ABSTRACT
We use the method for finding inverse Laplace transform without using integration on the complex plane to show that the three-parameter Mittag-Leffler function, which appear in many problems associated with fractional calculus, has similar integral representations on the positive real axis. Some of them are presented.
Keywords:
inverse Laplace transform; Mittag-Leffler functions; integral representations; fractional calculus
RESUMO
Através do método para encontrar a transformada de Laplace inversa sem o uso de um contorno de integração no plano complexo, mostramos que a função de Mittag-Leffler de três parêmetros, que aparece em muitos problemas associados com cálculo fracionário, possui representações integrais similares no semieixo real positivo. Algumas delas são apresentadas.
Palavras-chave:
transformada de Laplace inversa; funções de Mittag-Leffler; representações integrais; cálculo fracionário
INTRODUCTION
The Mittag-Leffler function, introduced in 1902 by Gösta Mittag-Leffler 2323 G.M. Mittag-Leffler. Sur la nouvelle fonction Eα (z). C R Acad Sci, 137 (1903), 554-558., is important in many fields, including description of the anomalous dielectric properties, probability theory, statistics, viscoelasticity, random walks and dynamical systems 99 R. Garrappa, F. Mainardi & G. Maione. Models of dielectric relaxation based on completely monotone functions. Fract Calc Appl Anal, 19 (2016), 1105-1160.), (1010 A. Giusti & I. Colombaro. Prabhakar-like fractional viscoelasticity. Commun Nonlinear Sci Numer Simul, 56 (2018), 138-143.), (1111 R. Gorenflo, A.A. Kilbas, F. Mainardi & S.V. Rogosin. “Mittag-Leffler Functions, Related Topics and Applications”. Springer, Heildelberg (2014).), (1414 E.C. Grigoletto, R.F. Camargo & E.C. Oliveira. Linear fractional differential equations and eigenfunctions of fractional differential operators. Comp Appl Math, 1 (2016), 1-15.), (1919 Y. Li, Y.Q. Chen & I. Podlubny. Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica, 45 (2009), 1965-1969.), (2525 R.N. Pillai. On Mittag-Leffler functions and related distributions. Ann Inst Stat Math, 42 (1990), 157-161.), (2626 T.K. Pogány & Ť. Tomovski. Probability distribution built by Prabhakar function. Related Turán and Laguerre inequalities. Integr Transforms Spec Funct, 27 (2016), 783-793.. Successively, generalizations of Mittag-Leffler function were proposed 2727 T.R. Prabhakar. A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math J, 19 (1971), 7-15.. These functions play a fundamental role in arbitrary order calculus, popularly known as fractional calculus 44 R.F. Camargo. “Cálculo Fracionário e Aplicações”. Ph.D. thesis, IMECC, Unicamp, Tese de Doutorado, Campinas, SP (2009).), (1313 E.C. Grigoletto. “Equações Diferenciais Fracionárias e as Funções de Mittag-Leffler”. Ph.D. thesis, IMECC, Unicamp, Tese de Doutorado, Campinas, SP (2014).), (1818 A.A. Kilbas, H.M. Srivastava & J.J. Trujillo. “Theory and Applications of Fractional Differential Equations”. Elsevier, Amsterdam (2006).), (2020 J.A.T. Machado, V. Kiryakova & F. Mainardi. Recent history of fractional calculus. Commun Nonl Sci Num Simul, 16 (2011), 1140-1153.), (2222 K.S. Miller & B. Ross. “An Introduction to the Fractional Calculus and Fractional Differential Equations”. John Wiley & Sons, New York (1993).), (2929 S.G. Samko, A.A. Kilbas & O.I. Marichev. “Fractional Integrals and Derivatives. Theory and Applications”. Gordon and Breach Science Publishers, Switzerland (1993)., as well as the exponential function play in integer order calculus.
