# Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Sinc Methods of Approximation

#### 2.1. Sinc Basis

**Definition**

**1.**

**Definition**

**2.**

**Theorem**

**1.**

#### 2.2. Indefinite Integral Approximation

**Theorem**

**2.**

#### 2.3. Convolution Integrals

#### 2.4. Inverse Laplace Transform

## 3. Numerical Examples

#### 3.1. Debye Model

#### 3.2. One Parameter Fractional Relaxation Equation

#### 3.3. Two Parameter Fractional Relaxation Equation

#### 3.4. Three Parameter Fractional Relaxation Equation: Prabhakar Function

#### 3.5. Scarpi’s Variable-Order Fractional Calculus

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Stenger, F.; Baumann, G.; Koures, V.G. Computational Methods for Chemistry and Physics, and Schrödinger in 3 + 11. In Advances in Quantum Chemistry; Elsevier: Amsterdam, The Netherlands, 2015; Volume 71, pp. 265–298. ISBN 978-0-12-802824-7. [Google Scholar]
- Baumann, G.; Stenger, F. Sinc-Approximations of Fractional Operators: A Computing Approach. Mathematics
**2015**, 3, 444–480. [Google Scholar] [CrossRef][Green Version] - Baumann, G.; Stenger, F. Fractional Calculus and Sinc Methods. Fract. Calc. Appl. Anal.
**2011**, 14, 568–622. [Google Scholar] [CrossRef] - Schmeisser, G.; Stenger, F. Sinc Approximation with a Gaussian Multiplier. Sampl. Theory Signal Image Process.
**2007**, 6, 199–221. [Google Scholar] [CrossRef] - Talbot, A. The Accurate Numerical Inversion of Laplace Transforms. IMA J. Appl. Math.
**1979**, 23, 97–120. [Google Scholar] [CrossRef] - López-Fernández, M.; Palencia, C. On the Numerical Inversion of the Laplace Transform of Certain Holomorphic Mappings. Appl. Numer. Math.
**2004**, 51, 289–303. [Google Scholar] [CrossRef] - López-Fernández, M.; Palencia, C.; Schädle, A. A Spectral Order Method for Inverting Sectorial Laplace Transforms. SIAM J. Numer. Anal.
**2006**, 44, 1332–1350. [Google Scholar] [CrossRef][Green Version] - Weideman, J.A.C.; Trefethen, L.N. Parabolic and Hyperbolic Contours for Computing the Bromwich Integral. Math. Comp.
**2007**, 76, 1341–1357. [Google Scholar] [CrossRef][Green Version] - Garrappa, R. Numerical Evaluation of Two and Three Parameter Mittag-Leffler Functions. SIAM J. Numer. Anal.
**2015**, 53, 1350–1369. [Google Scholar] [CrossRef][Green Version] - Baumann, G. Mathematica for Theoretical Physics, 2nd ed.; Springer: New York, NY, USA, 2005; ISBN 978-0-387-01674-0. [Google Scholar]
- Glöckle, W.G.; Nonnenmacher, T.F. A Fractional Calculus Approach to Self-Similar Protein Dynamics. Biophys. J.
**1995**, 68, 46–53. [Google Scholar] [CrossRef][Green Version] - Baumann, G.; Südland, N.; Nonnenmacher, T.F. Anomalous Relaxation and Diffusion Processes in Complex Systems. Transp. Theory Stat. Phys.
**2000**, 29, 157–171. [Google Scholar] [CrossRef] - Mainardi, F. Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models; Imperial College Press: London, UK; Hackensack, NJ, USA, 2010; ISBN 978-1-84816-329-4. [Google Scholar]
- Gorenflo, R.; Kilbas, A.A.; Mainardi, F.; Rogosin, S.V. Mittag-Leffler Functions, Related Topics and Applications; Springer Monographs in Mathematics; Springer: Berlin/Heidelberg, Germany, 2014; ISBN 978-3-662-43929-6. [Google Scholar]
- Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations. In North-Holland Mathematics Studies, 1st ed.; Elsevier: Amsterdam, The Netherlands; Boston, MA, USA, 2006; ISBN 978-0-444-51832-3. [Google Scholar]
- Mittag-Leffler, M.G. Sur l’intégrale de Laplace-Abel. C. R. Acad. Sci. Paris (Ser. II)
**1902**, 136, 937–939. [Google Scholar] - Wiman, A. Über den Fundamentalsatz in der Teorie der Funktionen E
_{α}(x). Acta Math.**1905**, 29, 191–201. [Google Scholar] [CrossRef] - Prabhakar, T.R. A Singular Integral Equation with a Generalized Mittag-Leffler Function in the Kernel. Yokohama Math. J.
**1971**, 19, 7–15. [Google Scholar] - Garrappa, R.; Popolizio, M. Fast methods for the computation of the Mittag-Leffler function. In Handbook of Fractional Calculus with Applications; Kochubei, A.N., Luchko, Y.F., Karniadakis, G., Tarasov, V.E., Petráš, I., Baleanu, D., Lopes, A.M., Eds.; De Gruyter: Berlin, Germany; Boston, MA, USA, 2019; Volume 3, pp. 329–346. ISBN 978-3-11-057081-6. [Google Scholar]
- Giusti, A.; Colombaro, I.; Garra, R.; Garrappa, R.; Polito, F.; Popolizio, M.; Mainardi, F. A Practical Guide to Prabhakar Fractional Calculus. Fract. Calc. Appl. Anal.
