \left(\delta -\gamma +1\right)}{\gamma \Gamma \left(\delta -\alpha -\gamma +2\right)}}$ with $\delta >-1$.

Remark 5. 

According to [19], all obtained results via the MK derivative are in agreement with the FrFrD’s results of order α in the sense of C with a power law, as in Equation (1).

According to Definition 2, we obtain the following [19]:

Theorem 4. 

Suppose that $0<\alpha ,\gamma \le 1,$ $\delta >-1$, and assume that $f,h$ are MK $\alpha ,\gamma$-differentiable at a point $t>0$. Then, we obtain

(i)

${}^{MK}D^{\alpha ,\gamma }\left(af+bh\right)\left(t\right)=a {}^{MK}D^{\alpha ,\gamma }f\left(t\right)+b {}^{MK}D^{\alpha ,\gamma }h\left(t\right)$, $\forall a,b\in R$.

(ii)

${}^{MK}D^{\alpha ,\gamma }\left(\xi \right)=0,$ $\forall$constant functions such that $f\left(t\right)=\xi$.

(iii)

${}^{MK}D^{\alpha ,\gamma }\left(fh\right)\left(t\right)=f\left(t\right){}^{MK}D^{\alpha ,\gamma }h\left(t\right)+h\left(t\right){}^{MK}D^{\alpha ,\gamma }f\left(t\right)$.

(iv)

${}^{MK}D^{\alpha ,\gamma }\left({\displaystyle \frac{f}{h}}\right)\left(t\right)={\displaystyle \frac{h\left(t\right){}^{MK}D^{\alpha ,\gamma }f\left(t\right)-f\left(t\right){}^{MK}D^{\alpha ,\gamma }h\left(t\right)}{{\left[h\left(t\right)\right]}^2}}$.

Since the MK derivative is local in its nature, results from mathematical analysis and chain rule can be formulated via the help of MK calculus [19]. This property is presented in the following theorem.

Theorem 5 (Chain Rule). 

Assume that $0<\alpha ,\gamma \le 1,$ $\delta >-1$, $h$ is an MK $\alpha ,\gamma$-differentiable at $t>0$ and $f$ is differentiable at $h\left(t\right)$, then

$${}^{MK}D^{\alpha ,\gamma }\left(f\circ h\right)\left(t\right)=f^{\prime }\left(h\left(t\right)\right){}^{MK}D^{\alpha ,\gamma }h\left(t\right).$$

(11)

Remark 6. 

According to Theorem 5, the MK derivative of order $\alpha$ of some elementary functions can be expressed as

(i)

${}^{MK}D^{\alpha ,\gamma }\left({\displaystyle \frac{\gamma }{\left(\alpha +\gamma -1\right)\Gamma \left(\alpha \right)}}{t}^{\alpha +\gamma -1}\right)=1$.

(ii)

${}^{MK}D^{\alpha ,\gamma }\left(e^{{\displaystyle \frac{\gamma }{\left(\alpha +\gamma -1\right)\Gamma \left(\alpha \right)}}{t}^{\alpha +\gamma -1}}\right)=e^{{\displaystyle \frac{\gamma }{\left(\alpha +\gamma -1\right)\Gamma \left(\alpha \right)}}{t}^{\alpha +\gamma -1}}$.

(iii)

${}^{MK}D^{\alpha ,\gamma }\left(sin\left({\displaystyle \frac{\gamma }{\left(\alpha +\gamma -1\right)\Gamma \left(\alpha \right)}}{t}^{\alpha +\gamma -1}\right)\right)=cos\left({\displaystyle \frac{\gamma }{\left(\alpha +\gamma -1\right)\Gamma \left(\alpha \right)}}{t}^{\alpha +\gamma -1}\right).$

(iv)

${}^{MK}D^{\alpha ,\gamma }\left(cos\left({\displaystyle \frac{\gamma }{\left(\alpha +\gamma -1\right)\Gamma \left(\alpha \right)}}{t}^{\alpha +\gamma -1}\right)\right)=-sin\left({\displaystyle \frac{\gamma }{\left(\alpha +\gamma -1\right)\Gamma \left(\alpha \right)}}{t}^{\alpha +\gamma -1}\right).$

Remark 7. 

From the differentiability property of the MK derivative, which is $\alpha ,\gamma$-differentiability, and by assuming that $h\left(t\right)>0$, then Equation (11) can be represented as

$${}^{MK}D^{\alpha ,\gamma } \left(f\circ h\right)\left(t\right)={\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}h{\left(t\right)}^{\alpha +\gamma -2}{}^{MK}D^{\alpha ,\gamma } f\left(h\left(t\right)\right) {}^{ MK}D^{\alpha ,\gamma }h\left(t\right),$$

(12)

where $M\left(\alpha ,\gamma ,\delta \right)={\displaystyle \frac{\Gamma \left(\delta -\gamma +1\right)}{\gamma \Gamma \left(\delta -\alpha -\gamma +2\right)}}$, and $\delta >-1$.

The MK $\alpha ,\gamma$-integral of a function $f$ starting at $k\ge 0$, can be recalled as formulated in [22].

Definition 3. 

${}^{MK}I_{\alpha ,\gamma }^k\left(f\right)\left(t\right)={\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_k^t{\displaystyle \frac{f\left(x\right)}{x^{2-\alpha -\gamma }}}·dx$, such that this integral is the well-known Riemann improper integral, $M\left(\alpha ,\gamma ,\delta \right)={\displaystyle \frac{\Gamma \left(\delta -\gamma +1\right)}{\gamma \Gamma \left(\delta -\alpha -\gamma +2\right)}}, 0<\alpha ,\gamma \le 1$, and $\delta >-1$.

From Definition 3, we obtain the following:

Theorem 6. 

${}^{MK}D^{\alpha ,\gamma }{}^{MK}I_{\alpha ,\gamma }^k\left(f\right)\left(t\right)=f\left(t\right)$, for $t\ge m$, such that$f$ is any continuous function in the domain of ${}^{MK}I_{\alpha ,\gamma }^k$.

Theorem 7. 

Suppose that $k>0$, $0<\alpha ,\gamma \le 1$, $\delta >-1$, and $f$ is a continuous real-valued function (RVF) on $\left[k,q\right]$. Let $H$ be any RVF with the property: ${}^{MK}D^{\alpha ,\gamma }H\left(t\right)=f\left(t\right)$ for all $t\in \left[k,q\right]$. Then

$${}^{MK}I_{\alpha ,\gamma }^k\left(f\right)\left(q\right)=H\left(q\right)-H\left(k\right).$$

(13)

Finally, it is also necessary to mention the objectives of our research regarding the notion of MK Volterra InEq, proposed in [20]. Thus, it is generally given as follows:

$$h\left(t\right)w\left(t\right)=f\left(t\right)+{\displaystyle \frac{\beta }{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{t}V\left(t,\tau \right)w\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}},$$

(14)

where $V\left(t,\tau \right)$ is the InEq’s kernel, $w\left(t\right)$ is an unknown function, $f\left(t\right)$ is a perturbation known function, $\beta$ is non-zero real parameter, $M\left(\alpha ,\gamma ,\delta \right)={\displaystyle \frac{\Gamma \left(\delta -\gamma +1\right)}{\gamma \Gamma \left(\delta -\alpha -\gamma +2\right)}}, 0<\alpha ,\gamma \le 1$, and $\delta >-1$.

If $\alpha =\gamma =1$ is substituted in Equation (14), we obtain the usual known Volterra InEq [1,2,3]. However, if $h\left(t\right)=1$, then Equation (14) is expressed as

$$w\left(t\right)=f\left(t\right)+{\displaystyle \frac{\beta }{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{t}V\left(t,\tau \right)w\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}},$$

(15)

which is the Volterra InEq of the second kind¸ while if $h\left(t\right)=0$, then Equation (14) is written as

$$f\left(t\right)+{\displaystyle \frac{\beta }{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{t}V\left(t,\tau \right)w\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}=0,$$

(16)

which is the MK Volterra InEq of the first kind.

3. A New Fractal–Fractional Laplace Transformation

In this section, a new notion of fractal–fractional Laplace transformation is proposed; in this work, it is referred to as Martínez–Kaabar fractal–fractional Laplace transformation (for simplicity, we will use the term MK Laplace transformation in all upcoming results). All its main properties will be established.

Definition 4. 

