Fractional Operators Associated with the -Extended Mathieu Series by Using Laplace Transform
In this paper, our leading objective is to relate the fractional integral operator known as -transform with the -extended Mathieu series. We show that the -transform turns to the classical Laplace transform; then, we get the integral relating the Laplace transform stated in corollaries. As corollaries and consequences, many interesting outcomes are exposed to follow from our main results. Also, in this paper, we have converted the -transform into a classical Laplace transform by changing the variable ; then, we get the integral involving the Laplace transform.
Fractional calculus is a fast-growing field of mathematics that shows the relations of fractional-order derivatives and integrals. Fractional calculus is an effectual subject to study many complex real-world systems. In recent years, many researchers have calculated the properties, applications, and extensions of fractional integral and differential operators involving the various special functions.
Fernandez and Baleanu  showed that many other named models of fractional calculus can fit within the class of operators defined by Prabhakar and that this class contains both singular and nonsingular operators together. They also characterized completely the cases in which these operators are singular or nonsingular and the cases in which they can be written as finite or infinite sums of Riemann–Liouville differ integrals, to obtain finally a catalogue of subclasses with different types of properties.
Atangana and Baleanu  proposed a new fractional derivative with nonlocal and nonsingular kernel. They presented some useful properties of the new derivative and applied it to solve the fractional heat transfer model.
Atangana and Koca  presented relationship of derivatives with some integral transform operators. New results are presented. They applied the derivative to a simple nonlinear system. They showed in detail the existence and uniqueness of the system solutions of the fractional system. They obtained a chaotic behavior which was not obtained by local derivative.
Atangana and Baleanu  extended the model of the movement of subsurface water via the geological formation called aquifer using a newly proposed derivative with fractional order. An alternative derivative to that of Caputo-Fabrizio with fractional order was presented. The relationship between both derivatives was presented. The new equation was solved analytically using some integral transforms. The exact solution is therefore compared to experimental data obtained from the settlement of the University of the Free State in South Africa. The numerical simulation shows the agreement of the experimental data with an analytical solution for some values of fractional order.
Manzoor et al.  used a Beta operator with Caputo (MSM) fractional differentiation of extended Mittag-Leffler function in terms of Beta function. They applied the Beta operator on the right-sided MSM fractional differential operator and on the left-sided MSM fractional differential operator. They also applied the Beta operator on the right-sided MSM fractional differential operator with Mittag-Leffler function and the left-sided MSM fractional differential operator with Mittag-Leffler function.
Yavuz  investigated the novel solutions of fractional-order option pricing models and their fundamental mathematical analyses. The main novelties of this paper are the analysis of the existence and uniqueness of European-type option pricing models providing to give fundamental solutions to them and a discussion of the related analyses by considering both the classical and generalized Mittag-Leffler kernels. Yavuz and Abdeljawad  presented a fundamental solution method for nonlinear fractional regularized long-wave (RLW) models. Since analytical methods cannot be applied easily to solve such models, numerical or semianalytical methods have been extensively considered in the literature.
Jena et al.  applied two-hybrid techniques, namely, -homotopy analysis Elzaki transform method (-HAETM) and iterative Elzaki transform method (IETM) to obtain the numerical solutions of time-fractional Navier-Stokes equations in polar coordinate described in the Caputo sense. -HAETM is the combination of the homotopy analysis method and Elzaki transform method, and IETM is the combination of two reliable methods, i.e., iterative method and Elzaki transform method.
Yavuz and Sene  address the solution of the incompressible second-grade fluid models. Fundamental qualitative properties of the solution are primarily studied for proving the adequacy of the physical interpretations of the proposed model. They used the Liouville-Caputo fractional derivative with its generalized version that gives more comprehensive physical results in the analysis and investigations.
