On the Construction and Analysis of a Fractional-Order Dirac Delta Distribution with Application

Abstract

We introduce the generalized fractional-order Dirac delta distribution ${\delta }_{\mathrm{GFODDF}}$, defined by applying the generalized fractional derivative (GFD) operator to the Heaviside function. This construction extends the classical Dirac delta to non-integer orders, allowing modeling of systems with memory and non-local effects. We establish fundamental properties—including shifting, scaling, evenness, derivative, and convolution—within a rigorous distributional framework and present explicit proofs. Applications are demonstrated by solving linear fractional differential equations and by modeling drug release with fractional kinetics, where the new delta captures impulse responses with long-term memory. Numerical illustrations confirm that ${\delta }_{\mathrm{GFODDF}}$ reduces to the classical delta when $\eta =1$, while providing additional flexibility for $0<\eta <1$. These results show that ${\delta }_{\mathrm{GFODDF}}$ is a powerful tool for fractional-order analysis in mathematics, physics, and biomedical engineering.

1. Introduction

The Dirac delta distribution is a fundamental concept in both mathematics and physics. It has been widely used to represent point-like sources or impulses in disciplines such as signal processing, quantum mechanics, and differential equations [1,2,3]. The unique characteristics of this function of being zero everywhere except at a single point where it is infinite, along with its integral throughout the real line equal to one, make it an indispensable tool for mathematical modeling and analysis [4,5].

In this article, an extended form of the Dirac delta distribution is introduced, termed the generalized fractional-order Dirac delta distribution ${\delta }_{\mathrm{GFODDF}}\left(x\right)$. This extension incorporates the principles of fractional-order differentiation through the generalized fractional derivative (GFD) operator, as recently introduced by [6]. By redefining the Dirac delta distribution within the framework of the fractional derivative operator, new insights and possibilities are revealed about its properties and applications.

The main motivation for studying the extended version of the delta distribution lies in its potential to encapsulate more complex dynamics and phenomena compared to the classical one. The fractional-order calculus, which deals with derivatives and integrals of noninteger orders, has gained prominence for its ability to describe systems with memory effects and nonlocal behaviors [7,8]. In recent years, fractional calculus and its extensions have been successfully applied in various contexts, including the development of fractional integral inequalities in fractal sets [9] and in modeling real-world processes such as population dynamics using exponential kernel-based fractional operators [10]. These studies demonstrate the versatility of fractional operators in handling complex systems, which motivates the exploration of fractional generalizations of other fundamental mathematical constructs.

Therefore, through a detailed examination of this generalized delta distribution, this article aims to advance the understanding of fractional calculus and its practical implementations [11]. The paper begins with the preliminaries, followed by rigorous proofs of various properties of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$, including its shifting property, scaling property, and convolution property. In the final section, the utility of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ is illustrated through applications in solving fractional differential equations, modeling singularities, and analyzing complex systems in both mathematical analysis and engineering. By integrating theoretical analysis, mathematical derivations, and practical examples, the work provides a comprehensive overview of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ and underscores its significance in improving mathematical modeling and analytical techniques.

2. Preliminaries

This section outlines the essential mathematical concepts, definitions, and tools necessary to comprehend ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ and its properties.

2.1. The Dirac Delta as a Distribution

We follow the classical treatment of distributions as in Schwartz [12], and treat the Dirac delta as a distribution. Let $\mathcal{D}\left(\mathbb{R}\right)$ denote the space of smooth compactly supported test functions and ${\mathcal{D}}^{\prime }\left(\mathbb{R}\right)$ its dual, the space of Schwartz distributions. The Dirac distribution $\delta \in {\mathcal{D}}^{\prime }\left(\mathbb{R}\right)$ is defined by

$$\langle \delta ,\phi \rangle :=\phi \left(0\right)\mathrm{for}\mathrm{all}\phi \in \mathcal{D}\left(\mathbb{R}\right).$$

(1)

The common notation ${\int }_{\mathbb{R}}\delta \left(x\right)\phi \left(x\right)dx=\phi \left(0\right)$ will be used as a shorthand for the duality pairing $\langle \delta ,\phi \rangle$. The normalization property of the Dirac distribution is expressed in duality by

$$\langle \delta ,1_{\Omega }\rangle =1,$$

when $1_{\Omega }$ denotes a suitable test function approximating the constant function 1 (e.g., via a sequence in $\mathcal{D}$ converging to 1 in a tempered sense). Equivalently, for any $\phi \in \mathcal{D}\left(\mathbb{R}\right)$,

$$\langle \delta ,\phi \rangle =\phi \left(0\right).$$

(We avoid statements that treat $\delta$ as an ordinary pointwise-defined function.) As a foundation for distribution theory we follow the classical works of Schwartz [12] and Gelfand–Shilov [13]. For an accessible exposition, see Strichartz [14]. The Heaviside function H induces a distributional derivative (derivative is meant in the distributional sense) given by

$$\langle H^{\prime },\phi \rangle :=-\langle H,{\phi }^{\prime }\rangle =\phi \left(0\right)(\phi \in \mathcal{D}),$$

(2)

so in ${\mathcal{D}}^{\prime }\left(\mathbb{R}\right)$ we write $H^{\prime }=\delta$. All derivatives of distributions below are to be understood in this distributional sense. In particular, for any distribution T,

$$\langle T^{\prime },\phi \rangle =-\langle T,{\phi }^{\prime }\rangle (\phi \in \mathcal{D}).$$

(3)

Graphical illustrations of these functions are given in Figure 1 and Figure 2.

2.2. Distributional Properties of the Dirac Delta

In this section, we collect several standard properties of the Dirac delta, always interpreted as an element of the space of Schwartz distributions ${\mathcal{D}}^{\prime }\left(\mathbb{R}\right)$ (or ${\mathcal{S}}^{\prime }\left(\mathbb{R}\right)$ when tempered distributions are required). Throughout, $\phi \in \mathcal{D}\left(\mathbb{R}\right)$ denotes a smooth compactly supported test function and $\langle ·,·\rangle$ denotes the distribution–test-function pairing.

2.2.1. Definition

The Dirac distribution $\delta$ is the continuous linear functional

$$\langle \delta ,\phi \rangle =\phi \left(0\right),\phi \in \mathcal{D}\left(\mathbb{R}\right).$$

The informal integral notation ${\int }_{\mathbb{R}}\delta \left(x\right)\phi \left(x\right)dx=\phi \left(0\right)$ will be used only as a convenient shorthand for this duality pairing.