Assume that $0<\alpha ,\gamma \le 1$, $\delta >-1$, and $f:[0,\infty )\to R$ is an RVF. Then, the MK Laplace transformation of order $\alpha ,\gamma$ is expressed as

$$F_{\alpha ,\gamma }\left(v\right)=L_{MK}\left[f\left(t\right)\right]\left(v\right)={\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{+\infty }{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}f\left(t\right){\displaystyle \frac{dt}{t^{2-\alpha -\gamma }}}.$$

(17)

Now, let us establish a sufficient condition that confirms the existence of the MK Laplace transformation. To prove this condition, the following definition is needed:

Definition 5 (Generalized Fractal–Fractional Exponential Order (GFrFrEO)). 

Assume that $0<\alpha ,\gamma \le 1$, $\delta >-1$, and $f:[0,\infty )\to R$ is an RVF. The function $f$ is considered to be of GFrFrEO if $\exists$ constants $K>0$, $\mu ϵ R$ and $T>0$, such that

$$\left|f\left(t\right)\right|\le K{e}^{{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}\mu {\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}} , \forall t\ge T.$$

(18)

Remark 8. 

Furthermore, to establish this existence result of the MK Laplace transformation, we assume the following condition for $\alpha$ and $\gamma$ (parameters): $0<\alpha ,\gamma \le 1$, with $\alpha +\gamma -1>0$.

Theorem 8. 

If $f$ is a piece-wise continuous function (PWCF) on $[0,\infty )$ and GFrFrEO, then $L_{MK}\left[f\left(t\right)\right]\left(v\right)$ exists for $v>\mu$.

Proof. 

Since $f$ is GFrFrEO, $\exists$ constants $K_1>0$, $\mu ϵ R$ and $T>0$, such that

$$\left|f\left(t\right)\right|\le K_1{e}^{{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}\mu {\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}} , for t\ge T.$$

Furthermore, $f$ is a PWCF on $\left[0,T\right]$ and hence bounded there as

$$\left|f\left(t\right)\right|\le K_2 , for t\in \left[0,T\right].$$

This implies that a constant $K$ can be selected to be sufficiently large, such that

$$\left|f\left(t\right)\right|\le k{e}^{{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}\mu {\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}} , for t\ge 0.$$

Therefore,

$$\left|{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\lambda \right)}}{{\displaystyle \int }}_0^T{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}f\left(t\right){\displaystyle \frac{dt}{t^{2-\alpha -\gamma }}}\right|\le {\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^T\left|{\displaystyle \frac{e^{-\frac{1}{M\left(\alpha ,\gamma ,\delta \right)}v\frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}f\left(t\right)}{t^{2-\alpha -\gamma }}}\right|dt\le {\displaystyle \frac{K}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^T{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}\left(v-\mu \right){\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}{\displaystyle \frac{dt}{t^{2-\alpha -\gamma }}}=-{\displaystyle \frac{K}{v-\mu }}\left(e^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}\left(v-\mu \right){\displaystyle \frac{T^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}-1\right).$$

Letting $T\to +\infty$, we see that

$${\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^T{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}f\left(t\right){\displaystyle \frac{dt}{t^{2-\alpha -\gamma }}}\le {\displaystyle \frac{K}{v-\mu }}, v>\mu .$$

□

Then, the linearity of the integral operator $L_{MK}$ will be studied, which can obviously be seen from the $L_{MK}$ definition and the MK integral operator’s properties.

Theorem 9. 

Assume that $0<\alpha ,\gamma \le 1$, with $\alpha +\gamma -1>0$, $\delta >-1$, $f,h:[0,\infty )\to R$ are RVFs, and ${\beta }_1,\beta \in R$. If $L_{GFrFr}\left[f\left(t\right)\right]\left(v\right)$ and $L_{GFrFr}\left[h\left(t\right)\right]\left(v\right)$ exist, then,

$$L_{GFrFr}\left[{\beta }_{1}f\left(t\right)+{\beta }_{2}h\left(t\right)\right]\left(v\right)={\beta }_1{L}_{GFrFr}\left[f\left(t\right)\right]\left(v\right)+{\beta }_2{L}_{GFrFr}\left[h\left(t\right)\right]\left(v\right).$$

(19)

Now, we show the main properties of the $L_{MK}$ integral transformation through the following theorem:

Theorem 10. 

Let $0<\alpha ,\gamma \le 1$, with $\alpha +\gamma -1>0$, $\delta >-1$. So, we obtain

(i)

$L_{MK}\left[\psi \right]\left(v\right)={\displaystyle \frac{\psi }{v}}$, where $\psi$ is any real number and $v>0$.

(ii)

$L_{MK}\left[t^d\right]\left(v\right)={\displaystyle \frac{{\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}^{\frac{d}{\alpha +\gamma -1}}}{v^{1+\frac{d}{\alpha +\gamma -1}}}}\Gamma \left(1+{\displaystyle \frac{d}{\alpha +\gamma -1}}\right)$, where the gamma function is represented by $\Gamma \left(c,x\right)={{\displaystyle \int }}_x^{+\infty }{t}^{c-1}{e}^{-t}dt$, $\Gamma \left(c,0\right)=\Gamma \left(c\right)$, and $d>-1$.

(iii)

$L_{MK}\left[e^{{\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}}\right]\left(v\right)={\displaystyle \frac{1}{v-\psi }}$, where$\psi$ is any real number and $v>\psi$.

(iv)

$L_{MK}\left[e^{{\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}}f\left(t\right)\right]\left(v\right)=F_{\alpha ,\gamma }\left(v-\psi \right)$, with $L_{MK}\left[f\left(t\right)\right]\left(v\right)=F_{\alpha ,\gamma }\left(v\right)$ for $v>\mu$, where $\psi$ is a real number and $v>\mu +\psi$.

(v)

If $L_{MK}\left[f\left(t\right)\right]\left(v\right)=F_{\alpha ,\gamma }\left(v\right)$ for $v>\mu$, then $L_{MK}\left[f\left(\psi t\right)\right]\left(v\right)={\displaystyle \frac{1}{{\psi }^{\alpha +\gamma -1}}}{F}_{\alpha ,\gamma }\left({\displaystyle \frac{v}{{\psi }^{\alpha +\gamma -1}}}\right)$, for $\psi >0$ and ${\displaystyle \frac{v}{{\psi }^{\alpha +\gamma -1}}}>\mu$.

(vi)

$L_{MK}\left[sin\left({\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)\right]\left(v\right)={\displaystyle \frac{\psi }{v^2+{\psi }^2}}$, for $\psi >0$ and $v>0$.

(vii)

$L_{MK}\left[cos\left({\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)\right]\left(v\right)={\displaystyle \frac{v}{v^2+{\psi }^2}}$, for $\psi >0$ and $v>0$.

(viii)

$L_{MK}\left[sinh\left({\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)\right]\left(v\right)={\displaystyle \frac{\psi }{v^2-{\psi }^2}}$, for any $\psi$ real number and $v>\left|\psi \right|.$

(ix)

$L_{MK}\left[cosh\left({\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)\right]\left(v\right)={\displaystyle \frac{v}{v^2-{\psi }^2}}$, for any $\psi$ real number and $v>\left|\psi \right|.$

Proof. 

(i)

Directly using the definition.

(ii)

Via the change of variable, the following is obtained:

$${\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{+\infty }{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}{t}^d{\displaystyle \frac{dt}{t^{2-\alpha -\gamma }}}={\displaystyle \frac{{\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}^{\frac{d}{\alpha +\gamma -1}}}{v^{1+\frac{d}{\alpha +\gamma -1}}}}\Gamma \left(1+{\displaystyle \frac{d}{\alpha +\gamma -1}}\right).$$

(iii)

Consider $f\left(t\right)=e^{{\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}}$, with $\psi \in R$. Then,

$${\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{+\infty }{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}{e}^{{\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}}{\displaystyle \frac{dt}{t^{2-\alpha -\gamma }}}={\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{+\infty }{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}\left(v-\psi \right){\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}{\displaystyle \frac{dt}{t^{2-\alpha -\gamma }}}={\displaystyle \frac{1}{v-\psi }}.$$

(iv)

Suppose $L_{MK}\left[f\left(t\right)\right]\left(v\right)=F_{\alpha ,\gamma }\left(v\right)$ for $v>\mu$. So, we have

$${\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{+\infty }{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}{e}^{{\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}}f\left(t\right){\displaystyle \frac{dt}{t^{2-\alpha -\gamma }}}={\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{+\infty }{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,V\right)}}\left(v-\psi \right){\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}f\left(t\right){\displaystyle \frac{dt}{t^{2-\alpha -\gamma }}}=F_{\alpha ,\gamma }\left(v-\psi \right),v-\psi >\mu .$$

(v)

