On fractional Caputo operator for the generalized glucose supply model via incomplete Aleph function

1. Origination

The branch of mathematics known as the study of special functions has been around for a very long time; its united and rather complete thesis comes from 17th century.^1,2,3 From the perspective of applied scientists and mathematicians who actually employ differential equations, the use of special functions as a key tool for precise analysis is seen. Special functions are certain fine operations that are significant in fine analysis, functional analysis, figure, medicines, or other processes. They have more or less fixed titles and memos. Bessel functions, gamma and beta functions, hypergeometric and confluent hypergeometric functions. Some very important divisions of special functions^{4,5,6,7,8,9,10,11,12} include Legendre functions, integral functions, probability integrals, lively classes of orthogonal polynomials in one or more variables, elliptic functions and integrals, and Riemann zeta functions. It is possible to create a math for related drivers that generalizes the classical bone while also defining real or complex number powers of isolation driver D and the integration driver J in a variety of ways. Fractional mathematics has grown in popularity and importance over the past forty years, partly because of its shown uses in several apparently distinct and vast areas of knowledge and engineering.^13,14,15,16 It does really include a variety of characteristics that may be useful for dealing with differential, integral, and differ-integral equations as well as other problems. Numerous scientific and industrial fields,^{17,18,19,20,21,22,23} including optics, visco-elasticity, fluid mechanics, electro-chemistry, natural population models, and signal recycling, use fractional mathematics. It is used to imitate mechanical and physical operations so fractional differential equations can be eloquently described.

The current work will analyze a diabetic individual^22,24 using fractional calculus and an essential unique function called Aleph. Nowadays, diabetes is proving to be a lethal condition. An insulin and blood sugar level imbalance is the main cause of diabetes. Additionally, there are two types of diabetes. Type I diabetes is more serious and prevalent than Type II diabetes. Type-I diabetes affects 10% of diabetics whereas Type-II diabetes affects the remaining 90%, according to a research. Therefore, it was crucial to investigate this contradiction from a scientific standpoint.

Normal human bodies maintain blood sugar levels within acceptable ranges. Glycogenesis is the process through which excess glucose is converted into glycogen and stored in the liver and muscle cells. However, anytime the blood glucose level falls within the range of normal glucose levels, the glycogenesis is a metabolic process that causes the blood glucose level to return to the normal range in an acceptable amount of time. Glycogen stored in the liver and muscles is transformed into glucose-1-phosphate and eventually glucose-6-phosphate during glycogenosis. While the process of creating glucose from non-carbohydrate sources is known as gluconeogenesis. Thus, all three of these metabolic activities keep the blood sugar level within generally acceptable ranges. The metabolic mechanisms are thus engaged whenever the blood sugar level exceeds certain limits that the glucose level returns to normal within an acceptable amount of time. This time period is longer for diabetes patients than it is for average people. A glucose tolerance test can be used to determine the blood glucose level.

2. Pre-Requisites

Fractional calculus is an important and essential tool in mathematical field nowadays. It has lots of practical and real-world applications in almost every field of life like engineering, science, medical science, etc. Followings are the basic information about the functions and operators used to prepare the presented paper.

2.1. Riemann–Liouville fractional derivative

The left inverse of Riemann–Liouville fractional integral of exponent $\alpha ,m-1<\alpha <m,m\in N$, is known as Riemann–Liouville fractional derivative and is explained as

where f is locally integrable function, $\Re (\mu )$ represents real portion of complex number $\mu \in C$ and the term $[\Re (\mu )]$ refers largest integer in $\Re (\mu )$. Contrarily, we see that Caputo gave a different definition for fractional derivative.

2.2. Caputo fractional derivative

The Caputo derivative²⁵ is another way for computing fractional derivatives. Caputo first mentioned it in his paper from 1967. While resolving differential equations, it is not necessary to specify the fractional order starting constraints using Caputo’s approach, in contrast to Riemann–Liouville fractional derivative. The definition of Caputo is demonstrated as follows :

and

2.3. Aleph function

Since Aleph function is the highest generalized known function till now, rest all are some particular cases of Aleph function so we have taken this function rather than taking some particular function for our analysis. The Aleph function^26,27 is defined as follows:

for all $\mathrm{\text{z}}=0$ here $\omega =\sqrt{-1}$ and The integration path $\Im ={\Im }_{\omega r\infty }$, $r\in R$ goes from $\gamma -\omega \infty$ to $\gamma +\omega \infty$ is s.t. poles of Gamma functions $\mathrm{\Gamma }(1-a_j-A_{j}s)$, $j=\overline{1,n}$ do not overlap with pole of Gamma function $\mathrm{\Gamma }(b_j+B_{j}s)$, $j=\overline{1,m}$. The parameters $p_i,q_i$ are non-negative integers satisfying for $0\le n\le p_i$, $0\le m\le q_i$, $c_i>0$ for $i=\overline{1,r}$. The parameters $A_j,B_j,A_{ji},B_{ji}$ are positive numbers and $a_j,b_j,a_{ji},b_{ji}$ are complex.