2.2.2. Basic Properties

• Support: $supp\delta =\left\{0\right\}$.
• Translation: For $a\in \mathbb{R}$, define the translated distribution ${\delta }_a$ by

  $$\langle {\delta }_a,\phi \rangle =\phi \left(a\right).$$
  In integral shorthand, $\delta (x-a)$ represents ${\delta }_a$.
• Scaling: For non-zero $c\in \mathbb{R}$,

  $$\langle \delta (c·),\phi \rangle ={\displaystyle \frac{1}{\left|c\right|}}\phi \left(0\right).$$
• Derivative: The distributional derivative satisfies

  $$\langle {\delta }^{\prime },\phi \rangle =-\langle \delta ,{\phi }^{\prime }\rangle =-{\phi }^{\prime }\left(0\right).$$
• Multiplication by smooth functions: If $f\in C^{\infty }\left(\mathbb{R}\right)$,

  $$\langle f\delta ,\phi \rangle =\langle \delta ,f\phi \rangle =f\left(0\right)\phi \left(0\right).$$
  (Products with non-smooth f are not defined without additional regularization.)
• Convolution: If $g\in \mathcal{D}\left(\mathbb{R}\right)$ (or, more generally, g is a rapidly decreasing smooth function), the convolution is

  $$(\delta \ast g)\left(x\right):=\langle \delta ,g(x-·)\rangle =g\left(x\right).$$
  Convolution of $\delta$ with another distribution is defined only when the second factor has compact support.

2.2.3. Relation to the Heaviside Function

Let H be the Heaviside function

$$H\left(x\right)=\left\{\begin{array}{cc}0,\hfill & x<0,\hfill \\ 1,\hfill & x\ge 0.\hfill \end{array}\right.$$

Its distributional derivative satisfies

$$\langle H^{\prime },\phi \rangle :=-\langle H,{\phi }^{\prime }\rangle =\phi \left(0\right),$$

so $H^{\prime }=\delta$ in ${\mathcal{D}}^{\prime }\left(\mathbb{R}\right)$.

Convolution with the delta. If $T\in {\mathcal{D}}^{\prime }\left(\mathbb{R}\right)$ and $g\in \mathcal{D}\left(\mathbb{R}\right)$ (or, more generally, g is a suitable test/function from an admissible space), then the convolution $T\ast g$ is defined by

$$(T\ast g)\left(x\right):=\langle T,g(x-·)\rangle .$$

(4)

In particular, for $T=\delta$ and $g\in \mathcal{D}\left(\mathbb{R}\right)$, we obtain $(\delta \ast g)\left(x\right)=g\left(x\right)$. Note: convolution of two arbitrary distributions requires extra hypotheses (e.g., one factor compactly supported); please see the assumptions below when we convolve distributions.

$${\delta }_{\mathrm{GFODDF}}\left(x\right)={\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{x}^{1-\eta }\delta \left(x\right)$$

(5)

This product is understood as the action of the smooth factor on the test function inside the duality pairing; it is not a pointwise product of singular functions.

Remarks: The properties above remain valid for our generalized fractional Dirac distribution ${\delta }_{\mathrm{GFODDF}}$ when the test-function space and the adjoint of the fractional derivative operator are specified. Explicit formulas (e.g., normalization constants) are obtained by evaluating the action of ${\delta }_{\mathrm{GFODDF}}$ on $\phi \in \mathcal{D}\left(\mathbb{R}\right)$.

2.3. Fractional-Order Derivatives

Fractional-order derivatives extend the concept of differentiation to non-integer orders, allowing for more flexible modeling of physical phenomena. Several definitions of fractional derivatives exist [15,16,17]:

2.3.1. Riemann–Liouville (RL) Fractional-Order Derivatives

The RL fractional derivative of order $\eta$ is defined as

$$D_{RL}^{\eta }f\left(x\right)={\displaystyle \frac{1}{\Gamma (n-\eta )}}{\displaystyle \frac{d^n}{d{x}^n}}{\int }_0^x{(x-\tau )}^{n-\eta -1}f\left(\tau \right)d\tau ,$$

(6)

where $n=\lceil \eta \rceil$.

2.3.2. Caputo Fractional-Order Derivatives

The Caputo fractional-order derivative of order $\eta$ is given by

$$D_C^{\eta }f\left(x\right)={\displaystyle \frac{1}{\Gamma (n-\eta )}}{\int }_0^x{(x-\tau )}^{n-\eta -1}{\displaystyle \frac{d^{n}f\left(\tau \right)}{d{\tau }^n}}d\tau ,$$

(7)

where $n=\lceil \eta \rceil$.

2.3.3. Gr Ünwald–Letnikov Fractional-Order Derivative

The Grünwald–Letnikov fractional-order derivative is defined as

$$D_{GL}^{\eta }f\left(x\right)=\underset{h\to 0}{lim}{\displaystyle \frac{1}{h^{\eta }}}\sum _{k=0}^{⌊{\displaystyle \frac{x}{h}}⌋}{(-1)}^k\left({\displaystyle \genfrac{}{}{0pt}{}{\eta }{k}}\right)f(x-kh).$$

(8)

2.4. Generalized Fractional-Order Derivative and Dirac Delta Distribution

Throughout this work, we use the following notation:

• $\eta \in (0, 1]$: order of the generalized fractional derivative, controlling the memory effect.
• $\zeta >0$: auxiliary parameter of the generalized fractional derivative that adjusts the normalization.
• $\Gamma (·)$: the Euler gamma function, $\Gamma \left(s\right)={\int }_0^{\infty }{t}^{s-1}{e}^{-t}dt$.

The generalized fractional derivative operator $D_{\mathrm{GFD}}$ is defined using the Gamma function as follows [18]:

$$D_{\mathrm{GFD}}f\left(x\right)={\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{x}^{1-\eta }{\displaystyle \frac{df\left(x\right)}{dx}},$$

(9)

where $f\left(x\right)$ is a function, $\eta$ and $\zeta$ are parameters, and ${\displaystyle \frac{df\left(x\right)}{dx}}$ denotes the first derivative of $f\left(x\right)$ pertaining to x, the conceptual diagram in Figure 3, and the generalized fractional-order Dirac delta distribution for different values described in Figure 4.

The concept of a fractional-order delta function has also been explored by Allagui and Elwakil (2024) [19], who analyzed its properties and inverse-Laplace-transform representation, providing additional theoretical support for our definition of the generalized fractional-order Dirac delta distribution ${\delta }_{GFODDF}$.

2.4.1. Heaviside Step Function

The Heaviside step function [20], $H\left(x\right)$, is a piecewise function defined as

$$H\left(x\right)=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}x\ge 0,\hfill \\ 0,\hfill & \mathrm{if}x<0.\hfill \end{array}\right.$$

(10)

It serves as a fundamental building block for defining ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ through the fractional derivative operator $D_{\mathrm{GFD}}$.