Yavuz  analyzed the behaviors of two different fractional derivative operators defined in the last decade. One of them is defined with the normalized sinc function (NSF) and the other one is defined with the Mittag-Leffler function (MLF). Both of them have a nonsingular kernel. Yavuz and Bonyah  examined the schistosomiasis fractional-order dynamic model via exponential law kernel sense and Mittag-Leffler kernel in Liouville–Caputo sense. Some special solutions for two operators are obtained using the iterative scheme through Laplace transform and Sumudu–Picard integration technique. The uniqueness and existence of solution for both operators are established. The numerical solutions for both operators approve that the desirable results can be obtained when the alpha value is less than one.
Yang  addressed a class of the fractional derivatives of constant and variable orders for the first time. Fractional-order relaxation equations of constants and variable orders in the sense of Caputo type are modeled from a mathematical point of view. The comparative results of the anomalous relaxation among the various fractional derivatives are also given. Yang  proposed the general Riemann-Liouville and Caputo-Liouville fractional derivatives with nonsingular power-law kernels, for the first time. New general laws of deformation within the framework of the general fractional derivatives are considered in detail. The creep and relaxation behaviors of the general fractional-order Voigt and Maxwell models are also obtained with the use of the Laplace transform.
The series is known as the Mathieu series. The first person to present such a sequence was Mathieu . Emersleben  in elegant form gives an essential meaning of the integral demonstration
The above series is also written in the expressions of Riemann-Zeta function by Choi and Srivastava  as given below: where and .
The generalized form of the Mathieu series by Cerone and Lenard  is given below as we have where and ; also, it can be written in the expression of Riemann-Zeta function by Pogány et al.  as given below: where and .
Taking the above equation in mind, let the -extended Mathieu series by Pogany and Parmar  be well-defined as where , , , and stands for the -extended Riemann zeta function by Chaudhry et al.  which is defined as where and .
Shah et al.  studied a compartmental mathematical model for the transmission dynamics of the novel coronavirus-19 under Caputo fractional order derivative. By using the fixed-point theory of Schauder and Banach, they established some necessary conditions for existence of at least one solution to model under investigation and its uniqueness. After the existence, a general numerical algorithm based on the Haar collocation method is established to compute the approximate solution of the model.
Sher et al.  studied the novel coronavirus (2019-nCoV or COVID-19) which is a threat to the whole world nowadays. They considered a fractional-order epidemic model which describes the dynamics of COVID-19 under nonsingular kernel type of fractional derivative. An attempt is made to discuss the existence of the model using the fixed-point theorem of Banach and Krasnoselskii type.
Definition 1. As indicated in Pohlen , let and be two power series; then, the Hadamard product of power series is defined as where where and stand for radius of convergence of the above series and , respectively. Therefore, in general, it is to be noted that if the one power series is the analytical function, then the series of Hadamard products is also the same as an analytical function.
Definition 2. The Gaussian hypergeometric function or ordinary hypergeometric function by Rainville  well-defined like is the special function which is represented by the hypergeometric series for “” neither zero nor a negative integer; then, the above notation is Also, and .
Definition 3. The Laplace transform of the function on interval by Sneddon  is defined as where and .
Definition 4. The Elzaki transform for the function of exponential order by Elzaki  is considered the function in the set ; we get For the function which is constant in set , , it must be a finite number, so maybe and are finite or infinite at that time the Elzaki transform denoted through operator as given below: where .
Definition 5. Let the function be integrable with a finite interval, ; if there exists a real number “,” then each of the following statements holds true, so, as, approaches to a finite limit like ; also, here, , approaches to a finite limit like ; then, the -transform,
exists whenever , .
The power function of the transform by Kumar  and Nadir and Khan  is given below:
2. The -Transform Associated with the -Extended Mathieu Series
Here, we have evaluated the -transform associated with the -extended Mathieu series and some of its certain cases in the form of corollaries.
Theorem 6. Let the -extended Mathieu series be given in (6) as where , , and and is known as the -extended Riemann zeta function. Now, by applying the -transform on the -extended Mathieu series, we want to show that where , , , , and is the Gaussian hypergeometric function as defined in the book of special function by Rainville .