Suppose $L_{MK}\left[f\left(t\right)\right]\left(v\right)=F_{\alpha ,\gamma }\left(v\right)$ for $v>\mu$. So, through the change of variable $\psi t=u$, we obtain

$${\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{+\infty }{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}f\left(\psi t\right){\displaystyle \frac{dt}{t^{2-\alpha -\gamma }}}={\displaystyle \frac{1}{{\psi }^{\alpha +\gamma -1}}}{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{+\infty }{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{\displaystyle \frac{v}{{\psi }^{\alpha +\gamma -1}}}{\displaystyle \frac{u^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}f\left(u\right){\displaystyle \frac{du}{u^{2-\alpha -\gamma }}}={\displaystyle \frac{1}{{\psi }^{\alpha +\gamma -1}}}{F}_{\alpha ,\gamma }\left({\displaystyle \frac{1}{{\psi }^{\alpha +\gamma -1}}}\right),{\displaystyle \frac{v}{{\psi }^{\alpha +\gamma -1}}}>\mu .$$

(vi)

Using the fact that

$${\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{\displaystyle \int }{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,V\right)}}v{\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}sin\left({\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right){\displaystyle \frac{dt}{t^{2-\alpha -\gamma }}}=-{\displaystyle \frac{\psi }{v^2+{\psi }^2}}{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}\left(cos\left({\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)+{\displaystyle \frac{v}{\psi }}sin\left({\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)\right),$$

we can obtain the required result.

(vii)

Similar to the previous one, using

$${\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{\displaystyle \int }{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}cos\left({\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\lambda \right)\left(\alpha +\gamma -1\right)}}\right){\displaystyle \frac{dt}{t^{2-\alpha -\gamma }}}={\displaystyle \frac{\psi }{v^2+{\psi }^2}}{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}\left(sin\left({\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)-{\displaystyle \frac{v}{\psi }}cos\left({\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)\right).$$

(viii)

As

$$L_{MK}\left[sinh\left({\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)\right]\left(v\right)={\displaystyle \frac{1}{2}}\left(L_{MK}\left[e^{{\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}}\right]\left(v\right)-L_{MK}\left[e^{-{\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}}\right]\left(v\right)\right),$$

it is easy to obtain the required result.

(ix)

Similarly, as

$$L_{MK}\left[cosh\left({\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)\right]\left(v\right)={\displaystyle \frac{1}{2}}\left(L_{MK}\left[e^{{\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}}\right]\left(v\right)+L_{MK}\left[e^{-{\displaystyle \frac{\psi t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}}\right]\left(v\right)\right).$$

□

Then, the connection between the MK Laplace transformation of a function and the MK $\alpha ,\gamma$-derivative is established as follows:

Theorem 11. 

Suppose that $f\left(t\right)$ is continuous and ${}^{MK}D^{\alpha ,\gamma }f\left(t\right)$ is piece-wise continuous for all $t\ge 0$. Suppose further that $f\left(t\right)$ is the GFrFrEO. Then, $L_{GFrFr}\left[{}^{MK}D^{\alpha ,\gamma }f\left(t\right)\right]\left(v\right)$ exists for $v>\mu$, and moreover,

$$L_{MK}\left[{}^{MK}D^{\alpha ,\gamma }f\left(t\right)\right]\left(v\right)=v{L}_{MK}\left[f\left(t\right)\right]\left(v\right)-f\left(0\right).$$

(20)

Proof. 

By using Equation (17), we obtain

$$L_{MK}\left[{}^{MK}D^{\alpha ,\gamma }f\left(t\right)\right]\left(v\right)={\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{+\infty }{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}{}^{MK}D^{\alpha ,\gamma }f\left(t\right){\displaystyle \frac{dt}{t^{2-\alpha -\gamma }}}.$$

Now, by using classical integration by parts, we obtain

$$L_{MK}\left[{}^{MK}D^{\alpha ,\gamma }f\left(t\right)\right]\left(v\right)=\underset{\tau \to +\infty }{\lim }{\left[e^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}f\left(t\right)\right]}_0^{\tau }+{\displaystyle \frac{v}{M\left(\alpha ,\gamma ,\lambda \right)}}{{\displaystyle \int }}_0^{+\infty }{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}f\left(t\right){\displaystyle \frac{dt}{t^{2-\alpha -\gamma }}}=\underset{\tau \to +\infty }{\lim }\left(e^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{{\tau }^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}f\left(\tau \right)-f\left(0\right)\right)+v{L}_{MK}\left[f\left(t\right)\right]\left(v\right).$$

If $0<\alpha ,\gamma \le 1$, with $\alpha +\gamma -1>0$, $\delta >-1$, and given that $f\left(t\right)$ is GFrFrEO, $\underset{\tau \to +\infty }{\lim }{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{{\tau }^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}f\left(\tau \right)=0$ whenever $v>\mu$.

Hence, for $v>\mu$,

$$L_{MK}\left[{}^{MK}D^{\alpha ,\gamma }f\left(t\right)\right]\left(v\right)=v{L}_{MK}\left[f\left(t\right)\right]\left(v\right)-f\left(0\right).$$

□

By employing the relevant condition again to the function $f$ and its MK $\alpha ,\gamma$-derivative, the MK Laplace transformation of the $n$ consecutive MK $\alpha ,\gamma$-derivatives can be written as ${}^{MK}D^{\alpha ,\gamma }\left({}^{MK}D^{\alpha ,\gamma }\left(\cdots \left({}^{M\mathrm{K}}D^{\alpha ,\gamma }f\left(t\right)\right)\right)\right)$, which can be formulated via the help of the previously mentioned theorem. We conclude with the following corollary:

Corollary 1. 

Assume that $f,{}^{MK}D^{\alpha ,\gamma }f, \dots ,{}^{MK}D_{\left(n-1\right)}^{\alpha ,\gamma }f$ are continuous, and ${}^{MK}D_{\left(n\right)}^{\alpha ,\gamma }f$ is PWCF for all $t\ge 0$. Suppose further that $f,{}^{MK}D^{\alpha ,\gamma }f, \dots ,{}^{MK}D_{\left(n-1\right)}^{\alpha ,\gamma }f\left(t\right)$ are GFrFrEO. Then, $L_{GFrFr}\left[{}^{MK}D_{\left(n\right)}^{\alpha ,\gamma }f\left(t\right)\right]\left(v\right)$ exists for $v>\mu$ and is given by

$$L_{MK}\left[{}^{MK}D_{\left(n\right)}^{\alpha ,\gamma }f\left(t\right)\right]\left(v\right)=v^n{L}_{MK}\left[f\left(t\right)\right]\left(v\right)-{\displaystyle \sum }_{k=0}^{n-1}{v}^{n-k-1}{}^{MK}D_{\left(k\right)}^{\alpha ,\gamma }f\left(0\right).$$

(21)

Hence, ${}^{MK}D_{\left(k\right)}^{\alpha ,\gamma }f\left(t\right)$ means the application of the MK $\alpha ,\gamma$-derivative k times.

It is known that the convolution product of two $L$-transformable functions is considered an essential result of classical Laplace transformation. Therefore, we will construct that result for MK Laplace transformation as follows:

Theorem 12. 

Let $0<\alpha ,\gamma \le 1$, with $\alpha +\gamma -1>0$, $\delta >-1$, and $f,h:[0,\infty )\to R$ be RVFs. If $L_{MK}\left[f\left(t^{\alpha +\gamma -1}\right)\right]\left(v\right)$ and $L_{MK}\left[h\left(t\right)\right]\left(v\right)$ exist, then we obtain

$$L_{MK}\left[f\ast h\right]\left(v\right)=L_{MK}\left[f\left(t^{\alpha +\gamma -1}\right)\right]\left(v\right)·L_{MK}\left[h\left(t\right)\right]\left(v\right),$$

(22)

where

$$\left(f\ast h\right)\left(t\right)={\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{t}f\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)h\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}.$$

(23)

Proof. 