2.4. Multi-variable Aleph function

The multiple contour integral is used to define the multi-variable Aleph function²¹:

with $\omega =\sqrt{-1}$ and

$$\begin{array}{l}A=\{{(a_j;{\alpha }_j^{(1)},\dots ,{\alpha }_j^{(r)})}_{1,n}\},\\ \{{\tau }_i{(a_{ji};{\alpha }_{ji}^{(1)},\dots ,{\alpha }_{ji}^{(r)})}_{n+1,p_i}\},\{{(c_j^{(1)};{\gamma }_j^{(1)})}_{1,n_1}\},\\ \{{\tau }_{i^{(1)}}{(c_{j{i}^{(1)}}^{(1)};{\gamma }_{j{i}^{(1)}}^{(1)})}_{n_1+1,p_{i^{(1)}}}\},\dots ,\{{(c_j^{(r)};{\gamma }_j^{(r)})}_{1,n_r}\},\{{\tau }_{i^{(r)}}{(c_{j{i}^{(r)}}^{(r)};{\gamma }_{j{i}^{(r)}}^{(r)})}_{n_r+1,p_{i^{(r)}}}\},\end{array}$$(2.5)

$$\begin{array}{l}B=\{{\tau }_i{(b_{ji};{\beta }_{ji}^{(1)},\dots ,{\beta }_{ji}^{(r)})}_{m+1,q_i}\},\{{(d_j^{(1)};{\delta }_j^{(1)})}_{1,m_1}\},\\ \{{\tau }_{i^{(1)}}{(d_{j{i}^{(1)}}^{(1)};{\delta }_{j{i}^{(1)}}^{(1)})}_{m_1+1,q_{i^{(1)}}}\},\dots ,\{{(d_j^{(r)};{\delta }_j^{(r)})}_{1,m_r}\},\{{\tau }_{i^{(r)}}{(d_{j{i}^{(r)}}^{(r)};{\delta }_{j{i}^{(r)}}^{(r)})}_{m_r+1,q_{i^{(r)}}}\},\end{array}$$(2.6)

and Extending the pertinent multi-variable H-function criterion below will result if numerous Mellin–Barnes-type contours are absolutely converged : where and $k=1,\dots ,r$, $i=1,\dots ,R$ and $i^{(k)}=1,\dots ,R^{(k)}$.

Complex numbers $z_i$ do not have a zero value. In this, the multi-variable Aleph function’s existence and absolute convergence conditions are discussed. This might be written as

and where $k=1,2,\dots ,r$; ${\alpha }_k=min[\mathrm{\text{Re}}(d_j^{(k)}/{\delta }_j^{(k)}]$, $j=1,\dots ,m_k$ and ${\beta }_k=max[\mathrm{\text{Re}}((c_j^{(k)}-1)/{\gamma }_j^{(k)}]$, $j=1,\dots ,n_k$.

2.5. Caputo derivative of incomplete Aleph function

Now, we define Caputo derivative of Incomplete Aleph function ${}^{(\mathrm{\Gamma })}{\aleph }_{p_i,q_i,{\rho }_i;r}^{m,n}(z{x}^{\delta }{t}^{\theta })$ for $n-1<\alpha <n\in N$ as below:

provided that $\theta >0$, $\zeta >0$, $1+\zeta \theta >n$, $|arg(z)|<\frac{1}{\pi }{k}_i$ and $\Re (L_i+1)<0$. Similarly, we can get the Caputo derivative of ${}^{(\gamma )}{\aleph }_{p_i,q_i,{\rho }_i;r}^{m,n}(z{x}^{\delta }{t}^{\theta })$ for $n-1<\alpha <n$ as below: provided that $\theta >0$, $\zeta >0$, $1+\zeta \theta >n$, $|arg(z)|<\frac{1}{\pi }{k}_i$ and $\Re (L_i+1)<0$.

If we talk about the novelty of my work, then till now no one has found the solution of blood glucose level in terms of Aleph function which is the highest known generalized function till now.