Proof. Consider Due to uniform convergence, we have changed the order of integration and summation: Now, changing by , we get Here, by using equation (16), we get Also, using (18), we have By Rainville , we have Now, by using (6) and (10), we have So, here, we have proved where , , , , and is the Gaussian hypergeometric function.
Corollary 7. Let , , and ; then, the following relation holds true; we have where , , , and is the Gaussian hypergeometric function.
Corollary 8. Let , , and ; then, the following relation also holds true: where , , , and is the Gaussian hypergeometric function.
Remark 9. It is to be noted that reduces to when , further for ; in the above equations, we have the following relation, . So, here, we have seen that the -transform holds true for the -extended Mathieu series, the generalized Mathieu series, and also for the Mathieu series.
3. Special Cases
Here, we have converted the -transform into a classical Laplace transform by changing the variable ; then, we get integral involving Laplace transform as given below.
Corollary 10. Let , , , and ; then, the Laplace transform formula holds true and establishes the following result:
Proof. Due to uniform convergence, we have changed the order of integration and summation: Now, by changing to , we get Using equation (16), we get Now, by Rainville , we get Using (6) and (10), we obtained Hence, we arrived at the required result where , , and , and .
Corollary 11. Let , , , and ; then, the Laplace transform formula holds true and we have
Corollary 12. Let , , , and ; then, the Laplace transform formula holds true as
Remark 13. Thus, it is to be prominent that the -transform is immediately reduced to the Elzaki transform by shifting the variable into .
It is noted that the -extended Mathieu series is more general in nature and various generalized types of Mathieu series defined in literature can easily be derived through the extended form. Similarly, the -transform defined by Kumar’s  fractional integral operator assists us in converting the table of the Laplace transform and the Elzaki transform into the corresponding transform and vice versa.
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
A. Fernandez and D. Baleanu, “Classes of operators in fractional calculus: a case study,” Mathematical Methods in the Applied Sciences, vol. 44, 2021.View at: Publisher Site | Google Scholar
A. Atangana and D. Baleanu, “New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model,” Thermal Science, vol. 20, no. 2, pp. 763–769, 2016.View at: Publisher Site | Google Scholar
A. Atangana and I. Koca, “Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order,” Chaos, Solitons & Fractals, vol. 89, pp. 447–454, 2016.View at: Publisher Site | Google Scholar
A. Atangana and D. Baleanu, “Caputo–Fabrizio derivative applied to groundwater flow within confined aquifer,” Journal of Engineering Mechanics, vol. 143, no. 5, 2017.View at: Publisher Site | Google Scholar
T. Manzoor, A. Khan, K. G. Wubneh, and H. A. Kahsay, “Beta operator with Caputo-Type Marichev-Saigo-Maeda fractional differential operator of extended Mittag-Leffler function,” Advances in Mathematical Physics, vol. 2021, Article ID 5560543, 9 pages, 2021.View at: Publisher Site | Google Scholar
M. Yavuz, “European option pricing models described by fractional operators with classical and generalized Mittag‐Leffler kernels,” Numerical Methods for Partial Differential Equations, vol. 37, pp. 1–23, 2020.View at: Publisher Site | Google Scholar
M. Yavuz and T. Abdeljawad, “Nonlinear regularized long-wave models with a new integral transformation applied to the fractional derivative with power and Mittag-Leffler kernel,” Advances in Difference Equations, vol. 2020, no. 1, Article ID 367, 2020.View at: Publisher Site | Google Scholar
M. Jena, S. Chakraverty, and M. Yavuz, “Two-hybrid techniques coupled with an integral transformation for Caputo time-fractional Navier- stokes equations,” Progress in Fractional Differentiation and Applications, vol. 6, no. 3, pp. 201–213, 2020.View at: Publisher Site | Google Scholar
M. Yavuz and N. Sene, “Approximate solutions of the model describing fluid flow using generalized -Laplace transform method and heat balance integral method,” Axioms, vol. 9, no. 4, p. 123, 2020.View at: Publisher Site | Google Scholar
M. Yavuz, “Characterizations of two different fractional operators without singular kernel,” Mathematical Modelling of Natural Phenomena, vol. 14, no. 3, p. 302, 2019.View at: Publisher Site | Google Scholar
M. Yavuz and E. Bonyah, “New approaches to the fractional dynamics of schistosomiasis disease model,” Physica A: Statistical Mechanics and its Applications, vol. 525, no. 1, pp. 373–393, 2019.View at: Publisher Site | Google Scholar
X.-J. Yang, “Fractional derivatives of constant and variable orders applied to anomalous relaxation models in heat transfer problems,” Thermal Science, vol. 21, no. 3, pp. 1161–1171, 2017.View at: Publisher Site | Google Scholar
X.-J. Yang, “New rheological problems involving general fractional derivatives with nonsingular power-law kernels,” Proceedings of the Romanian Academy Series A-Mathematics Physics Technical Sciences Information Science, vol. 19, no. 1, pp. 45–52, 2018.View at: Google Scholar