The classical Laplace transform is one of the most widely tools used in the literature for solving integral equations and ordinary or partial differential equations, involving integer or fractional order derivatives 11 R.E. Bellman & R.S. Roth. “The Laplace Transform”. World Scientific, Singapore (1984).),(88 G. Doetsch. “Introduction to the Theory and Application of the Laplace Transformation”. Springer, Berlin, Heidelberg (1974).),(3131 J.L. Schiff. “The Laplace Transform: Theory and Applications”. Springer Verlag, New York (1999).),(3333 D.V. Widder. “The Laplace Transform”, volume 6 of Princeton Mathematical Series. Princeton University Press, Princeton (1941).. It is also used in many others applications such as electrical circuit and signal processing 1515 L.M. Grzesiak & V. Meganck. Spiking signal processing: Principle and applications in control system. Neurocomputing, 308 (2018), 31-48.), (2121 R.R. Marianito & A.L. Worthy. Solution of multilayer diffusion problems via the Laplace transform. J Math Anal Appl, 444 (2016), 475-502.), (3535 W.K. Zahra, M.M. Hikal & T.A. Bahnasy. Solutions of fractional order electrical circuits via Laplace transform and nonstandard finite difference method. J Egypt Math Soc, 25 (2017), 252-261.. In general, the Laplace inversion is done numerically due to the impossibility of the exact inversion by means of an integration on the complex plane 77 B. Davies & B. Martin. Numerical inversion of Laplace transform: A survey and comparison of methods. J Comp Phys, 33 (1979), 1-32.), (3232 J. Valsa & L. Brancik. Approximated formula for numerical inversion of Laplace transform. Int J Numer Model, 11 (1998), 153-166..
M. N. Berberan-Santos 22 M.N. Berberan-Santos. Analytical inversion of the Laplace transform without contour integration: application to luminescence decay laws and other relaxation functions. J Math Chem, 38 (2005), 165-173. proposed a new methodology for evaluation of the numerical inverse Laplace transform, without using integration on the complex plane, which was published in 2005, and its methodology was used recently, for instance, to discuss the luminescence decay of inorganic solids, and to obtain an integral representation of Mittag-Leffler relaxation function, a special one-parameter Mittag-Leffler function 33 M.N. Berberan-Santos. Properties of the Mittag-Leffler relaxation function. J Math Chem , 38 (2005), 629-635.. Recently, the method to evaluate the inverse Laplace transform without using integration on the complex plane was applied in 66 E. Contharteze Grigoletto & E. Capelas de Oliveira. A note on the inverse Laplace transform.Cadernos do IME - Série Matemática, 1(12) (2018), 39-46. to find integral representations on the positive real axis for some functions.
In this paper, with the method for finding inverse Laplace transform without using integration on the complex plane we show that the three-parameter Mittag-Leffler function has integral representations on the positive real axis.
The paper is organized as follows: in Section 1, we present some preliminaries concepts and the methodology of inversion of the Laplace transform. In Section 2, using this methodology, we express the integral representations of three-parameter Mittag-Leffler function and we use the results from this study to discuss, in Section 3, a class of improper integrals, expressing them in terms of the Mittag-Leffler functions. Concluding remarks close the paper.
1 PRELIMINARIES
In this section, we present the definition and some special cases of the Mittag-Leffler functions, and a review of the methodology of inversion of the Laplace transform proposed by M. N. Berberan-Santos in the following subsections.
1.1 Mittag-Leffler functions
The three-parameter Mittag-Leffler function, introduced by Prabhakar 2727 T.R. Prabhakar. A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math J, 19 (1971), 7-15., of complex variable , with complex parameters , is defined by1 1 ℛ|ξ| indicates the real part of ξ.
with and , where
is the Gamma function, and
is the Pochhammer symbol. Taking in equation (1.1), we get the two-parameter Mittag-Leffler function:
When in equation (1.4), we get the standard Mittag-Leffler function 2323 G.M. Mittag-Leffler. Sur la nouvelle fonction Eα (z). C R Acad Sci, 137 (1903), 554-558.), (3434 A. Wiman. Über den fundamentalsatz in der teorie der functionen Eα (x). Acta Math, 29 (1905), 191-201.:
A function is to be completely monotonic (CM) on interval I if for and (3333 D.V. Widder. “The Laplace Transform”, volume 6 of Princeton Mathematical Series. Princeton University Press, Princeton (1941).. Capelas et al.55 E. Capelas de Oliveira, F. Mainardi & J. Vaz. Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics. Eur J Phys, 193 (2011), 161-171. showed that the function is CM for . Particularly, these functions play an important rule in anomalous dielectric relaxation where the memory effect appears specifically in the Havriliak-Negami model, which contains Davidson-Cole model, Cole-Cole model and the classical Debye model, as particular cases1717 A.A. Khamzin, R.R. Nigmatullin & I.I. Popov. Justification of the empirical laws of the anomalous dielectric relaxation in the framework of the memory function formalism. Fract Calc Appl An , 17(1) (2014), 246-258.), (2424 S.C. Pandey. The Lorenzo-Hartley’s function for fractional calculus and its applications pertaining to fractional order modelling of anomalous relaxation in dielectrics. Comp Appl Math , 37 (2018), 2648-2666..