There are four sections throughout the entire document. The introduction of fractional calculus and special functions, as well as a brief history, are covered in Sec. 1. The prerequisites for the topic are covered in the second section. The model under examination is covered in the next section. We summarized our findings in Sec. 4, and in the final part, we thanked the writers and researchers whose papers and research effort had a positive impact on our conclusions. The references are so included in the last section.

3. Mathematical Model and Solution

In this section, at first we are going to explain the mathematical model under consideration and by which glucose tolerance test is governed and analyzed. The liver serves as a deep compartment for the glucose tolerance test, providing blood with glucose at a consistent rate k. The mathematical expression for $Y(t)$, the glucose quantity in the blood at any moment t, is as follows :

where $\lambda$ is the constant of proportionality known as transfer coefficient. $\lambda$ is assumed to be independent of t. This transfer coefficient may be taken as rate of change of sugar per unit amount of sugar.

Further, we change glucose equation (3.1) by using Caputo fractional operator :

Now we are going to find the solution of Eq. (3.2) by using the incomplete Aleph function. For this purpose, we take $Y(x,t)$ as a function that can be represented in terms of incomplete Aleph function ${}^{(\mathrm{\Gamma })}{\aleph }_{p_i,q_i,{\rho }_i;r}^{m,n}(z{x}^{\delta }{t}^{\theta })$ given as Now the Caputo fractional derivative of above Eq. (3.3) for $0<\alpha <1$ is given as provided that $\theta >0$, $\zeta >0$, $1+\zeta \theta >n$, $|arg(z)|<\frac{1}{\pi }{k}_i$ and $\Re (L_i+1)<0$. Now using Eqs. (3.3) and (3.4) in Eq. (3.2), we get following result: provided that $\theta >0$, $\zeta >0$, $1+\zeta \theta >n$, $|arg(z)|<\frac{1}{\pi }{k}_i$ and $\Re (L_i+1)<0.$ Similarly, we can find the result in terms of ${}^{(\gamma )}{\aleph }_{p_i,q_i,{\rho }_i;r}^{m,n}(z{x}^{\delta }{t}^{\theta })$ as provided that $\theta >0$, $\zeta >0$, $1+\zeta \theta >n$, $|arg(z)|<\frac{1}{\pi }{k}_i$ and $\Re (L_i+1)<0$.

Equations (3.5) and (3.6) give the solution of glucose level in human blood in terms of incomplete Aleph function. Now, we check the validity of the applied methodology, we have also found some particular cases by using some specific values and functions which are already discovered by renowned mathematicians and researchers as well. The particular cases are mentioned below:

3.1. Particular cases

(a)	If we put ${\rho }_1={\rho }_2={\rho }_3=\cdots ={\rho }_r=1$ and replace incomplete Aleph function by Aleph function then we get the result obtained by Chaurasia and Jain.
(b)	If we put $r={\rho }_1={\rho }_2={\rho }_3=\cdots ={\rho }_r=1$ and replace incomplete Aleph function by Aleph function then we get the result obtained by Srivastava and Srivastava.
(c)	If we put $\alpha =1$ and replace incomplete Aleph function by Aleph function then we get the result obtained by Dubey.
(d)	The Aleph function may be reduced to numerous other straightforward single-variable functions, including the H-function, G-function, and E-function, by specializing the parameters.28,29,30,31,32,33,34,35

4. Conclusion

In this study, we have analyzed the human blood glucose level using a novel approach. Given how significant a problem it is globally and how much study has been done on it. Here, the glucose level has been analyzed using the Caputo fractional operator in conjunction with the incomplete Aleph function. Additionally, we have obtained a few noteworthy and significant outcomes that support the applicability of our methodology. The paper’s results, which are obtained in a general form, include a variety of intriguing scenarios and are highly beneficial for the literature on applied mathematics and related subjects.

Availability of Statistics and Materials

Availability of statistics is already cited in paper.

Funding Information

No funding available.

Statement of Disagreement

According to researchers, there are no conflicts of interest to disclose about the paper that is being presented.

Author’s Contribution

The study was directed by Ravi Shanker Dubey, who also analyzed the findings. Himani Agarwal organized the necessary research materials while Manvendra Narayan Mishra prepared the paper and carried out all the mathematical computations. The draft was read, corrected, and polished by all authors.

ORCID

Himani Agarwal  https://orcid.org/0009-0002-6699-6835

Manvendra Narayan Mishra  https://orcid.org/0000-0002-6471-3574

Ravi Shanker Dubey  https://orcid.org/0000-0002-3019-4438