By applying MK Laplace transformation to Equation (23), we have

$$\begin{array}{c}{L}_{MK}\left[f\ast h\right]\left(v\right)\hfill \\ ={\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{+\infty }{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}\left({\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{t}f\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)h\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}\right){\displaystyle \frac{dt}{t^{2-\alpha -\gamma }}}\hfill \\ ={\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{+\infty }{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{{\tau }^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}h\left(\tau \right)({\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_{\tau }^{+\infty }{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)}{\alpha +\gamma -1}}}f(t^{\alpha +\gamma -1}\hfill \\ -{\tau }^{\alpha +\gamma -1}){\displaystyle \frac{dt}{t^{2-\alpha -\gamma }}}){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}.\hfill \end{array}$$

Now, consider the change of variables, $t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}={\epsilon }^{\alpha +\gamma -1}$. So,

$$\begin{array}{c}{L}_{MK}\left[f\ast h\right]\left(v\right)\hfill \\ ={\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{+\infty }{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{{\tau }^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}h\left(\tau \right)\left({\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{+\infty }{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{{\epsilon }^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}f\left({\epsilon }^{\alpha +\gamma -1}\right){\displaystyle \frac{dϵ}{{\epsilon }^{2-\alpha -\gamma }}}\right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}},\hfill \end{array}$$

which can be written as

$$L_{MK}\left[f\ast h\right]\left(v\right)=L_{MK}\left[f\left(t^{\alpha +\gamma -1}\right)\right]\left(v\right)·L_{MK}\left[h\left(t\right)\right]\left(v\right).$$

□

The MK Laplace transformation of the MK $\alpha ,\gamma$-integral is in connection with the MK Laplace transformation of its integrand, as follows:

Theorem 13. 

Assume that $h$ is a GFrFrEO and continuous for $t\ge 0$. Then, we obtain

$$L_{MK}\left[{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{t}h\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}\right]\left(v\right)={\displaystyle \frac{1}{v}}{L}_{MK}\left[h\left(t\right)\right]\left(v\right).$$

(24)

Proof. 

If we take $f\left(t\right)=1$ and $L_{MK}\left[f\left(t\right)\right]\left(v\right)={\displaystyle \frac{1}{v}}$ from Theorem 12, our result follows easily.

□

We now present some qualitative properties of the MK Laplace transformation of PWCFs on $[0,\infty )$ and GFrFrEO.

Theorem 14. (Behavior of the MK Laplace Transformation at Infinity). 

Assume that $f$ is PWCF on $[0,\infty )$ and GFrFrEO. Then, we obtain

$$\underset{v\to +\infty }{\lim }{L}_{MK}\left[f\left(t\right)\right]\left(v\right)=0.$$

(25)

Proof. 

Since $f$ is GFrFrEO, $\exists t_0,K_1, \mu$, such that $\left|f\left(t\right)\right|\le K_1{e}^{{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}\mu {\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}$ for $t\ge t_0$. Also, $f$ is a PWCF on $\left[0,t_0\right]$, and hence $f$ is bounded, so $\exists$ $K_1$ such that $\left|f\left(t\right)\right|\le K_2$ for $t\ge t_0$. Choosing $K=max\left\{K_1, K_2\right\}$ we have $\left|f\left(t\right)\right|\le K{e}^{{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}\mu {\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}$ for $t\ge 0$. Now, we obtain

$$\begin{array}{cc}\left|L_{MK}\left[f\left(t\right)\right]\left(v\right)\right|\hfill & =\left|{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{+\infty }{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}f\left(t\right){\displaystyle \frac{dt}{t^{2-\alpha -\gamma }}}\right|\hfill \\ & =\underset{T\to +\infty }{\lim }\left|{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^T{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}v{\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}f\left(t\right){\displaystyle \frac{dt}{t^{2-\alpha -\gamma }}}\right|\hfill \\ & \le {\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}\underset{T\to +\infty }{\lim }{{\displaystyle \int }}_0^T\left|{\displaystyle \frac{e^{-\frac{1}{M\left(\alpha ,\gamma ,\delta \right)}v\frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}f\left(t\right)}{t^{2-\alpha -\gamma }}}\right|dt\hfill \\ & \le {\displaystyle \frac{k}{M\left(\alpha ,\gamma ,\delta \right)}}\underset{T\to +\infty }{\lim }{{\displaystyle \int }}_0^T{e}^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}\left(v-\mu \right){\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}{\displaystyle \frac{dt}{t^{2-\alpha -\gamma }}}\hfill \\ & ={\displaystyle \frac{K}{\mu -v}}\underset{T\to +\infty }{\lim }\left(e^{-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}\left(v-\mu \right){\displaystyle \frac{t^{\alpha +\gamma -1}}{\alpha +\gamma -1}}}-1\right)={\displaystyle \frac{K}{v-\mu }}.\hfill \end{array}$$

From the previous inequality, it follows easily into Equation (25).

□

Theorem 15 (Initial Value Theorem). 

Suppose that $f\left(t\right)$ is MK $\alpha ,\gamma$-differentiable and ${}^{MK}D^{\alpha ,\gamma }f\left(t\right)$ is PWCF for all $t\ge 0$. Suppose further that $f\left(t\right),{}^{MK}D^{\alpha ,\gamma }f\left(t\right)$ are GFrFrEO. Then, we obtain

$$\underset{v\to +\infty }{\lim }v{L}_{MK}\left[f\left(t\right)\right]\left(v\right)=f\left(0\right).$$

(26)

Proof. 

From Theorem 11, we write

$$L_{MK}\left[{}^{MK}D^{\alpha ,\gamma }f\left(t\right)\right]\left(v\right)=v{L}_{MK}\left[f\left(t\right)\right]\left(v\right)-f\left(0\right).$$

However, by applying the previous theorem to ${}^{MK}D^{\alpha ,\gamma }f\left(t\right)$, we obtain

$$\underset{v\to +\infty }{\lim }{L}_{MK}\left[{}^{MK}D^{\alpha ,\gamma }f\left(t\right)\right]\left(v\right)=0,$$

and therefore

$$\underset{v\to +\infty }{\lim }{L}_{MK}\left[{}^{MK}D^{\alpha ,\gamma }f\left(t\right)\right]\left(v\right)=\underset{v\to +\infty }{\lim }\left(v{L}_{MK}\left[f\left(t\right)\right]\left(v\right)-f\left(0\right)\right)=0.$$

This concludes our proof. □

The following result relates the MK Laplace transformation to the usual Laplace transformation.

Theorem 16. 

Let $0<\alpha ,\gamma \le 1$, with $\alpha +\gamma -1>0$, $\delta >-1$, and $f:[0,\infty )\to R$ is a function such that $L_{MK}\left[f\left(t\right)\right]\left(v\right)=F_{\alpha ,\gamma }\left(v\right)$. Then, we obtain

$$F_{\alpha ,\gamma }\left(v\right)=L\left[f\left({\displaystyle \frac{t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)\right]\left(v\right),$$

(27)

where $L\left[f\left(u\right)\right]\left(v\right)={{\displaystyle \int }}_0^{+\infty }{e}^{-vu}f\left(u\right)du$.

Proof. 

The proof follows easily by setting $u={\displaystyle \frac{t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\lambda \right)\left(\alpha +\gamma -1\right)}}$.

□

Now, let us define the inverse MK Laplace transformation:

Definition 6. 

For a PWCF on $[0,\infty )$ and generalized exponential order $f\left(t\right)$ whose MK Laplace transformation is $F_{\alpha ,\gamma }\left(v\right)$, we name $f\left(t\right)$ the inverse MK Laplace transformation of $F_{\alpha ,\gamma }\left(v\right)$ and express it as $f\left(t\right)=L_{MK}^{-1} \left[F_{\alpha ,\gamma }\left(v\right)\right]\left(t\right)$. Symbolically, we obtain

$$f\left(t\right)=L_{MK}^{-1} \left[F_{\alpha ,V}\left(v\right)\right]\left(t\right)\iff F_{\alpha ,V}\left(v\right)=L_{MK}\left[f\left(t\right)\right]\left(v\right).$$

(28)

The inverse GFrFr Laplace transformation satisfies the property of linearity as follows:

Theorem 17. 

Given two generalized Laplace transformations, $F_{\alpha ,\gamma }\left(v\right)$ and $H_{\alpha ,\gamma }\left(v\right)$, we obtain

$$L_{MK}^{-1} \left[{\beta }_1{F}_{\alpha ,\gamma }\left(v\right)+{\beta }_2{H}_{\alpha ,\gamma }\left(v\right)\right]\left(t\right)={\beta }_1{L}_{MK}^{-1} \left[F_{\alpha ,\gamma }\left(v\right)\right]\left(t\right)+{\beta }_2{L}_{MK}^{-1} \left[H_{\alpha ,\gamma }\left(v\right)\right]\left(t\right).$$

Proof. 