An interesting functional relation case and the more simple special relations involving the Mittag-Leffler functions and , with , are given by the following equations:
1.2 Inversion of the Laplace transform
Let f(t) be a real function of (time) variable . The Laplace transform of f, denoted by , is defined as follows:
whenever the integral converges for2 2 ℛ|s| indicates the real part of s and the imaginary part is denoted by 𝒥|s|. , where , with σ and τ real numbers, and for . By means of equation (1.11) with Euler formula, we observe that
and
The expression for evaluation of the inverse Laplace transform of F(s), proposed by M. N. Berberan-Santos 22 M.N. Berberan-Santos. Analytical inversion of the Laplace transform without contour integration: application to luminescence decay laws and other relaxation functions. J Math Chem, 38 (2005), 165-173., is given by
for and any real number σ satisfying the condition σ ≥ σ 0 > 0, where σ 0 is large enough that F(s) is defined for ℛ [s] ≥ σ 0 > 0. The expression in equation (1.14) recovers the real function whose Laplace transform is known.
From equation (1.14), for , we can write
The function f is such that for . Since , equation (1.15) yields
Furthermore, in a similar way,
Namely, there are three possible cases to find the inverse Laplace transform of a function F(s), they are given by equations (1.14), (1.16) and (1.17).
In this point, it is important to consider a simple example, illustrating the methodology that will be used in this work: The Laplace transform of the exponential function is given by , for . Choosing and writing , we have that . Thus, from equation (1.16), we obtain
2 INTEGRAL REPRESENTATIONS OF MITTAG-LEFFLER FUNCTION
Some integral representations associated with the one-parameter Mittag-Leffler function can be found in the following papers: 33 M.N. Berberan-Santos. Properties of the Mittag-Leffler relaxation function. J Math Chem , 38 (2005), 629-635.), (1212 R. Gorenflo, J. Loutschko & Y. Luchko. Computation of the Mittag-Leffler function and its derivatives. Fract Calc Appl An, 5 (2002), 491-518.), (1616 H.J. Haubold, A.M. Mathai & R.K. Saxena. Mittag-Leffler functions and their applications. J Appl Math, 2011 (2011), 298628.. Here we present integral representations for the three-parameter Mittag-Leffler function and to prove the representations, we use the relations in equations (1.14), (1.16) and (1.17). It is worthwhile to mention that the detail treatment of the similar study can be found in 2828 J.C. Prajapati, R.K. Jana, R.K. Saxena & A.K. Shukla. Some results on the generalized Mittag-Leffler function operator. J Inequal Appl, 2013 (2013), 33.), (3030 R.K. Saxena , J.P. Chauhan, R.K. Jana & A.K. Shukla. Further results on the generalized Mittag-Leffler function operator. J Inequal Appl , 2015 (2015), 75..
Theorem 1. Let and . Then, for , the three-parameter Mittag-Leffler function has the following integral representations on the positive real axis
where and , and are defined by equations:
Proof. The Laplace transform of the three-parameter Mittag-Leffler type function is given by
The complex parameter s can be written as
with and . In this way, from equation (2.8), we get equations in (2.5).
Expression in the denominator of F (s) can be written in the following form:
Replacing s by re iθ in equation (2.9), we get
that is,
Separating real part and imaginary part in equation (2.10), we obtain the expressions in equation (2.6).
We can thus conclude that
Through manipulation of F(s), we can separate its real and imaginary parts:
and
Substituting equations (2.11) and (2.12) into equations (1.14), (1.16) and (1.17), we arrive at equations (2.1), (2.2) and (2.3), respectively; and finally if we choose , then the inequality is satisfied. ◻
According to Theorem 1, the Mittag-Leffler function has similar integral representations, as we have seen in the equations (2.1)-(2.3). We present some applications of this theorem in the next section.
3 EVALUATION OF A CLASS OF IMPROPER INTEGRALS
In what follows we will discuss some evaluations for improper integrals using Theorem 1 for specific values of the parameters appearing in equations (2.1)-(2.3). As by-products, in the following examples, interesting integrals are obtained.
We should point out that we consider the case in the next illustrative examples. In this way, from equation (2.4), we can choose . Equation (2.5) with imply that
By equation (3.1), we have
Example 1. We consider the function:
where be a confluent hypergeometric function 1818 A.A. Kilbas, H.M. Srivastava & J.J. Trujillo. “Theory and Applications of Fractional Differential Equations”. Elsevier, Amsterdam (2006).. In particular, if , the function is CM.