E. L. Mathieu, Traite de physique Mathematique VI-VII: Theorie de Pelasticite dessolides, Gauthier-villars, 1890.
O. Emersleben, “Über die Reihe .,” Mathematische Annalen, vol. 125, no. 1, pp. 165–171, 1952.View at: Publisher Site | Google Scholar
J. Choi and H. M. Srivastava, “Mathieu series and associated sums involving the Zeta functions,” Computers & Mathematics with Applications, vol. 59, no. 2, pp. 861–867, 2010.View at: Publisher Site | Google Scholar
P. Cerone and C. T. Lenard, “On integral forms of generalised Mathieu series,” Journal of Inequalities in Pure and Applied Mathematics, vol. 10, no. 4, pp. 1–11, 2003.View at: Google Scholar
T. K. Pogány, H. M. Srivastava, and Ž. Tomovski, “Some families of Mathieu -series and alternating Mathieu -series,” Applied Mathematics and Computation, vol. 173, no. 1, pp. 69–108, 2006.View at: Publisher Site | Google Scholar
T. K. Pogany and R. K. Parmar, “On -extended Mathieu series. Rad Hazu,” Mathematicke Znanosti, vol. 22, pp. 107–117, 2018.View at: Google Scholar
M. A. Chaudhry, A. Qadir, M. T. Boudjelkha, M. Rafique, and S. M. Zubair, “Extended Riemann zeta functions,” The Rocky Mountain journal of mathematics, vol. 3, no. 1, pp. 1237–1263, 2001.View at: Google Scholar
K. Shah, Z. A. Khan, A. Ali, R. Amin, H. Khan, and A. Khan, “Haar wavelet collocation approach for the solution of fractional order COVID-19 model using Caputo derivative,” Alexandria Engineering Journal, vol. 59, no. 5, pp. 3221–3231, 2020.View at: Publisher Site | Google Scholar
M. Sher, K. Shah, Z. A. Khan, H. Khan, and A. Khan, “Computational and theoretical modeling of the transmission dynamics of novel COVID-19 under Mittag-Leffler power law,” Alexandria Engineering Journal, vol. 59, no. 5, pp. 3133–3147, 2020.View at: Publisher Site | Google Scholar
T. Pohlen, The Hadamard Product and Universal Power Series, Dissertation, Universitat Trier, Trier, 2009.
E. D. Rainville, Special Functions, The Macmillan Company, New York, NY, USA, 1971.
I. N. Sneddon, The Use of Integral Transforms, McGraw Hill, India, 1979.
T. M. Elzaki, “The new integral transform Elzaki transform,” Global Journal of Pure and Applied Mathematics, vol. 7, no. 1, pp. 57–64, 2011.View at: Google Scholar
D. Kumar, “Solution of fractional kinetic equation by a class of integral transform of pathway type,” Journal of Mathematical Physics, vol. 54, no. 4, p. 043509, 2013.View at: Publisher Site | Google Scholar
A. Nadir and A. Khan, “Certain fractional operators of extended Mittag-Leffler function,” Journal of Inequality and Special Functions, vol. 10, no. 1, pp. 12–26, 2019.View at: Google Scholar