Suppose that $L_{MK}\left[f\left(t\right)\right]\left(v\right)=F_{\alpha ,\gamma }\left(v\right)$, and $L_{MK}\left[h\left(t\right)\right]\left(v\right)=H_{\alpha ,\gamma }\left(v\right)$. Since

$$L_{MK}\left[{\beta }_{1}f\left(t\right)+{\beta }_{2}h\left(t\right)\right]\left(v\right)={\beta }_1{L}_{MK}\left[f\left(t\right)\right]\left(v\right)+{\beta }_2{L}_{MK}\left[h\left(t\right)\right]\left(v\right)={\beta }_1{F}_{\alpha ,\gamma }\left(v\right)+{\beta }_2{H}_{\alpha ,\gamma }\left(v\right),$$

we have

$$L_{MK}^{-1} \left[{\beta }_1{F}_{\alpha ,\gamma }\left(v\right)+{\beta }_2{H}_{\alpha ,\gamma }\left(v\right)\right]\left(t\right)={\beta }_1{L}_{MK}^{-1} \left[F_{\alpha ,\gamma }\left(v\right)\right]\left(t\right)+{\beta }_2{L}_{MK}^{-1} \left[H_{\alpha ,\gamma }\left(v\right)\right]\left(t\right).$$

□

4. The Technique of MK Laplace Transformation for the Solution of Fractal–Fractional Integral Equations

The technique of Laplace transformation is a powerful classical technique to solve initial value problems and InEqs [1,2]. In this section, our main objective is to introduce the method of Laplace transformation in the context of the new concept of Laplace transformation, proposed in Definition 4. As in the classical case, we show its applicability and effectiveness using two important classes of MK integral equations: the Volterra InEq and the AInEq.

4.1. The MK Volterra InEq of Second Kind

Consider the following MK Volterra InEq of the second kind:

$$w\left(t\right)=f\left(t\right)+{\displaystyle \frac{\beta }{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{t}V\left(t,\tau \right)w\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}},$$

(29)

where $V\left(t,\tau \right)$ is the integral equation’s kernel, $w\left(t\right)$ is an unknown function, $f\left(t\right)$ is a perturbation known function, $\beta$ is a non-zero real parameter, and $M\left(\alpha ,\gamma ,\delta \right)={\displaystyle \frac{\Gamma \left(\delta -\gamma +1\right)}{\gamma \Gamma \left(\delta -\alpha -\gamma +2\right)}}, 0<\alpha ,\gamma \le 1$, with $\mathsf{\alpha }+\mathsf{\gamma }-1>0$, and $\delta >-1$. However, if the kernel, $V\left(t,\tau \right)$, depends on the difference $t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}$, then it is named the $\alpha ,\gamma$-difference kernel, and the integral ${\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{t}V\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)w\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}$ represents convolution. Thus, we obtain

$$w\left(t\right)=f\left(t\right)+{\displaystyle \frac{\beta }{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{t}V\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)y\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}=f\left(t\right)+\beta \left(V\ast w\right)\left(t\right).$$

(30)

First, by taking the MK Laplace transformation on both sides of Equation (30), we obtain

$$L_{MK}\left[w\left(t\right)\right]\left(v\right)=L_{MK}\left[f\left(t\right)\right]\left(v\right)+\beta L_{MK}\left[\left(V\ast w\right)\left(t\right)\right]\left(v\right).$$

Now, by applying Theorem 12, we obtain

$$L_{MK}\left[w\left(t\right)\right]\left(v\right)=L_{MK}\left[f\left(t\right)\right]\left(v\right)+\beta L_{MK}\left[V\left(t^{\alpha +\gamma -1}\right)\right]\left(v\right)·L_{MK}\left[w\left(t\right)\right]\left(v\right).$$

By rearranging all of terms appropriately, the above equation becomes

$$L_{MK}\left[w\left(t\right)\right]\left(v\right)={\displaystyle \frac{L_{MK}\left[f\left(t\right)\right]\left(v\right)}{1-\beta L_{MK}\left[V\left(t^{\alpha +\gamma -1}\right)\right]\left(v\right)}}, \beta L_{MK}\left[V\left(t^{\alpha +\gamma -1}\right)\right]\left(v\right)\ne 1.$$

Finally, the solution $\mathrm{w}\left(\mathrm{t}\right)$ results from applying the inverse MK Laplace transformation of both sides as

$$w\left(t\right)=L_{MK}^{-1} \left[{\displaystyle \frac{L_{MK}\left[f\left(t\right)\right]\left(v\right)}{1-\beta L_{MK}\left[V\left(t^{\alpha +\gamma -1}\right)\right]\left(v\right)}}\right]\left(t\right).$$

(31)

Next, let us apply this introduced technique to several interesting illustrative examples, as follows:

Example 1. 

Consider the following MKFF Volterra InEq of the second kind with $f\left(t\right)={\displaystyle \frac{t^{3\left(\alpha +\gamma -1\right)}}{\Gamma \left(3\left(\alpha +\gamma \right)-2\right)}}, \beta =-1, V\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)=t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}$,

$$w\left(t\right)={\displaystyle \frac{t^{3\left(\alpha +\gamma -1\right)}}{\Gamma \left(3\left(\alpha +\gamma \right)-2\right)}}-{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^t\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)y\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}.$$

(32)

By employing the MK Laplace transformation for both sides to Equation (32), we obtain

$$L_{MK}\left[w\left(t\right)\right]\left(v\right)={\displaystyle \frac{\left(3!\right){\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}^3}{\Gamma \left(3\left(\alpha +\gamma \right)-2\right)}}·{\displaystyle \frac{1}{v^4}}-M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)·{\displaystyle \frac{1}{v^2}}·L_{MK}\left[w\left(t\right)\right]\left(v\right).$$

By rearranging all of terms appropriately, the above equation becomes

$$L_{MK}\left[w\left(t\right)\right]\left(v\right)={\displaystyle \frac{\left(3!\right){\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}^3}{\Gamma \left(3\left(\alpha +\gamma \right)-2\right)}}·{\displaystyle \frac{1}{v^2\left(v^2+M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}}={\displaystyle \frac{\left(3!\right){\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}^2}{\Gamma \left(3\left(\alpha +\gamma \right)-2\right)}}\left({\displaystyle \frac{1}{v^2}}-{\displaystyle \frac{1}{\left(v^2+M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}} \right).$$

Now, by applying inverse MK Laplace transformation of both sides, we obtain

$$w\left(t\right)={\displaystyle \frac{\left(3!\right){\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}^2}{\Gamma \left(3\left(\alpha +\gamma \right)-2\right)}}{t}^{\alpha +\gamma -1}-{\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}^{{\displaystyle \frac{3}{2}}}\sin \left(\sqrt{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}{t}^{\alpha +\gamma -1}\right).$$

Remark 9. 

If $\alpha =\gamma =1$, then Equation (32) is the following classical Volterra InEq:

$$w\left(t\right)={\displaystyle \frac{t^3}{3!}}-{{\displaystyle \int }}_0^t\left(t-\tau \right)w\left(\tau \right)d\tau ,$$

which represents the exact solution given by $w\left(t\right)=t-\sin \left(t\right)$.

Example 2. 

Consider the following MKFF Volterra InEq of the second kind with $f\left(t\right)=sin\left({\displaystyle \frac{t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)+cos\left({\displaystyle \frac{t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right), \beta =2,V\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)=sin\left({\displaystyle \frac{t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)$,

$$w\left(t\right)=sin\left({\displaystyle \frac{t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)+cos\left({\displaystyle \frac{t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)+{\displaystyle \frac{2}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{t}sin\left({\displaystyle \frac{t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)w\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}.$$

(33)

Taking MK Laplace transformation for both sides to Equation (33), we obtain

$$L_{MK}\left[w\left(t\right)\right]\left(v\right)={\displaystyle \frac{1}{v^2+1}}+{\displaystyle \frac{v}{v^2+1}}+{\displaystyle \frac{2}{v^2+1}}{L}_{MK}\left[w\left(t\right)\right]\left(v\right).$$

By rearranging all of terms appropriately, the above equation becomes

$$L_{MK}\left[w\left(t\right)\right]\left(v\right)={\displaystyle \frac{1}{v-1}}.$$

Now, by employing inverse MK Laplace transformation on both sides, we obtain

$$w\left(t\right)=e^{{\displaystyle \frac{t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}}.$$

Remark 10. 