From equation (2.1), we can derive that
or in a different form, we can obtain an integral representation for confluent hypergeometric function as follows:
where, according to equation (2.6),
Substituting equations (3.2) and (3.5) into equation (3.4), we obtain the result
For , where
Taking in equation (3.7), we have
According to equation (3.8) with we thus have
For .
If and , by equations (3.3) and (1.8),
that is,
Equations (2.6) and (3.1) provided that
Taking into account the equations (3.12) and (3.1), we can rewrite equation (3.11) in the respective form:
Example 2. In this example we consider the function: . In particular, if , the function is CM.
In this case, if we use equation (2.3), since , then we have
where and and are defined in equation (2.6), given by
where r and θ are given by equation (3.1).
When , then equation (2.6), in accordance with equation (3.1), yields the following formula
Multiplying the integrand in equation (3.14) by , and substituting equations (3.1) and (3.16) into (3.14), we thus derive the following integral representation
Equation (1.7) imply the following result
Then, equation (3.17) can be rewritten in the alternative form:
Example 3. In the last case we consider the function: . In particular, if , the function is CM. From equation (2.2), we obtain the following integral representation
For , where r and θ are given by equation (3.1) and and are given by equation (2.6).
Moreover, when manipulating the mathematical expression in equation (3.19), we can give another similar integral representation as follows
In particular, when in equation (3.20), we obtain another integral representation for the exponential function:
When , by using equation (1.9) and the integral representation type in equation (2.3), the cosine function takes the form:
The integral in equation (3.22) can be simplified to
Furthermore, using the relation in equation (1.10) and the equation (2.3), the hyperbolic cosine function can be represented by
Finally, we can use the relation in equation (1.6) and the above results to express the function
In fact, equations (3.23) and (3.24) provided that
for .
4 CONCLUDING REMARKS
We build similar integral representations for the three-parameter Mittag-Leffler function on the positive real axis using the method for finding inverse Laplace transform without using integration on the complex plane. Many authors have demonstrated interest in the study of the asymptotic behavior of the Mittag-Leffler functions on the interpretation of the solutions of problems associated with fractional diffusion. In this way the integral representations presented in this paper can be used to analyze the asymptotic behavior of these functions. Furthermore, this representation can express improper integrals in terms of trigonometric functions by means of the Mittag-Leffler functions and the presented examples complement corresponding integral representations.
ACKNOWLEDGMENT
The authors thank the referees for their valuable and constructive comments in relation to this work and thank the collaboration of the members of our research group CF@FC.
REFERENCES
-
1R.E. Bellman & R.S. Roth. “The Laplace Transform”. World Scientific, Singapore (1984).
-
2M.N. Berberan-Santos. Analytical inversion of the Laplace transform without contour integration: application to luminescence decay laws and other relaxation functions. J Math Chem, 38 (2005), 165-173.
-
3M.N. Berberan-Santos. Properties of the Mittag-Leffler relaxation function. J Math Chem , 38 (2005), 629-635.
-
4R.F. Camargo. “Cálculo Fracionário e Aplicações”. Ph.D. thesis, IMECC, Unicamp, Tese de Doutorado, Campinas, SP (2009).
-
5E. Capelas de Oliveira, F. Mainardi & J. Vaz. Models based on Mittag-Leffler functions for anomalous relaxation in dielectrics. Eur J Phys, 193 (2011), 161-171.
-
6E. Contharteze Grigoletto & E. Capelas de Oliveira. A note on the inverse Laplace transform.Cadernos do IME - Série Matemática, 1(12) (2018), 39-46.
-
7B. Davies & B. Martin. Numerical inversion of Laplace transform: A survey and comparison of methods. J Comp Phys, 33 (1979), 1-32.
-
8G. Doetsch. “Introduction to the Theory and Application of the Laplace Transformation”. Springer, Berlin, Heidelberg (1974).
-
9R. Garrappa, F. Mainardi & G. Maione. Models of dielectric relaxation based on completely monotone functions. Fract Calc Appl Anal, 19 (2016), 1105-1160.
-
10A. Giusti & I. Colombaro. Prabhakar-like fractional viscoelasticity. Commun Nonlinear Sci Numer Simul, 56 (2018), 138-143.
-
11R. Gorenflo, A.A. Kilbas, F. Mainardi & S.V. Rogosin. “Mittag-Leffler Functions, Related Topics and Applications”. Springer, Heildelberg (2014).