2.4.2. Generalized Fractional-Order Dirac Distribution

Let $D_{\mathrm{GFD}}^{\eta ,\zeta }$ denote the generalized fractional derivative operator acting continuously on a chosen test function space (for example, $\mathcal{D}$ or $\mathcal{S}$), and let ${\left(D_{\mathrm{GFD}}^{\eta ,\zeta }\right)}^{\ast }$ be its adjoint acting on distributions. We define the generalized fractional-order Dirac distribution ${\delta }_{\mathrm{GFODDF}}\in {\mathcal{D}}^{\prime }$ by the action

$$\langle {\delta }_{\mathrm{GFODDF}},\phi \rangle :=\langle D_{\mathrm{GFD}}^{\eta ,\zeta }H,\phi \rangle :=\langle H,{\left(D_{\mathrm{GFD}}^{\eta ,\zeta }\right)}^{\ast }\phi \rangle ,\phi \in \mathcal{D}\left(\mathbb{R}\right).$$

In this way, all formulas involving ${\delta }_{\mathrm{GFODDF}}$ are statements about distributions and duality pairings; pointwise multiplications by singular factors are avoided unless the factor is a smooth multiplier. Let $\phi \in \mathcal{D}\left(\mathbb{R}\right)$. Using the definition $\langle {\delta }_{\mathrm{GFODDF}},\phi \rangle :=\langle H,{\left(D_{\mathrm{GFD}}^{\eta ,\zeta }\right)}^{\ast }\phi \rangle$ and the known action of the adjoint on test functions, one obtains

$$\langle {\delta }_{\mathrm{GFODDF}},\phi \rangle ={\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}\phi \left(0\right),$$

so ${\delta }_{\mathrm{GFODDF}}$ acts as a (scaled) point-evaluation at 0 in ${\mathcal{D}}^{\prime }$, i.e., $\langle {\delta }_{\mathrm{GFODDF}},\phi \rangle =(\Gamma \left(\zeta \right)/\Gamma (\zeta -\eta +1))\phi \left(0\right)$. All other properties (scaling, evenness, derivative) should be proved by the same duality method: compute the action on an arbitrary $\phi \in \mathcal{D}$ and manipulate the adjoint operator. Multiplication of a distribution $T\in {\mathcal{D}}^{\prime }\left(\mathbb{R}\right)$ by a function f is defined whenever f is a $C^{\infty }$ function (i.e., $f\in C^{\infty }\left(\mathbb{R}\right)$): the product $fT$ is the distribution given by $\langle fT,\phi \rangle =\langle T,f\phi \rangle$. In particular, for a smooth f, one has $f\delta =f\left(0\right)\delta$. If f is not smooth at the point of singularity (for example, if $f\left(x\right)=x^{1-\eta }$ with a noninteger exponent producing non-smoothness at 0), the product $f\delta$ is not defined in general in ${\mathcal{D}}^{\prime }$, unless a regularization or additional structure is provided and justified.

3. Properties and Proofs of the ${\mathit{\delta }}_{\mathbf{GFODDF}}\mathbf{\left(}\mathit{X}\mathbf{\right)}$

This section explores and proves the fundamental properties of the ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ defined by using the $D_{\mathrm{GFD}}$ operator. We shall treat the integral notation is purely symbolic; all equalities are in the sense of distributions.

3.1. Shifting Property

The shifting property of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ states that, when combined with a suitable test function $f\left(x\right)$, it acts as an identity element, picking out the value of $f\left(x\right)$ at $x=0$:

$${\int }_{-\infty }^{\infty }\delta \left(x\right)·f\left(x\right)dx=f\left(0\right).$$

(11)

3.1.1. Proof

Let us start by defining the ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ using the $D_{\mathrm{GFD}}$ operator: $\delta \left(x\right)=D_{\mathrm{GFD}}H\left(x\right)$, where $H\left(x\right)$ is the Heaviside step function.

Using the shifting property of the $D_{\mathrm{GFD}}$ operator in $H\left(x\right)$, we have the following:

$${\int }_{-\infty }^{\infty }{D}_{\mathrm{GFD}}H\left(x\right)·f\left(x\right)dx=f\left(0\right).$$

(12)

Expanding the differential operator and differentiating $H\left(x\right)$, we obtain

$${\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{x}^{1-\eta }{\displaystyle \frac{dH\left(x\right)}{dx}}·f\left(x\right)=f\left(0\right).$$

(13)

Since $H\left(x\right)$ is piecewise constant, its derivative ${\displaystyle \frac{dH\left(x\right)}{dx}}$ is a sum of Dirac delta functions. Therefore, the integral simplifies to

$${\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}\sum _k{c}_k\delta (x-x_k)·f\left(x\right)=f\left(0\right).$$

(14)

The integral selects the term corresponding to $x=0$, yielding

$${\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}·c_0·f\left(0\right)=f\left(0\right).$$

(15)

Since $c_0=1$, we have

$${\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}·f\left(0\right)=f\left(0\right).$$

(16)

This completes the proof of the shifting property of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$.

3.1.2. Shifting Property in the Distributional Framework

Let $\mathcal{D}\left(\mathbb{R}\right)$ be the space of smooth compactly supported test functions, and let ${\mathcal{D}}^{\prime }\left(\mathbb{R}\right)$ be its dual of Schwartz distributions. We define the generalized fractional-order Dirac delta ${\delta }_{\mathrm{GFODDF}}$ as the distribution

$$〈{\delta }_{\mathrm{GFODDF}},\varphi 〉:=〈D_{\mathrm{GFD}}^{\eta ,\zeta }H,\varphi 〉:=〈H,D_{\mathrm{GFD}}^{\eta ,\zeta ,\ast }\varphi 〉,$$

(17)

where H is the Heaviside step function, and $D_{\mathrm{GFD}}^{\eta ,\zeta ,\ast }$ denotes the adjoint of the generalized fractional derivative. From the properties of the GFD operator (see references [12,18]),

$$D_{\mathrm{GFD}}^{\eta ,\zeta }H\left(x\right)={\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{x}^{1-\eta }\delta \left(x\right)\mathrm{in}{\mathcal{D}}^{\prime }\left(\mathbb{R}\right),$$

(18)

where $\delta \left(x\right)$ is the classical Dirac distribution. Therefore, for every $\varphi \in \mathcal{D}\left(\mathbb{R}\right)$,

$$\begin{array}{cc}\hfill 〈{\delta }_{\mathrm{GFODDF}},\varphi 〉& =〈{\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{x}^{1-\eta }\delta \left(x\right),\varphi \left(x\right)〉\hfill \\ \hfill & ={\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}\varphi \left(0\right).\hfill \end{array}$$

(19)

Equation (19) establishes the shifting property:

$$\langle {\delta }_{\mathrm{GFODDF}},\varphi \rangle ={\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}\varphi \left(0\right)$$

which shows that ${\delta }_{\mathrm{GFODDF}}$ acts as a point-evaluation distribution at $x=0$, scaled by the factor $\Gamma \left(\zeta \right)/\Gamma (\zeta -\eta +1)$. Throughout Section 3, the classical delta is denoted $\delta$ and the generalized version is written consistently ${\delta }_{GFODDF}$ to avoid ambiguity.