If   $\alpha =\gamma =1$, then the Equation (33) becomes the following classical Volterra InEq:

$$w\left(t\right)=\sin \left(t\right)+\cos \left(t\right)+2{{\displaystyle \int }}_0^{t}sin\left(t-\tau \right)w\left(\tau \right)d\tau .$$

4.2. The MK Volterra InEq of First Kind

Consider the following MK Volterra InEq of the first kind:

$$f\left(t\right)={\displaystyle \frac{\beta }{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{t}V\left(t,\tau \right)w\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}},$$

(34)

where $V\left(t,\tau \right)$ is the InEq’s kernel, $w\left(t\right)$ is an unknown function, $f\left(t\right)$ is a perturbation known function), $\beta$ is a non-zero real parameter, and $M\left(\alpha ,\gamma ,\delta \right)={\displaystyle \frac{\Gamma \left(\delta -\gamma +1\right)}{\gamma \Gamma \left(\delta -\alpha -\gamma +2\right)}}, 0<\alpha ,\gamma \le 1$, with $\mathsf{\alpha }+\mathsf{\gamma }-1>0$, and $\delta >-1$. However, if the kernel, $V\left(t,\tau \right)$, depends on the difference $t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}$, then it is named the $\alpha ,\gamma$-difference kernel, and the integral ${\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{t}V\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)w\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}$ represents convolution. Thus, we obtain

$$f\left(t\right)={\displaystyle \frac{\beta }{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^{t}V\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)w\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}=f\left(t\right)+\left(V\ast w\right)\left(t\right).$$

(35)

First, by employing the MK Laplace transformation on both sides of Equation (35), we obtain

$$L_{MK}\left[f\left(t\right)\right]\left(v\right)=\beta L_{MK}\left[\left(V\ast w\right)\left(t\right)\right]\left(v\right).$$

Now, by applying Theorem 12, we obtain

$$L_{MK}\left[f\left(t\right)\right]\left(v\right)=\beta L_{MK}\left[V\left(t^{\alpha +\gamma -1}\right)\right]\left(v\right)L_{MK}\left[w\left(t\right)\right]\left(v\right).$$

By rearranging all of terms appropriately, the above equation becomes

$$L_{MK}\left[w\left(t\right)\right]\left(v\right)={\displaystyle \frac{L_{MK}\left[f\left(t\right)\right]\left(v\right)}{ \beta L_{MK}\left[V\left(t^{\alpha +\gamma -1}\right)\right]\left(v\right)}}, L_{MK}\left[V\left(t^{\alpha +\gamma -1}\right)\right]\left(v\right)\ne 0.$$

Finally, the solution: $w\left(t\right)$ is results from by applying the inverse MK Laplace transformation of both sides as

$$w\left(t\right)=L_{MK}^{-1} \left[{\displaystyle \frac{L_{MK}\left[f\left(t\right)\right]\left(v\right)}{\beta L_{MK}\left[V\left(t^{\alpha +\gamma -1}\right)\right]\left(v\right)}}\right]\left(t\right).$$

(36)

Next, we apply this introduced technique to several interesting illustrative examples.

Example 3. 

Consider the following MK Volterra InEq of the first kind with $f\left(t\right)=e^{{\displaystyle \frac{t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}}-sin\left({\displaystyle \frac{t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)-cos\left({\displaystyle \frac{t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right), \beta =2,V\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)=e^{{\displaystyle \frac{\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}}$,

$$e^{{\displaystyle \frac{t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}}-sin\left({\displaystyle \frac{t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)-cos\left({\displaystyle \frac{t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)={\displaystyle \frac{2}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^t{e}^{{\displaystyle \frac{\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}}w\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}.$$

(37)

By applying the MK Laplace transformation for both sides to Equation (37), we obtain

$${\displaystyle \frac{1}{v-1}}-{\displaystyle \frac{1}{v^2+1}}-{\displaystyle \frac{v}{v^2+1}}={\displaystyle \frac{2}{v-1}}{L}_{MK}\left[w\left(t\right)\right]\left(v\right),$$

or equivalently,

$${\displaystyle \frac{2}{\left(v-1\right)\left(v^2+1\right)}}={\displaystyle \frac{2}{v-1}}{L}_{MK}\left[w\left(t\right)\right]\left(v\right).$$

By rearranging all of terms appropriately, the above equation becomes

$$L_{MK}\left[w\left(t\right)\right]\left(v\right)={\displaystyle \frac{1}{v^2+1}}.$$

Now, by employing the inverse MK Laplace transformation of both sides, we obtain

$$w\left(t\right)=sin\left({\displaystyle \frac{t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right).$$

Remark 11. 

If  $\alpha =\gamma =1$, then Equation (37) becomes the following classical Volterra InEq:

$$e^t-\sin \left(t\right)-\cos \left(t\right)=2{{\displaystyle \int }}_0^t{e}^{\left(t-\tau \right)}w\left(\tau \right)d\tau ,$$

which represents the exact solution given by $y\left(t\right)=\sin \left(t\right)$.

Example 4. 

Consider the following MK Volterra InEq of the second kind with $f\left(t\right)=-2+{\displaystyle \frac{t^{2\left(\alpha +\gamma -1\right)}}{2{\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}^2}}+{\displaystyle \frac{t^{3\left(\alpha +\gamma -1\right)}}{3{\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}^3}}+sinh\left({\displaystyle \frac{t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)+2cosh\left({\displaystyle \frac{t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right), \beta =1,V\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)={\displaystyle \frac{t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}+2$

$$-2+{\displaystyle \frac{t^{2\left(\alpha +\gamma -1\right)}}{2{\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}^2}}+{\displaystyle \frac{t^{3\left(\alpha +\gamma -1\right)}}{3{\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}^3}}+sinh\left({\displaystyle \frac{t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)+2cosh\left({\displaystyle \frac{t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right)={\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^t\left({\displaystyle \frac{t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}+2\right)w\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}.$$

(38)

By applying the MK Laplace transformation for both sides to Equation (38), we obtain

$$-{\displaystyle \frac{2}{v}}+{\displaystyle \frac{1}{v^3}}+{\displaystyle \frac{2}{v^4}}+{\displaystyle \frac{1}{v^2-1}}+{\displaystyle \frac{2v}{v^2-1}}=\left({\displaystyle \frac{1}{v^2}}+{\displaystyle \frac{2}{v}}\right)L_{MK}\left[w\left(t\right)\right]\left(v\right),$$

or equivalently,

$${\displaystyle \frac{v^4+3v^3+2v^2-v-2}{v^4\left(v^2-1\right)}}=\left({\displaystyle \frac{1}{v^2}}+{\displaystyle \frac{2}{v}}\right)L_{MK}\left[w\left(t\right)\right]\left(v\right).$$

By rearranging all of terms appropriately, the above equation becomes

$$L_{MK}\left[w\left(t\right)\right]\left(v\right)={\displaystyle \frac{v^3+v^2-1}{v^2\left(v^2-1\right)}}={\displaystyle \frac{1}{v^2}}+{\displaystyle \frac{v}{v^2-1}}.$$

Now, by employing the inverse MK Laplace transformation of both sides, we obtain

$$w\left(t\right)={\displaystyle \frac{t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}+cosh\left({\displaystyle \frac{t^{\alpha +\gamma -1}}{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}\right).$$

Remark 12. 

If $\alpha =\gamma =1$, then Equation (38) is the following classical Volterra InEq:

$$-2+{\displaystyle \frac{t^2}{2}}+{\displaystyle \frac{t^3}{3!}}+\sin \mathrm{h}\left(t\right)+2\cos \left(t\right)={{\displaystyle \int }}_0^t\left(t-\tau +2\right)w\left(\tau \right)d\tau ,$$

which represents the exact solution given by $w\left(t\right)=t+\cos \mathrm{h}\left(t\right)$.

4.3. The MK Abel InEq

Consider the following MK Abel InEq:

$$f\left(t\right)={\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^t{\displaystyle \frac{1}{\sqrt{t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}}}}w\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}},$$

(39)

where $w\left(t\right)$ is an unknown function, $f\left(t\right)$ is a perturbation known function, and $M\left(\alpha ,\gamma ,\delta \right)={\displaystyle \frac{\Gamma \left(\delta -\gamma +1\right)}{\gamma \Gamma \left(\delta -\alpha -\gamma +2\right)}}, 0<\alpha ,\gamma \le 1$, with $\mathsf{\alpha }+\mathsf{\gamma }-1>0$, $\delta >-1$, and $t>0$. The MK Abel InEq is also known as the MK Volterra InEq of the first kind. In addition, the kernel $V\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)$ in InEq is

$$V\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)={\displaystyle \frac{1}{\sqrt{t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}}}},$$

where

$$V\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)\to \infty , \mathrm{as} \tau \to t.$$

If $\alpha =\beta =1$, then Equation (39) becomes the classical AInEq [1,2,9].