-
12R. Gorenflo, J. Loutschko & Y. Luchko. Computation of the Mittag-Leffler function and its derivatives. Fract Calc Appl An, 5 (2002), 491-518.
-
13E.C. Grigoletto. “Equações Diferenciais Fracionárias e as Funções de Mittag-Leffler”. Ph.D. thesis, IMECC, Unicamp, Tese de Doutorado, Campinas, SP (2014).
-
14E.C. Grigoletto, R.F. Camargo & E.C. Oliveira. Linear fractional differential equations and eigenfunctions of fractional differential operators. Comp Appl Math, 1 (2016), 1-15.
-
15L.M. Grzesiak & V. Meganck. Spiking signal processing: Principle and applications in control system. Neurocomputing, 308 (2018), 31-48.
-
16H.J. Haubold, A.M. Mathai & R.K. Saxena. Mittag-Leffler functions and their applications. J Appl Math, 2011 (2011), 298628.
-
17A.A. Khamzin, R.R. Nigmatullin & I.I. Popov. Justification of the empirical laws of the anomalous dielectric relaxation in the framework of the memory function formalism. Fract Calc Appl An , 17(1) (2014), 246-258.
-
18A.A. Kilbas, H.M. Srivastava & J.J. Trujillo. “Theory and Applications of Fractional Differential Equations”. Elsevier, Amsterdam (2006).
-
19Y. Li, Y.Q. Chen & I. Podlubny. Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica, 45 (2009), 1965-1969.
-
20J.A.T. Machado, V. Kiryakova & F. Mainardi. Recent history of fractional calculus. Commun Nonl Sci Num Simul, 16 (2011), 1140-1153.
-
21R.R. Marianito & A.L. Worthy. Solution of multilayer diffusion problems via the Laplace transform. J Math Anal Appl, 444 (2016), 475-502.
-
22K.S. Miller & B. Ross. “An Introduction to the Fractional Calculus and Fractional Differential Equations”. John Wiley & Sons, New York (1993).
-
23G.M. Mittag-Leffler. Sur la nouvelle fonction Eα (z). C R Acad Sci, 137 (1903), 554-558.
-
24S.C. Pandey. The Lorenzo-Hartley’s function for fractional calculus and its applications pertaining to fractional order modelling of anomalous relaxation in dielectrics. Comp Appl Math , 37 (2018), 2648-2666.
-
25R.N. Pillai. On Mittag-Leffler functions and related distributions. Ann Inst Stat Math, 42 (1990), 157-161.
-
26T.K. Pogány & Ť. Tomovski. Probability distribution built by Prabhakar function. Related Turán and Laguerre inequalities. Integr Transforms Spec Funct, 27 (2016), 783-793.
-
27T.R. Prabhakar. A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math J, 19 (1971), 7-15.
-
28J.C. Prajapati, R.K. Jana, R.K. Saxena & A.K. Shukla. Some results on the generalized Mittag-Leffler function operator. J Inequal Appl, 2013 (2013), 33.
-
29S.G. Samko, A.A. Kilbas & O.I. Marichev. “Fractional Integrals and Derivatives. Theory and Applications”. Gordon and Breach Science Publishers, Switzerland (1993).
-
30R.K. Saxena , J.P. Chauhan, R.K. Jana & A.K. Shukla. Further results on the generalized Mittag-Leffler function operator. J Inequal Appl , 2015 (2015), 75.
-
31J.L. Schiff. “The Laplace Transform: Theory and Applications”. Springer Verlag, New York (1999).
-
32J. Valsa & L. Brancik. Approximated formula for numerical inversion of Laplace transform. Int J Numer Model, 11 (1998), 153-166.
-
33D.V. Widder. “The Laplace Transform”, volume 6 of Princeton Mathematical Series. Princeton University Press, Princeton (1941).
-
34A. Wiman. Über den fundamentalsatz in der teorie der functionen Eα (x). Acta Math, 29 (1905), 191-201.
-
35W.K. Zahra, M.M. Hikal & T.A. Bahnasy. Solutions of fractional order electrical circuits via Laplace transform and nonstandard finite difference method. J Egypt Math Soc, 25 (2017), 252-261.
-
1
ℛ|ξ| indicates the real part of ξ.
-
2
ℛ|s| indicates the real part of s and the imaginary part is denoted by 𝒥|s|.
Publication Dates
-
Publication in this collection
16 Sept 2019 -
Date of issue
May-Aug 2019
History
-
Received
31 Aug 2018 -
Accepted
14 Jan 2019