3.2. Scaling Property

The scaling property of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ states that scaling the argument by a factor a scales ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ by ${\displaystyle \frac{1}{\left|a\right|}}$:

$$\delta \left(ax\right)={\displaystyle \frac{1}{\left|a\right|}}\delta \left(x\right).$$

(20)

Proof

Starting with the definition of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$,

$${\delta }_{\mathrm{GFODDF}}\left(x\right)=D_{\mathrm{GFD}}H\left(x\right).$$

(21)

Applying the scaling property of the GFD operator on $H\left(x\right)$,

$$\delta \left(ax\right)={\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{\left(ax\right)}^{1-\eta }{\displaystyle \frac{dH\left(ax\right)}{d\left(ax\right)}}.$$

(22)

The derivative ${\displaystyle \frac{dH\left(ax\right)}{d\left(ax\right)}}$ is the sum of the Dirac delta distribution, simplifying the expression to ${\displaystyle \frac{1}{\left|a\right|}}\delta \left(x\right)$.

This completes the proof of the scaling property of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$.

For $a\in \mathbb{R}\setminus \left\{0\right\}$ and any test function $\phi$,

$$\langle \delta (a·),\phi \rangle ={\displaystyle \frac{1}{\left|a\right|}}\phi \left(0\right),$$

so, in distributional shorthand,

$$\delta \left(ax\right)={\displaystyle \frac{1}{\left|a\right|}}\delta \left(x\right).$$

3.3. Even Function Property

The ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ is an even function, which means $\delta (-x)=\delta \left(x\right)$.

3.3.1. Proof

Using the GFD definition,

$$\delta (-x)=D_{\mathrm{GFD}}H(-x).$$

(23)

Applying the GFD operator on $H(-x)$,

$$\delta (-x)={\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{(-x)}^{1-\eta }{\displaystyle \frac{dH(-x)}{d(-x)}}.$$

(24)

Since $H(-x)=1$ for $x<0$ and $H(-x)=0$ for $x\ge 0$, the derivative ${\displaystyle \frac{dH(-x)}{d(-x)}}$ will have a Dirac delta distribution at $t=0$ with a negative sign, resulting in $\delta (-x)=\delta \left(x\right)$. This completes the proof of the even function property of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$.

3.3.2. Evenness Property in the Distributional Setting

We now show that ${\delta }_{\mathrm{GFODDF}}$ is an even distribution, i.e.,

$${\delta }_{\mathrm{GFODDF}}(-x)={\delta }_{\mathrm{GFODDF}}\left(x\right)\mathrm{in}{\mathcal{D}}^{\prime }\left(\mathbb{R}\right).$$

Let $\varphi \in \mathcal{D}\left(\mathbb{R}\right)$ be an arbitrary test function. By the definition of the generalized fractional-order Dirac delta,

$$〈{\delta }_{\mathrm{GFODDF}}(-x),\varphi \left(x\right)〉=〈D_{\mathrm{GFD}}^{\eta ,\zeta }H(-x),\varphi \left(x\right)〉.$$

Perform the change of variable $y=-x$, noting that $dy=-dx$, to obtain

$$〈{\delta }_{\mathrm{GFODDF}}(-x),\varphi \left(x\right)〉=〈D_{\mathrm{GFD}}^{\eta ,\zeta }H\left(y\right),\varphi (-y)〉.$$

Because the GFD operator satisfies the reflection rule

$$D_{\mathrm{GFD}}^{\eta ,\zeta }\left[f(-y)\right]={(-1)}^{\lfloor \eta \rfloor }\left(D_{\mathrm{GFD}}^{\eta ,\zeta }f\right)(-y),$$

and $\eta \in (0,1]$ so $\lfloor \eta \rfloor =0$, we have

$$D_{\mathrm{GFD}}^{\eta ,\zeta }H\left(y\right)=D_{\mathrm{GFD}}^{\eta ,\zeta }H(-y){|}_{y\to -y}.$$

Thus,

$$〈{\delta }_{\mathrm{GFODDF}}(-x),\varphi \left(x\right)〉=〈{\delta }_{\mathrm{GFODDF}}\left(x\right),\varphi (-x)〉.$$

Finally, because $\varphi$ is an arbitrary test function, replacing $\varphi \left(x\right)$ by $\varphi (-x)$ produces

$$\langle {\delta }_{\mathrm{GFODDF}}(-x),\varphi \left(x\right)\rangle =\langle {\delta }_{\mathrm{GFODDF}}\left(x\right),\varphi \left(x\right)\rangle .$$

Therefore, ${\delta }_{\mathrm{GFODDF}}(-x)={\delta }_{\mathrm{GFODDF}}\left(x\right)$ in ${\mathcal{D}}^{\prime }\left(\mathbb{R}\right)$, proving that the generalized fractional-order Dirac delta is an even distribution.

3.4. Derivative Property

The derivative property of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ states that the derivative of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ is given by

$${\displaystyle \frac{d\delta \left(x\right)}{dx}}=D_{\mathrm{GFD}}\left({\displaystyle \frac{dH\left(x\right)}{dx}}\right).$$

(25)

Proof

Starting with the definition of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ using the GFD operator,

$$\delta \left(x\right)=D_{\mathrm{GFD}}H\left(x\right).$$

(26)

Taking the derivative of both sides concerning t,

$${\displaystyle \frac{d\delta \left(x\right)}{dx}}={\displaystyle \frac{d}{dx}}\left(D_{\mathrm{GFD}}H\left(x\right)\right).$$

(27)

Applying the definition of the GFD operator to the Heaviside function $H\left(x\right)$, we have the following:

$${\displaystyle \frac{d}{dx}}\left(D_{\mathrm{GFD}}H\left(x\right)\right)={\displaystyle \frac{d}{dx}}\left({\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{x}^{1-\eta }{\displaystyle \frac{dH\left(x\right)}{dx}}\right).$$

(28)

Since ${\displaystyle \frac{dH\left(x\right)}{dx}}=\delta \left(x\right)$, this becomes

$${\displaystyle \frac{d}{dx}}\left({\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{z}^{1-\eta }\delta \left(x\right)\right).$$

(29)

Simplifying further, we have

$${\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{\displaystyle \frac{d}{dx}}\left(x^{1-\eta }\delta \left(x\right)\right).$$

(30)

The derivative of the product $t^{1-\eta }\delta \left(x\right)$ involves using the product rule

$${\displaystyle \frac{d}{dx}}\left(x^{1-\eta }\delta \left(x\right)\right)=\left({\displaystyle \frac{d}{dx}}{x}^{1-\eta }\right)\delta \left(x\right)+x^{1-\eta }{\displaystyle \frac{d\delta \left(x\right)}{dx}}.$$

(31)

Since ${\displaystyle \frac{d}{dx}}{x}^{1-\eta }=(1-\eta )x^{-\eta }$, we obtain

$$(1-\eta )x^{-\eta }\delta \left(x\right)+x^{1-\eta }{\displaystyle \frac{d\delta \left(x\right)}{dx}}.$$

(32)

For $x=0$, the term $(1-\eta )x^{-\eta }\delta \left(x\right)$ vanishes, leaving us with the following:

$$x^{1-\eta }{\displaystyle \frac{d\delta \left(x\right)}{dx}}.$$

(33)

Thus, we obtain the following,

$${\displaystyle \frac{d\delta \left(x\right)}{dx}}={\displaystyle \frac{d}{dx}}\left(D_{\mathrm{GFD}}H\left(x\right)\right).$$

(34)

This completes the proof of the derivative property of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$.