First, by taking generalized Laplace transformation on both sides of Equation (39), we obtain

$$L_{MK}\left[f\left(t\right)\right]\left(v\right)=L_{MK}\left[t^{-{\displaystyle \frac{\alpha +\gamma -1}{2}}}\ast w\left(t\right)\right]\left(v\right).$$

Now, by applying Theorems 10 and 12, we obtain

$$L_{MK}\left[f\left(t\right)\right]\left(v\right)=L_{MK}\left[t^{-{\displaystyle \frac{\alpha +\gamma -1}{2}}}\right]\left(v\right)·L_{MK}\left[w\left(t\right)\right]\left(v\right)=\sqrt{{\displaystyle \frac{\pi }{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}}·{\displaystyle \frac{1}{v^{\frac{1}{2}}}}·L_{MK}\left[w\left(t\right)\right]\left(v\right).$$

By rearranging all of terms appropriately, the above equation becomes

$$L_{MK}\left[w\left(t\right)\right]\left(v\right)=\sqrt{{\displaystyle \frac{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}{\pi }}}·v^{{\displaystyle \frac{1}{2}}}·L_{MK}\left[f\left(t\right)\right]\left(v\right).$$

If we use Theorems 10 and 12 again, the above equation can be expressed as

$$\begin{array}{cc}{L}_{MK}\left[w\left(t\right)\right]\left(v\right)\hfill & ={\displaystyle \frac{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}{\pi }}·v·\left[\sqrt{{\displaystyle \frac{\pi }{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}}·{\displaystyle \frac{1}{v^{\frac{1}{2}}}}·L_{MK}\left[f\left(t\right)\right]\left(v\right)\right]\hfill \\ & ={\displaystyle \frac{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}{\pi }}·v·L_{MK}\left[t^{-{\displaystyle \frac{\alpha +\gamma -1}{2}}}\ast f\left(t\right)\right]\left(v\right)\hfill \\ & ={\displaystyle \frac{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}{\pi }}·v·L_{MK}\left[{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^t{\displaystyle \frac{1}{\sqrt{t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}}}}f\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}\right].\hfill \end{array}$$

We can write this equation as follows:

$$L_{MK}\left[w\left(t\right)\right]\left(v\right)={\displaystyle \frac{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}{\pi }}·v·L_{MK}\left[F\left(t\right)\right]\left(v\right),$$

(40)

where

$$F\left(t\right)={\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^t{\displaystyle \frac{1}{\sqrt{t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}}}}f\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}.$$

(41)

Now, by applying Theorem 11 on Equation (41), we obtain

$$L_{MK}\left[{}^{MK}D^{\alpha ,\gamma }F\left(t\right)\right]\left(v\right)=v{L}_{MK}\left[F\left(t\right)\right]\left(v\right)-F\left(0\right)=v{L}_{MK}\left[F\left(t\right)\right]\left(v\right).$$

By rearranging all of terms appropriately, the above equation becomes

$$L_{MK}\left[F\left(t\right)\right]\left(v\right)={\displaystyle \frac{1}{v}}·L_{MK}\left[{}^{MK}D^{\alpha ,\gamma }F\left(t\right)\right]\left(v\right).$$

(42)

From Equations (40) and (42), we obtain

$$L_{MK}\left[w\left(t\right)\right]\left(v\right)={\displaystyle \frac{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}{\pi }}·L_{MK}\left[{}^{MK}D^{\alpha ,\gamma }F\left(t\right)\right].$$

(43)

By applying the inverse MK Laplace transformation on both sides of the above equation, we obtain

$$w\left(t\right)={\displaystyle \frac{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}{\pi }}·{}^{MK}D^{\alpha ,\gamma }F\left(t\right).$$

(44)

Using Equations (41) and (44), we obtain

$$w\left(t\right)={\displaystyle \frac{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}{\pi }}·{}^{MK}D^{\alpha ,\gamma }\left({\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^t{\displaystyle \frac{1}{\sqrt{t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}}}}f\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}\right),$$

which is the required solution of the Equation (39).

Remark 13. 

By evaluating the MK $\alpha ,\gamma$-integral on the right side of the above equation and then taking the MK $\alpha ,\gamma$-derivative with respect to $t$, we obtain

$$w\left(t\right)={\displaystyle \frac{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}{\pi }}·\left({\displaystyle \frac{f\left(0\right)}{t^{\frac{\alpha +\gamma -1}{2}}}}+{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^t{\displaystyle \frac{1}{\sqrt{t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}}}}{}^{MK}D^{\alpha ,\gamma }f\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}\right).$$

(45)

Next, we will apply this introduced technique to the following interesting illustrative examples:

Example 5. 

Consider the following MK AInEq with $f\left(t\right)={\displaystyle \frac{3\pi }{8}}{t}^{2\left(\alpha +\gamma -1\right)}$:

$${\displaystyle \frac{3\pi }{8}}{t}^{2\left(\alpha +\gamma -1\right)}={\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^t{\displaystyle \frac{1}{\sqrt{t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}}}}w\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}.$$

(46)

By taking the MKFF Laplace transformation for both sides to Equation (46), we obtain

$${\displaystyle \frac{3\pi {\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}^2}{4}}·{\displaystyle \frac{1}{v^3}}=\sqrt{{\displaystyle \frac{\pi }{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}}·{\displaystyle \frac{1}{v^{\frac{1}{2}}}}·L_{MK}\left[w\left(t\right)\right]\left(v\right).$$

By rearranging all of terms appropriately, the above equation becomes

$$L_{MK}\left[w\left(t\right)\right]\left(v\right)={\displaystyle \frac{3\sqrt{\pi }{\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}^{\frac{5}{2}}}{4}}·{\displaystyle \frac{1}{v^{\frac{5}{2}}}}.$$

Now, by employing the inverse MK Laplace transformation of both sides, we obtain

$$\left(t\right)=M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)t^{{\displaystyle \frac{3\left(\alpha +\gamma -1\right)}{2}}}.$$

This MK InEq’s solution can be obtained by directly applying Equation (45).

Remark 14. 

If $\alpha =\gamma =1$, then Equation (46) becomes the following classical AInEq:

$${\displaystyle \frac{3\pi }{8}}{t}^2={{\displaystyle \int }}_0^t\sqrt{t-\tau } w\left(\tau \right)d\tau ,$$

which represents the exact solution given by $w\left(t\right)=t^{{\displaystyle \frac{3}{2}}}$.

Example 6. 

Consider the following MK AInEq with  $f\left(t\right)=2t^{{\displaystyle \frac{\alpha +\gamma -1}{2}}}+{\displaystyle \frac{8}{3}}{t}^{{\displaystyle \frac{3\left(\alpha +\gamma -1\right)}{2}}}$:

$$2t^{{\displaystyle \frac{\alpha +\gamma -1}{2}}}+{\displaystyle \frac{8}{3}}{t}^{{\displaystyle \frac{3\left(\alpha +\gamma -1\right)}{2}}}={\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^t{\displaystyle \frac{1}{\sqrt{t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}}}}w\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}.$$

(47)

By applying the MK Laplace transformation for both sides of Equation (47), we obtain

$$\sqrt{\pi }\left(\sqrt{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}·{\displaystyle \frac{1}{v^{\frac{3}{2}}}}+2{\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}^{{\displaystyle \frac{3}{2}}}·{\displaystyle \frac{1}{v^{\frac{5}{2}}}}\right)=\sqrt{{\displaystyle \frac{\pi }{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}}}·{\displaystyle \frac{1}{v^{\frac{1}{2}}}}·L_{MK}\left[w\left(t\right)\right]\left(v\right).$$

By rearranging all of terms appropriately, the above equation becomes

$$L_{MK}\left[w\left(t\right)\right]\left(v\right)=\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right){\displaystyle \frac{1}{v}}+2 {\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}^2{\displaystyle \frac{1}{v^2}}.$$

Now, by employing the inverse MK Laplace transformation of both sides, we obtain

$$w\left(t\right)=\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)\left(1+2t^{\alpha +\gamma -1}\right).$$

This MK InEq’s solution can be obtained by directly applying Equation (45).

Remark 15. 

If  $\alpha =\gamma =1$, then Equation (47) becomes the following classical AInEq:

$$2\sqrt{t}+{\displaystyle \frac{8}{3}}{t}^{{\displaystyle \frac{3}{2}}}={{\displaystyle \int }}_0^t\sqrt{t-\tau } w\left(\tau \right)d\tau ,$$

which represents the exact solution given by $w\left(t\right)=1+2t$.

4.4. The MK Generalized Abel InEq

Consider the following generalized MK AInEq:

$$f\left(t\right)={\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^t{\displaystyle \frac{1}{{\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)}^{\lambda }}}w\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}},$$

(48)

where $w\left(t\right)$ is an unknown function, $f\left(t\right)$ is a perturbation known function, and $M\left(\alpha ,\gamma ,\delta \right)={\displaystyle \frac{\Gamma \left(\delta -\gamma +1\right)}{\gamma \Gamma \left(\delta -\alpha -\gamma +2\right)}}, 0<\alpha ,\gamma \le 1$, with $\delta >-1$, $0<\lambda <1$, and $t>0$. In addition, the kernel, $V\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)$, in this InEq is

$$V\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)={\displaystyle \frac{1}{{\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)}^{\lambda }}},0<\lambda <1 .$$

If $\alpha =\beta =1$, then Equation (48) becomes the classical generalized AInEq [1,2].

