# Computable solution of fractional kinetic equations using Mathieu-type series

## Abstract

The Mathieu series appeared in the study of elasticity of solid bodies in the work of Émile Leonard Mathieu. Since then numerous authors have studied various problems arising from the Mathieu series in several diverse ways. In this line, our aim is to study the solution of fractional kinetic equations involving generalized Mathieu-type series. The generality of this series will help us to deduce results related to a fractional kinetic equation involving another form of Mathieu series. To obtain the solution, we use the Laplace transform technique. Besides, a graphical representation is given to observe the behavior of the obtained solutions.

## Keywords

Generalized fractional kinetic equation Mathieu-type series Laplace transform Sumudu transform## MSC

26A33 33A10 33E20 44A10## 1 Introduction and preliminaries

Fractional calculus (FC) is a useful mathematical tool to study fractional order integrals and derivatives. The fractional calculus has been developed and used in different areas of science and engineering. The concept of fractional differential equations and their applications have played a significant role in many diverse fields such as applied science, physics, biology, chemistry, and engineering. The kinetic equations designate a system of differential equations, which describe the rate of change of the chemical composition of a star for each order in terms of the reaction rates for production and destruction. During the last several decades, fractional kinetic equations in various forms have been broadly and usefully employed when describing and solving several important problems of physics and astrophysics (see, e.g., [2, 7, 8, 9, 10, 11, 13, 14, 15, 16, 20, 24, 25, 26, 27, 28, 29, 31] and the references therein). The special functions and their applications appear in the solutions of fractional integral and differential equations and are related to comprehensive problems in the several areas of mathematics and mathematical physics. In view of the effectiveness and great importance of the fractional kinetic equations in certain astrophysical problems, the authors develop a generalized form of the fractional kinetic equations along with Mathieu-type series. The broad generality of Mathieu-type series will allow us to deduce many special cases of the main results.

*i*at time \(t=0\) and \(c_{i}>0\). Neglecting index

*i*and integrating, (2) becomes

*Γ*as

*s*is real or complex parameter. The Laplace transform of the function \(f(t)\) is defined by

*f*and

*g*are at least piecewise continuous. One of the most significant properties possessed by the convolution in connection with the Laplace transform is that the Laplace transform of the convolution of two functions is the product of their transforms (see, e.g., [20, 30, 32, 33]).

### The Laplace Convolution Theorem

*If*

*f*

*and*

*g*

*are piecewise continuous on*\([0,\infty )\)

*and of exponential order*

*α*

*when*\(t \rightarrow \infty \),

*then*

*We find*

*by using the following well*-

*known identity*:

*in which*\(L^{-1}\)

*denotes the inverse Laplace transform*.

Recently, many papers investigated the solution of generalized fractional kinetic equations involving a variety of special functions by using Laplace and Sumudu transformations. For instance, for generalized fractional kinetic equations with \(f(t)\) in (13) replaced by certain forms of generalized Mittag-Leffler function (11), Saxena et al. [29] presented their corresponding solutions. Moreover, Chaurasia and Kumar [7] investigated the solution of generalized fractional kinetic equations involving M-series, and Choi and Kumar [9] gave their corresponding solutions involving Aleph function. Furthermore, Agarwal et al. [1, 2] investigated solutions of fractional kinetic equations with \(f(t)\) in (13) replaced by certain forms of *k*-Mittag-Leffler function and *k*-Bessel function. Subsequently, Nisar and Qi [22] presented solution of fractional kinetic equations with \(f(t)\) in (13) replaced by certain forms of generalized *k*-Bessel function.

*μ*replaced with \(\mu +1\) corresponds to the Mathieu series defined by Tomovski and Pogány [36]:

Due to the great importance of fractional kinetic equations involving special functions, in this paper, we aim to investigate solution of generalized fractional kinetic equations (13) with \(f(t)\) replaced by several generalized Mathieu-type series, by mainly using the Laplace and Sumudu transforms. The results presented here, being general, are also shown to reduce to fractional kinetic equations involving simpler special functions. The manuscript is organized as follows. In Sect. 2, the solution of fractional kinetic equations involving Mathieu-type series is established, and in Sect. 3, a graphical interpretation and nature of the solution are discussed. Section 4 is focused on some examples, and in Sect. 5 concluding remarks are given.