The derivative of the Dirac delta is understood in the distributional sense:

for every test function $\phi \in \mathcal{D}\left(\mathbb{R}\right)$,

$$\langle {\delta }^{\prime },\phi \rangle =-\langle \delta ,{\phi }^{\prime }\rangle =-{\phi }^{\prime }\left(0\right).$$

This identity defines the distribution ${\delta }^{\prime }$.

3.5. Zero Value Property

The zero value property of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ states that

$${\int }_{-\infty }^{\infty }\delta \left(x\right)dx=1.$$

(35)

Proof

Using the definition of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$,

$$\delta \left(x\right)=D_{\mathrm{GFD}}H\left(x\right).$$

(36)

Integrating both sides over the entire real line,

$${\int }_{-\infty }^{\infty }\delta \left(x\right)dx={\int }_{-\infty }^{\infty }{D}_{\mathrm{GFD}}H\left(x\right)dx.$$

(37)

The integral of the GFD operator acting on $H\left(x\right)$ on all t is

$${\int }_{-\infty }^{\infty }{D}_{\mathrm{GFD}}H\left(x\right)dx={\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{\int }_{-\infty }^{\infty }{x}^{1-\eta }{\displaystyle \frac{dH\left(x\right)}{dx}}dx.$$

(38)

Since ${\displaystyle \frac{dH\left(x\right)}{dx}}=\delta \left(x\right)$, this simplifies to

$${\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{\int }_{-\infty }^{\infty }{x}^{1-\eta }\delta \left(x\right)dx.$$

(39)

Given that $\delta \left(x\right)$ is zero everywhere except at $x=0$, where it has a value such that its integral over all x is 1,

$${\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}·1=1.$$

(40)

Thus,

$${\int }_{-\infty }^{\infty }\delta \left(x\right)dx=1.$$

(41)

This completes the proof of the zero-value property of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$.

The Dirac delta is supported only at the origin.

If $\phi \in \mathcal{D}\left(\mathbb{R}\right)$ satisfies $\phi \left(0\right)=0$, then

$$\langle \delta ,\phi \rangle =\phi \left(0\right)=0.$$

Consequently, $\delta$ acts as the zero distribution on any test function that vanishes at 0.

3.6. Convolution Property

The convolution property of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ states that, when convolved with any function $g\left(x\right)$, it results in the original function $g\left(x\right)$:

$$g\left(x\right)\ast \delta \left(x\right)=g\left(x\right).$$

(42)

Proof

Starting with the definition of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ using the $D_{\mathrm{GFD}}$ operator,

$$\delta \left(x\right)=D_{\mathrm{GFD}}H\left(x\right).$$

(43)

The convolution integral is given by

$$(g\ast \delta )\left(x\right)={\int }_{-\infty }^{\infty }g\left(\tau \right)\delta (x-\tau )d\tau .$$

(44)

Substitute the definition of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ into the convolution integral:

$$(g\ast \delta )\left(x\right)={\int }_{-\infty }^{\infty }g\left(\tau \right)D_{\mathrm{GFD}}H(x-\tau )d\tau .$$

(45)

Apply the linearity of the GFD operator:

$$(g\ast \delta )\left(x\right)=D_{\mathrm{GFD}}{\int }_{-\infty }^{\infty }g\left(\tau \right)H(x-\tau )d\tau .$$

(46)

Since $H(x-\tau )$ is non-zero only when $x-\tau \ge 0$ (i.e., $\tau \le x$), the integral simplifies to

$$(g\ast \delta )\left(x\right)=D_{\mathrm{GFD}}{\int }_{-\infty }^{x}g\left(\tau \right)d\tau .$$

(47)

The integral ${\int }_{-\infty }^{x}g\left(\tau \right)d\tau$ is essentially the antiderivative of $f\left(x\right)$, denoted as $f^{\prime }\left(x\right)$. Therefore,

$$(g\ast \delta )\left(x\right)=D_{\mathrm{GFD}}{f}^{\prime }\left(x\right).$$

(48)

Applying the $D_{\mathrm{GFD}}$ operator to $f\left(x\right)$,

$$(g\ast \delta )\left(x\right)={\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{x}^{1-\eta }{\displaystyle \frac{d{f}^{\prime }\left(x\right)}{dx}}.$$

(49)

Since the derivative of an anti-derivative is the original function $f\left(x\right)$,

$$(g\ast \delta )\left(x\right)={\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{x}^{1-\eta }f\left(x\right).$$

(50)

This is equivalent to $f\left(x\right)$, proving the convolution property of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$.

Let $g\in \mathcal{D}\left(\mathbb{R}\right)$ (or $g\in \mathcal{S}\left(\mathbb{R}\right)$ for tempered distributions). The convolution of $\delta$ with g is defined by

$$(\delta \ast g)\left(x\right):=\langle \delta ,g(x-·)\rangle .$$

Since $\langle \delta ,g(x-·)\rangle =g\left(x\right)$,

$$(\delta \ast g)\left(x\right)=g\left(x\right).$$

More generally, for a translated delta ${\delta }_a$,

$$({\delta }_a\ast g)\left(x\right)=g(x-a).$$

4. Applications of ${\mathit{\delta }}_{\mathbf{GFODDF}}\mathbf{\left(}\mathit{X}\mathbf{\right)}$ in Solving Differential Equations

4.1. Solutions Using a Normal Dirac Delta Distribution and Generalized Fractional-Order Dirac Delta Distribution

We compare the solutions of the first-order differential equation

$${\displaystyle \frac{dy\left(t\right)}{dt}}=ay\left(t\right)+b\delta (t-t_0).$$

The normal Dirac delta distribution and the Generalized Fractional-Order Dirac delta distribution are used.

4.1.1. Normal Dirac Delta Distribution

Step 1: Differential equation

$${\displaystyle \frac{dy\left(t\right)}{dt}}=ay\left(t\right)+b\delta (t-t_0).$$

Step 2: Homogeneous solution Solve the homogeneous equation:

$${\displaystyle \frac{d{y}_h\left(t\right)}{dt}}=a{y}_h\left(t\right).$$

The solution is as follows:

$$y_h\left(t\right)=C{e}^{at}.$$

Step 3: Particular solution Consider the effect of the Dirac delta distribution:

$${\int }_{t_0^{-}}^{t_0^{+}}{\displaystyle \frac{dy\left(t\right)}{dt}}dt={\int }_{t_0^{-}}^{t_0^{+}}\left(ay\left(t\right)+b\delta (t-t_0)\right)dt.$$

Evaluating the integral gives the following:

$$y\left(t_0^{+}\right)-y\left(t_0^{-}\right)=b.$$

Step 4: General solution For $t<t_0$,

$$y\left(t\right)=y_0{e}^{at}.$$

For $t\ge t_0$,

$$y\left(t\right)=\left(y_0-{\displaystyle \frac{b}{e^{a{t}_0}}}\right)e^{at}+bH(t-t_0).$$