First, by taking the generalized Laplace transformation on both sides of Equation (48), we obtain

$$L_{MK}\left[f\left(t\right)\right]\left(v\right)=L_{MK}\left[t^{-\lambda \left(\alpha +\gamma -1\right)}\ast w\left(t\right)\right]\left(v\right).$$

Now, by applying Theorems 10 and 12, we obtain

$$\begin{array}{cc}{L}_{MK}\left[f\left(t\right)\right]\left(v\right)\hfill & =L_{MK}\left[t^{-\lambda \left(\alpha +\gamma -1\right)}\right]\left(v\right)·L_{MK}\left[w\left(t\right)\right]\left(v\right)\hfill \\ & ={\displaystyle \frac{\Gamma \left(1-\lambda \right)}{{\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}^{\lambda }}}·{\displaystyle \frac{1}{v^{1-\lambda }}}·L_{MK}\left[w\left(t\right)\right]\left(v\right).\hfill \end{array}$$

By rearranging all of terms appropriately, the above equation becomes

$$L_{MK}\left[w\left(t\right)\right]\left(v\right)={\displaystyle \frac{{\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}^{\lambda }}{\Gamma \left(1-\lambda \right)}}·v^{1-\lambda }·L_{MK}\left[f\left(t\right)\right]\left(v\right).$$

If we again use Theorems 7 and 9, the above equation can be expressed as

$$\begin{array}{cc}{L}_{MK}\left[w\left(t\right)\right]\left(v\right)\hfill & ={\displaystyle \frac{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}{\Gamma \left(\lambda \right)\Gamma \left(1-\lambda \right)}}·v·\left[{\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}^{\lambda -1}·\Gamma \left(\lambda \right)·{\displaystyle \frac{1}{v^{\lambda }}}·L_{MK}\left[f\left(t\right)\right]\left(v\right)\right]\hfill \\ & ={\displaystyle \frac{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}{\Gamma \left(\lambda \right)\Gamma \left(1-\lambda \right)}}·v·L_{MK}\left[t^{\left(\lambda -1\right)\left(\alpha +\gamma -1\right)}\ast f\left(t\right)\right]\left(v\right)\hfill \\ & ={\displaystyle \frac{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}{\Gamma \left(\lambda \right)\Gamma \left(1-\lambda \right)}}·v·L_{MK}\left[{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^t{\displaystyle \frac{1}{{\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)}^{1-\lambda }}}f\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}\right].\hfill \end{array}$$

We can write this equation as

$$L_{MK}\left[w\left(t\right)\right]\left(v\right)={\displaystyle \frac{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)}{\Gamma \left(\lambda \right)\Gamma \left(1-\lambda \right)}}·v·L_{MK}\left[F\left(t\right)\right]\left(v\right),$$

(49)

where

$$F\left(t\right)={\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^t{\displaystyle \frac{1}{{\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)}^{1-\lambda }}}f\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}.$$

(50)

Using the facts

$$L_{MK}\left[{}^{MK}D^{\alpha ,\gamma }F\left(t\right)\right]\left(v\right)=v{L}_{MK}\left[F\left(t\right)\right]\left(v\right)-F\left(0\right),$$

and

$$\Gamma \left(\lambda \right)\Gamma \left(1-\lambda \right)={\displaystyle \frac{\pi }{sin\left(\lambda \pi \right)}},$$

in Equation (49), we obtain

$$L_{MK}\left[w\left(t\right)\right]\left(v\right)={\displaystyle \frac{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)sin\left(\lambda \pi \right)}{\pi }}·L_{MK}\left[{}^{MK}D^{\alpha ,\gamma }F\left(t\right)\right].$$

(51)

Applying the inverse MK Laplace transformation on both sides of Equation (51):

$$w\left(t\right)={\displaystyle \frac{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)sin\left(\lambda \pi \right)}{\pi }}·{}^{MK}D^{\alpha ,\gamma }F\left(t\right).$$

(52)

Using Equations (50) and (52), we obtain

$$w\left(t\right)={\displaystyle \frac{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)sin\left(\lambda \pi \right)}{\pi }}·{}^{MK}D^{\alpha ,\gamma }\left({\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^t{\displaystyle \frac{1}{{\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)}^{1-\lambda }}}f\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}\right),$$

which is the required solution of the Equation (48).

Remark 16. 

By evaluating the MK $\alpha ,\gamma$-integral on the right side of the above equation and then taking the MK $\alpha ,\gamma$-derivative with respect to $t$, we obtain

$$w\left(t\right)={\displaystyle \frac{M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)sin\left(\lambda \pi \right)}{\pi }}·\left({\displaystyle \frac{f\left(0\right)}{t^{\left(1-\lambda \right)\left(\alpha +\gamma -1\right)}}}+{\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^t{\displaystyle \frac{1}{{\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)}^{1-\lambda }}}{}^{MK}D^{\alpha ,\gamma }f\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}\right).$$

(53)

Next, we apply this introduced technique to the following interesting illustrative example.

Example 7. 

Consider the following generalized MK AInEq with $f\left(t\right)=\pi t^{\alpha +\gamma -1}$ and $\lambda ={\displaystyle \frac{2}{3}}$.

$$\pi t^{\alpha +\gamma -1}={\displaystyle \frac{1}{M\left(\alpha ,\gamma ,\delta \right)}}{{\displaystyle \int }}_0^t{\displaystyle \frac{1}{{\left(t^{\alpha +\gamma -1}-{\tau }^{\alpha +\gamma -1}\right)}^{\frac{2}{3}}}}w\left(\tau \right){\displaystyle \frac{d\tau }{{\tau }^{2-\alpha -\gamma }}}$$

(54)

By applying the MK Laplace transformation for both sides to Equation (54), we obtain

$$\pi ·M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)·{\displaystyle \frac{1}{v^2}}={\displaystyle \frac{\Gamma \left(\frac{1}{3}\right)}{{\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}^{\frac{2}{3}}}}·{\displaystyle \frac{1}{v^{\frac{1}{3}}}}·L_{MK}\left[w\left(t\right)\right]\left(v\right).$$

By rearranging all of terms appropriately, the above equation becomes

$$L_{MK}\left[w\left(t\right)\right]\left(v\right)={\displaystyle \frac{\pi {\left(M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)\right)}^{\frac{5}{3}}}{\Gamma \left(\frac{1}{3}\right)}}·{\displaystyle \frac{1}{v^{\frac{5}{3}}}}.$$

Now, by employing the inverse MK Laplace transformation of both sides, we obtain

$$w\left(t\right)={\displaystyle \frac{3\sqrt{3}}{4}}M\left(\alpha ,\gamma ,\delta \right)\left(\alpha +\gamma -1\right)t^{{\displaystyle \frac{2\left(\alpha +\gamma -1\right)}{3}}}.$$

This MK InEq’s solution can be obtained by directly applying Equation (53).

Remark 17. 

If $\alpha =\gamma =1$, then Equation (54) becomes the following classical generalized AInEq:

$$\pi t={{\displaystyle \int }}_0^t{\displaystyle \frac{1}{{\left(t-\tau \right)}^{\frac{2}{3}}}}w\left(\tau \right)d\tau ,$$

which represents the exact solution given by $w\left(t\right)={\displaystyle \frac{3\sqrt{3}}{4}}{t}^{{\displaystyle \frac{2}{3}}}$.

5. Conclusions

In this research work, the extension of the recently introduced MK calculus theory has been addressed in terms of integral transformations with applications in fractal–fractional InEqs. A new fractal–fractional Laplace transformation has been proposed, named MK Laplace transformation, which involves the MK α, γ-integral operator. For this new integral transformation, the main properties and results of the classical Laplace transformation are formulated and investigated. Based on the developed Laplace transformation, the method of MK Laplace transformation in the sense of the Volterra MK InEqs of the first and second kind is proposed. This proposed technique has been also applied to the MK AInEq and the MK generalized AInEq. The use of this technique has been illustrated with various interesting examples. The analysis of the results obtained in this research verifies that the proposed method, by including the MK α, γ-integral operator, is an efficient mathematical tool for obtaining solutions to various scientific scenarios that possess a fractal effect and include fractional InEqs. Finally, our results indicate a novel extension of the classical theory. In future studies, the newly proposed technique will be extended further to include MK double and triple Laplace transformation, with more applications in mathematical physics.