## 2 Solution of fractional kinetic equations

In this section, we obtain the solution of generalized fractional kinetic equation (13) involving generalized Mathieu-type series (25) by applying the Laplace transform technique.

### Theorem 2.1

*For all*\(\nu , c, h >{0}\); \(c\neq {h}\),

*if*\(\mu , \rho , \sigma , \tau , l \in {\mathbb{R}^{+}}\), \(|ht|\leq {1}\),

*then equation*

*has the solution*

### Proof

In view of relation (26), we state the following consequence of Theorem 2.1.

### Corollary

*For all*\(\nu , c, h >{0}\); \(c\neq {h}\),

*if*\(\mu , l \in {\mathbb{R} ^{+}}\), \(|ht|\leq {1}\),

*then equation*

*has the solution*

### Theorem 2.2

*Assume that*\(\nu , c, w >{0}\)

*and*\(\mu , \rho , \sigma , \tau , l \in {\mathbb{R}^{+}}\), \(|wt|\leq {1}\),

*then equation*

*has the solution*

### Proof

In view of relation (26), we establish the following consequence of Theorem 2.2.

### Corollary

*Assume that*\(\nu , \alpha , c, w >{0}\),

*and let*\(\mu , l \in {\mathbb{R} ^{+}}\), \(|wt|\leq {1}\),

*then equation*

*has the solution*

### Theorem 2.3

*For all*\(\nu >0\), \(c>{0}\), \(\mu , \rho , \sigma , \tau , l \in {\mathbb{R} ^{+}}\), \(|t|\leq {1}\),

*the equation*

*has the solution*

### Proof

This can be proved by the same procedure as in the proof of Theorem 2.1. So we omit all details. □

Using relation (26), we state the following consequence of Theorem 2.3.

### Corollary

*For all*\(\nu >0\), \(c>{0}\), \(\mu , l \in {\mathbb{R}^{+}}\), \(|t|\leq {1}\),

*the equation*

*has the solution*

## 3 Graphical representations

*ν*, the graph is increasing in nature, and if we choose larger values for

*ν*, then it is gradually decreasing. The graphical results confirm that the region of convergence of solutions depends continuously on the fractional parameter

*ν*. Hence, by observing the nature of the solutions for various parameters and time interval, it is concluded that \(N(t)\) can be negative as well as positive. A similar observation can be obtained for the solutions (34) and (39).

## 4 Example

*M*is a real finite number and \(\delta _{1}\) and \(\delta _{2}\) can be finite or infinite. For more details, see [38, 39].

*f*by means of the relation

Due to the importance of the above observation, in this section, we first define the Sumudu transform of Mathieu-type series (25) and further determine the solution of fractional kinetic equations by applying Sumudu transform technique as given in the following Examples 4.1, 4.2, and 4.3.

Using the same procedure of analysis as in Theorems 2.1, 2.2, and 2.3, we can find the solutions of generalized fractional kinetic equations involving Mathieu-type series, which are given in the following three examples.

### Example 4.1

### Example 4.2

### Example 4.3

## 5 Conclusion

In the current study, we have obtained the solution of generalized fractional kinetic equations involving generalized Mathieu-type series with the help of Laplace and Sumudu transform techniques. The results of Sect. 2 are general in character and are likely to find certain applications in the theory of fractional calculus and special functions. From the graphical analysis, we conclude that \(N(t) > 0\) or \(N(t) < 0\) for distinct values of the parameters. By suitably specializing the values of the parameters of generalized Mathieu-type series (25), our main results can yield several new solutions of generalized fractional kinetic equations corresponding to the Mathieu series given by various authors [6, 21, 34, 37]. Therefore, the investigated results in this paper would at once give many results involving diverse special functions occurring in the problems of astrophysics, mathematical physics, and engineering.

## Notes

### Acknowledgements

Not applicable.

### Availability of data and materials

Not applicable.

### Authors’ contributions

All authors contributed equally. All authors read and approved the final manuscript.

### Funding

Not applicable.

### Competing interests

The authors declare that they have no competing interest.

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