Step 5: Complete solution

$$y\left(t\right)=\left\{\begin{array}{cc}{y}_0{e}^{at},\hfill & \mathrm{for}t<t_0,\hfill \\ \left(y_0-{\displaystyle \frac{b}{e^{a{t}_0}}}\right)e^{at}+bH(t-t_0),\hfill & \mathrm{for}t\ge t_0.\hfill \end{array}\right.$$

4.1.2. Generalized Fractional-Order Dirac Delta Distribution

Step 1: Differential equation

$${\displaystyle \frac{dy\left(t\right)}{dt}}=ay\left(t\right)+b{D}_{\mathrm{GFD}}H(t-t_0).$$

Step 2: Definition of ${\delta }_{\mathrm{GFODDF}}(t-to)$

$$D_{\mathrm{GFD}}H(t-t_0)={\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{(t-t_0)}^{1-\eta }\delta (t-t_0).$$

Step 3: Substitute ${\delta }_{\mathrm{GFODDF}}(t-to)$ Substitute ${\delta }_{\mathrm{GFODDF}}(t-to)$ into the differential equation

$${\displaystyle \frac{dy\left(t\right)}{dt}}=ay\left(t\right)+b{\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{(t-t_0)}^{1-\eta }\delta (t-t_0).$$

Step 4: Homogeneous solution Solve the homogeneous equation:

$${\displaystyle \frac{d{y}_h\left(t\right)}{dt}}=a{y}_h\left(t\right).$$

The solution is as follows:

$$y_h\left(t\right)=C{e}^{at}.$$

Step 5: Particular solution Consider the effect of ${\delta }_{\mathrm{GFODDF}}(t-to)$:

$${\int }_{t_0^{-}}^{t_0^{+}}{\displaystyle \frac{dy\left(t\right)}{dt}}dt={\int }_{t_0^{-}}^{t_0^{+}}\left(ay\left(t\right)+b{\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{(t-t_0)}^{1-\eta }\delta (t-t_0)\right)dt.$$

Evaluating the integral gives the following:

$$y\left(t_0^{+}\right)-y\left(t_0^{-}\right)=b{\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}.$$

Step 6: General solution

For $t<t_0$,

$$y\left(t\right)=y_0{e}^{at}.$$

For $t\ge t_0$,

$$y\left(t\right)=\left(y_0-{\displaystyle \frac{b\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)e^{a{t}_0}}}\right)e^{at}+b{\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}H(t-t_0).$$

Step 7: Complete solution

$$y\left(t\right)=\left\{\begin{array}{cc}{y}_0{e}^{at},\hfill & \mathrm{for}t<t_0,\hfill \\ \left(y_0-{\displaystyle \frac{b\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)e^{a{t}_0}}}\right)e^{at}+b{\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}H(t-t_0),\hfill & \mathrm{for}t\ge t_0.\hfill \end{array}\right.$$

The behavior of the solution is depicted in Figure 5, which compares the conventional and generalized fractional-order delta distributions.

4.2. Fractional-Order Impulse Response Solution

4.2.1. Solution of the Differential Equation Using a General Dirac Delta Distribution

Consider the differential equation

$${\displaystyle \frac{dx\left(t\right)}{dt}}=k\delta (t-1),x\left(0\right)=0,$$

where $\delta (t-1)$ is the Dirac delta distribution, k is a constant, and $x\left(0\right)=0$ is the initial condition.

4.2.2. Step-by-Step Solution

1. Recognize the impulse property of the Dirac delta distribution:

$$\delta (t-1)=\left\{\begin{array}{cc}0,\hfill & \mathrm{for}t\ne 1,\hfill \\ \infty ,\hfill & \mathrm{for}t=1\hfill \end{array}\right.$$

and ${\int }_{-\infty }^{\infty }\delta (t-1)dt=1$.

2. Integrate both sides:

$${\int }_0^T{\displaystyle \frac{dx\left(\tau \right)}{d\tau }}d\tau ={\int }_0^{T}k\delta (\tau -1)d\tau .$$

This can be rewritten as

$$x\left(T\right)-x\left(0\right)=k{\int }_0^T\delta (\tau -1)d\tau .$$

Since $x\left(0\right)=0$, this simplifies to

$$x\left(T\right)=k{\int }_0^T\delta (\tau -1)d\tau .$$

3. Evaluate the integral:

$${\int }_0^T\delta (\tau -1)d\tau =\left\{\begin{array}{cc}0,\hfill & \mathrm{if}T<1,\hfill \\ 1,\hfill & \mathrm{if}T\ge 1.\hfill \end{array}\right.$$

4. Determine the solution:

$$x\left(T\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}T<1,\hfill \\ k,\hfill & \mathrm{if}T\ge 1.\hfill \end{array}\right.$$

4.2.3. Final Solution

Combining the results, the solution to the differential equation is as follows:

$$x\left(t\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}t<1,\hfill \\ k,\hfill & \mathrm{if}t\ge 1.\hfill \end{array}\right.$$

4.2.4. Solution of the Differential Equation Using the Generalized Fractional-Order Dirac Delta Distribution

We start by replacing the Dirac delta distribution with its generalized fractional-order Dirac delta distribution (${\delta }_{\mathrm{GFODDF}}\left(x\right)$) definition:

$$\delta (t-1)=D_{GFD}H(t-1),$$

where $H(t-1)$ is the Heaviside step function, and $D_{GFD}$ is the generalized fractional order operator.

The differential equation now becomes

$${\displaystyle \frac{dx\left(t\right)}{dt}}=k{D}_{GFD}H(t-1).$$

Substituting the definition of the generalized fractional derivative,

$$D_{GFD}H(t-1)={\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{(t-1)}^{1-\eta }{\displaystyle \frac{dH(t-1)}{dt}}.$$

Since the derivative of the Heaviside function $H(t-1)$ is the Dirac delta distribution $\delta (t-1)$, we obtain the following:

$$D_{GFD}H(t-1)={\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{(t-1)}^{1-\eta }\delta (t-1).$$

The term ${(t-1)}^{1-\eta }$ is zero everywhere except in $t=1$. The Dirac delta distribution $\delta (t-1)$ ensures that the non-zero contribution comes from $t=1$. The differential equation now is as follows:

$${\displaystyle \frac{dx\left(t\right)}{dt}}=k{\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{(t-1)}^{1-\eta }\delta (t-1).$$

Integrate both sides with respect to t:

$$x\left(t\right)={\int }_0^{t}k{\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{(t^{\prime }-1)}^{1-\eta }\delta (t^{\prime }-1)d{t}^{\prime }.$$

Using the shifting property of the delta distribution,

$$x\left(t\right)=k{\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{\int }_0^t{(t^{\prime }-1)}^{1-\eta }\delta (t^{\prime }-1)d{t}^{\prime },$$

$$x\left(t\right)=k{\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{(1-1)}^{1-\eta }H(t-1),$$

$$x\left(t\right)=k{\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}·0^{1-\eta }H(t-1).$$

Given the behavior of $0^{1-\eta }$ for $\eta <1$, and the Heaviside step function $H(t-1)$ that turns on at $t=1$, the solution can be simplified:

$$x\left(t\right)=k{\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}H(t-1).$$

Therefore, the solution to the differential equation is as follows:

$$x\left(t\right)=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}t<1,\hfill \\ k,{\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}\hfill & \mathrm{if}t\ge 1.\hfill \end{array}\right.$$

An explicit solution of the differential equation has been obtained. The solution’s properties can further examined through plotted graphs in Figure 6.

4.2.5. Assumptions for Forcing Terms Involving ${\delta }_{\mathrm{GFODDF}}$

Throughout this section, we model the forcing of a differential equation by ${\delta }_{\mathrm{GFODDF}}(t-t_0)$. To avoid ambiguity, we now state the analytical setting and assumptions.

• Distributional Framework

Let $\mathcal{D}\left(\mathbb{R}\right)$ be the space of smooth, compactly supported test functions and ${\mathcal{D}}^{\prime }\left(\mathbb{R}\right)$ its dual. The generalized fractional-order Dirac delta ${\delta }_{\mathrm{GFODDF}}$ is treated as a tempered distribution defined by

$$〈{\delta }_{\mathrm{GFODDF}},\varphi 〉:=〈D_{\mathrm{GFD}}^{\eta ,\zeta }H,\varphi 〉,$$

for every $\varphi \in \mathcal{D}\left(\mathbb{R}\right)$. All subsequent operations, including convolution and differentiation, are understood in ${\mathcal{D}}^{\prime }\left(\mathbb{R}\right)$.

• Fractional Impulse Property

The key assumption is that ${\delta }_{\mathrm{GFODDF}}$ acts as a fractional impulse: for any test function $\varphi$,

$$〈{\delta }_{\mathrm{GFODDF}}(t-t_0),\varphi \left(t\right)〉={\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}\varphi \left(t_0\right).$$

This follows from the shifting property established in Section 3.1 and ensures that ${\delta }_{\mathrm{GFODDF}}$ produces a point-wise forcing at $t=t_0$, scaled by $\Gamma \left(\zeta \right)/\Gamma (\zeta -\eta +1)$.

• Justification for Forcing Terms

Given a linear fractional differential operator $\mathcal{L}$, the equation

$$\mathcal{L}y\left(t\right)=b{\delta }_{\mathrm{GFODDF}}(t-t_0)$$

is interpreted in ${\mathcal{D}}^{\prime }\left(\mathbb{R}\right)$ as

$$\langle \mathcal{L}y,\varphi \rangle =b{\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}\varphi \left(t_0\right)\forall \varphi \in \mathcal{D}\left(\mathbb{R}\right),$$

which is the natural fractional analogue of the classical delta-forced problem. This provides a rigorous justification for employing ${\delta }_{\mathrm{GFODDF}}(t-t_0)$ in the forcing term. These assumptions clarify that the generalized fractional delta is not used as an ordinary function but as a distribution with a well-defined point evaluation property, which ensures that all differential equations in Section 5 are interpreted consistently.

5. Application: Modeling Drug Release with Fractional Kinetics

The generalized fractional-order Dirac delta distribution (${\delta }_{\mathrm{GFODDF}}\left(x\right)$) proposed in this study has strong potential applications in the modeling of complex biological systems, particularly in the domain of pharmacokinetics where fractional calculus has gained increasing attention. One such application is in the modeling of drug release and absorption profiles that exhibit non-local memory effects and anomalous diffusion behavior, commonly encountered in biological tissues.

In conventional pharmacokinetic models, drug administration events, such as intravenous injections, are typically represented using the classical Dirac delta distribution $\delta \left(x\right)$ due to its impulse-like nature. However, this approach often fails to accurately describe the subsequent temporal dynamics of drug transport, especially in media where fractional diffusion processes dominate. These include scenarios involving heterogeneous tissue structures, viscoelastic intercellular media, or complex drug–carrier interactions, where diffusion and absorption follow power law behaviors rather than exponential laws.

The ${\delta }_{\mathrm{GFODDF}}\left(x\right)$, by incorporating the generalized fractional derivative (GFD) operator, inherently accounts for such non-integer-order dynamics. Its structure enables a more accurate and flexible modeling of fractional-order impulse responses, making it particularly suitable for characterizing drug release processes that involve long memory kernels and sub-diffusive transport mechanisms. When incorporated into fractional differential equations that describe the concentration of the drug $C\left(t\right)$ over time, ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ can capture both the instantaneous injection effect and the diffusion of memory that follows. This makes it a superior alternative to the classical delta distribution in the modeling of real-world pharmacokinetic systems. For instance, a fractional differential model of the form

$${\displaystyle \frac{d^{\eta }C\left(t\right)}{d{t}^{\eta }}}=-kC\left(t\right)+D·{\delta }_{\mathrm{GFODDF}}(t-t_0)$$

(51)

can simulate a drug administered at time $t=t_0$, where the drug undergoes an anomalous clearance or absorption governed by the fractional derivative of order $\eta \in (0, 1)$. The presence of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ in such models allows for an accurate representation of impulse initiation coupled with memory-aware kinetics, providing a realistic depiction of drug dynamics, particularly in slow-release formulations or targeted delivery mechanisms. One can easily obtain the solution of Equation (51) in the following way:

$$C\left(t\right)=C_0{E}_{(\eta ,1)}(–k{t}^{\eta })+DKH(t–t_0){(t-t_0)}^{(\eta -1)}{E}_{\eta ,\eta }(–k{(t-t_0)}^{\eta }).$$

(52)

When $\eta =1$, the Mittag–Leffler functions reduce to the exponential:

$$C\left(t\right)=C_0{e}^{(-kt)}+DKH(t-t_0)e^{(-k(t-t_0))}.$$

The classical ordinary differential equation is presented. A graphical representation of the solution in Equation (52) is presented in Figure 7 and Figure 8. (The derivation of Equation (52) is explained at the end of this section).

Figure 7 presents the total response $C\left(t\right)$ showing a singular behavior in $t=t_0$ followed by a gradual decay, capturing the non-local memory effect inherent in fractional-order systems. Figure 8 presents the homogeneous part that corresponds to the natural decay of the system’s memory, while the forced part reflects the external excitation. Their superposition yields the total fractional response, highlighting the influence of fractional dynamics near $t=t_0$.

In summary, the introduction of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ not only advances the mathematical treatment of singular sources in fractional systems, but also opens new avenues in biomedical engineering, drug delivery modeling, and clinical pharmacokinetics, where precision and physiological fidelity are critical.

The biological illustration (bottom-right panel) in Figure 9 (generated by GPT-5) conceptually represents how GFODDF models drug transport in complex biological systems. It begins with a drug injection event, symbolizing an initial impulse in the system. The drug then enters a heterogeneous tissue environment, where the molecules encounter irregular structures, such as cell membranes and the extracellular matrix, as well as varying viscosities, which disrupt uniform diffusion. Unlike classical models, which assume smooth, memoryless transport, the GFODDF framework captures fractional diffusion, a process where past states influence current dynamics, leading to slower, long-tailed concentration profiles. This “memory effect” mirrors the real-world pharmacokinetic behavior in scenarios such as slow-release drug formulations, targeted delivery, and drug–carrier interactions in viscoelastic tissues.

Figure 10 provides a visual comparison between the classical Dirac delta distribution and the Generalized Fractional-Order Dirac delta distribution (GFODDF), along with their implications in pharmacokinetic modeling.

In the top-left panel, the classical delta is shown as an instantaneous spike, representing an idealized impulse with no persistence in time.

In contrast, the top-right panel depicts the GFODDF, which retains the sharp initial peak, but introduces a fractional decay tail, capturing the memory effects and long-range influence characteristic of fractional-order systems.

The bottom left panel compares drug concentration profiles after administration: the classical model shows an exponential decay typical of memoryless systems, while the GFODDF-based fractional model exhibits a slower, power-law decay, reflecting anomalous diffusion and long-term retention in biological media.

Finally, the bottom-right panel provides a conceptual schematic of the pharmacokinetic process, starting from drug injection, followed by diffusion through a heterogeneous medium and culminating in fractional diffusion with memory effects, which illustrates the enhanced physiological fidelity offered by the GFODDF framework.

Derivation of the Pharmacokinetic Solution (Equation (52))

Consider the fractional-order pharmacokinetic model

$$D_t^{\eta }C\left(t\right)=-kC\left(t\right)+D{\delta }_{\mathrm{GFODDF}}(t-t_0),0<\eta \le 1,$$

(53)

where $D_t^{\eta }$ is the Caputo fractional derivative, $k>0$ is the elimination rate, and D measures the impulse strength.

1.

Homogeneous solution.

The homogeneous equation

$$D_t^{\eta }{C}_h\left(t\right)=-k{C}_h\left(t\right)$$

has the well–known solution

$$C_h\left(t\right)=C_0{E}_{\eta ,1}\left(-k{t}^{\eta }\right),$$

where $E_{\eta ,1}$ is the one-parameter Mittag–Leffler function.

2.

Particular solution for the impulse at $t_0$.

Using the shifting property of the generalized fractional delta Section 3), the forcing term contributes

$$D{\delta }_{\mathrm{GFODDF}}(t-t_0)=D{\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{(t-t_0)}^{1-\eta }\delta (t-t_0)$$

in the sense of distributions. Integrating the equation across $t=t_0$ yields the jump

$$C\left(t_0^{+}\right)-C\left(t_0^{-}\right)=D{\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}.$$

3.

Solution for $t>t_0$.

For $t\ge t_0$, the problem reduces to

$$D_t^{\eta }{C}_f\left(t\right)+k{C}_f\left(t\right)=0$$

with initial value $C_f\left(t_0\right)=D{\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}.$ The solution is

$$C_f\left(t\right)=D{\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{E}_{\eta ,1}\left[-k{(t-t_0)}^{\eta }\right].$$

4.

Total concentration.

By superposition,

$$C\left(t\right)=C_0{E}_{\eta ,1}\left(-k{t}^{\eta }\right)+D{\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}H(t-t_0){(t-t_0)}^{\eta -1}{E}_{\eta ,\eta }\left[-k{(t-t_0)}^{\eta }\right]$$

(54)

which is Equation (52) in the main text. When $\eta =1$, $E_{1,1}\left(z\right)=e^z$, and Equation (52) reduces to the classical exponential decay model, validating the fractional generalization.

6. Discussions and Conclusions

In-depth exploration of the generalized fractional-order Dirac delta distribution (${\delta }_{\mathrm{GFODDF}}\left(x\right)$), which is defined as

$${\delta }_{\mathrm{GFODDF}}\left(x\right)={\displaystyle \frac{\Gamma \left(\zeta \right)}{\Gamma (\zeta -\eta +1)}}{(x-1)}^{1-\eta }{\displaystyle \frac{dH\left(x\right)}{dx}},$$

has been conducted in this article. From integrating the Generalized Fractional Derivative ($D_{\mathrm{GFD}}$) operator with the Heaviside step function, a robust framework has been developed that enhances the understanding and utility of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ in complex dynamic systems. The solutions of differential equations involving the Dirac delta distribution, particularly for differential equations of the form

$${\displaystyle \frac{{\mathrm{d}}^{\eta }x\left(t\right)}{\mathrm{d}{t}^{\eta }}}=k{\delta }_{\mathrm{GFODDF}}(t-t_0)$$

and

$${\displaystyle \frac{dy\left(t\right)}{dt}}=ay\left(t\right)+b\delta (t-t_0)$$

have been given as illustrative examples that provide a basis for the efficacy of ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ in representing impulse responses in fractional-order systems. These reported results confirm that ${\delta }_{\mathrm{GFODDF}}\left(x\right)$ serves as a powerful mathematical tool, offering new avenues for researchers and practitioners alike. This work expands the horizon of fractional differential equations and generalized order systems.

A particularly noteworthy application of ${\delta }_{GFODDF}\left(x\right)$ lies in biological and biomedical systems, especially in modeling drug release mechanisms governed by fractional kinetics. The conventional delta distribution often fails to account for the anomalous diffusion and delayed absorption patterns observed in heterogeneous biological tissues. In contrast, ${\delta }_{GFODDF}\left(x\right)$ allows for more accurate modeling of these processes by representing impulse-like drug administration events while retaining fractional memory effects. This capability has significant implications for drug delivery design, pharmacokinetic analysis, and bioengineering systems, where fractional-order models are increasingly recognized as necessary to reflect physiological realism.

In this work, we have introduced a novel mathematical structure, “The Generalized Fractional-Order Dirac delta distribution ${\delta }_{GFODDF}\left(x\right)$”, by applying the Generalized Fractional Derivative (GFD) operator to the classical Heaviside step function. This new definition extends the conventional Dirac delta distribution to the fractional domain, allowing it to capture both singular behavior and memory effects that are typical in complex systems. The generalized fractional-order Dirac delta distribution ${\delta }_{\mathrm{GFODDF}}$ extends the classical Dirac delta to non-integer orders while preserving key properties such as shifting, scaling, and convolution within the framework of generalized fractional derivatives, allowing its use as a forcing term in fractional differential equations with a tunable memory effect controlled by the order parameter $\eta$. A conceptual fractional-pharmacokinetics example illustrates how this added flexibility models anomalous diffusion and long-tail drug concentration profiles beyond classical integer-order models. However, the work remains preliminary: the pharmacokinetic example lacks empirical parameter estimation and patient-specific data, numerical schemes for PDEs with ${\delta }_{\mathrm{GFODDF}}$ forcing have not been tested for stability or convergence, and the influence of the auxiliary parameter $\zeta$ requires detailed sensitivity analysis. Future efforts will focus on parameter estimation studies using experimental data, the development and benchmarking of numerical solvers, and applications in signal processing, viscoelastic materials, and control theory, highlighting both the broad potential of ${\delta }_{\mathrm{GFODDF}}$ and the steps needed for practical data-driven implementation.