**2020**, 23, 9–54. [Google Scholar] [CrossRef][Green Version] - Bromwich, T.J.I. Normal Coordinates in Dynamical Systems. Proc. Lond. Math. Soc.
**1917**, s2-15, 401–448. [Google Scholar] [CrossRef][Green Version] - Ang, D.D.; Lund, J.; Stenger, F. Complex Variable and Regularization Methods of Inversion of the Laplace Transform. Math. Comp.
**1989**, 53, 589–608. [Google Scholar] [CrossRef] - Davies, B.; Martin, B. Numerical Inversion of the Laplace Transform: A Survey and Comparison of Methods. J. Comput. Phys.
**1979**, 33, 1–32. [Google Scholar] [CrossRef] - Debye, P. Zur Theorie der spezifischen Wärmen. Ann. Phys.
**1912**, 344, 789–839. [Google Scholar] [CrossRef][Green Version] - Baumann, G. Mathematica for Theoretical Physics; Springer: New York, NY, USA, 2013; Volumes I and II. [Google Scholar]
- Südland, N.; Baumann, G.; Nonnenmacher, T.F. Fractional Driftless Fokker-Planck Equation with Power Law Diffusion Coefficients. In Computer Algebra in Scientific Computing CASC 2001, Proceedings of the Fourth International Workshop on Computer Algebra in Scientific Computing, Konstanz, Germany, 22–26 September 2001; Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V., Eds.; Springer: Berlin/Heidelberg, Germany, 2001; p. 513. ISBN 978-3-642-56666-0. [Google Scholar]
- Kilbas, A.A.; Saigo, M.; Saxena, R.K. Solution of Volterra Integro-Differential Equations with Generalized Mittag-Leffler Function in the Kernels. J. Integral Equ. Appl.
**2002**, 14, 377–396. [Google Scholar] [CrossRef] - Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. In Mathematics in Science and Engineering; Academic Press: San Diego, CA, USA, 1999; ISBN 978-0-12-558840-9. [Google Scholar]
- Roy, S.D. On the Realization of a Constant-Argument Immittance or Fractional Operator. IEEE Trans. Circ. Theory
**1967**, 14, 264–274. [Google Scholar] [CrossRef] - Stenger, F. Numerical Methods Based on Sinc and Analytic Functions; Springer Series in Computational Mathematics; Springer: New York, NY, USA, 1993; Volume 20, ISBN 978-1-4612-7637-1. [Google Scholar]
- Stenger, F.; El-Sharkawy, H.A.; Baumann, G. The Lebesgue Constant for Sinc Approximations. In New Perspectives on Approximation and Sampling Theory: Festschrift in Honor of Paul Butzer’s 85th Birthday; Zayed, A.I., Schmeisser, G., Eds.; Applied and Numerical Harmonic Analysis; Springer International Publishing: Cham, Switzerland, 2014; ISBN 978-3-319-08800-6. [Google Scholar]
- Stenger, F. Handbook of Sinc Numerical Methods. In Chapman & Hall/CRC Numerical Analysis and Scientific Computing; CRC Press: Boca Raton, FL, USA, 2011; ISBN 978-1-4398-2158-9. [Google Scholar]
- Baumann, G. New Sinc Methods of Numerical Analysis: Festschrift in Honor of Frank Stenger’s 80th Birthday; Springer Nature: Cham, Switzerland, 2021; ISBN 978-3-030-49715-6. [Google Scholar]
- Qian, L.; Creamer, D.B. Localization of the Generalized Sampling Series and Its Numerical Application. SIAM J. Numer. Anal.
**2006**, 43, 2500–2516. [Google Scholar] [CrossRef] - Qian, L. On the Regularized Whittaker-Kotel’nikov-Shannon Sampling Formula. Proc. Am. Math. Soc.
**2003**, 131, 1169–1176. [Google Scholar] [CrossRef] - Butzer, P.L.; Stens, R.L. A Modification of the Whittaker-Kotelnikov-Shannon Sampling Series. Aeq. Math.
**1985**, 28, 305–311. [Google Scholar] [CrossRef] - Shen, X.; Zayed, A.I. (Eds.) Multiscale Signal Analysis and Modeling; Springer: New York, NY, USA, 2013; ISBN 978-1-4614-4144-1. [Google Scholar]
- Stenger, F. Collocating Convolutions. Math. Comput.
**1995**, 64, 211–235. [Google Scholar] [CrossRef] - Han, L.; Xu, J. Proof of Stenger’s Conjecture on Matrix I(-1) of Sinc Methods. J. Comput. Appl. Math.
**2014**, 255, 805–811. [Google Scholar] [CrossRef] - Gray, R.M. Toeplitz and Circulant Matrices. A Review; Now: Boston, MA, USA, 2006. [Google Scholar]
- Trench, W.F. Spectral Evolution of a One-Parameter Extension of a Real Symmetric Toeplitz Matrix. SIAM J. Matrix Anal. Appl.
**1990**, 11, 601–611. [Google Scholar] [CrossRef] - Grenander, U.; Szegö, G. Toeplitz Forms and Their Applications, 2nd (textually unaltered) ed.; Chelsea Pub. Co.: New York, NY, USA, 1984; ISBN 978-0-8284-0321-4. [Google Scholar]
- Lubich, C. Convolution Quadrature and Discretized Operational Calculus. II. Numer. Math.
**1988**, 52, 413–425. [Google Scholar] [CrossRef] - Post, E. Generalized Differentiation. Trans. AMS
**1930**, 32, 723–781. [Google Scholar] [CrossRef] - Khamzin, A.A.; Nikitin, A.S. Trap-Controlled Fractal Diffusion Model of an Atypical Dielectric Response. Chem. Phys.
**2021**, 111163. [Google Scholar] [CrossRef] - Bia, P.; Caratelli, D.; Mescia, L.; Cicchetti, R.; Maione, G.; Prudenzano, F. A Novel FDTD Formulation Based on Fractional Derivatives for Dispersive Havriliak-Negami Media. Signal Process.
**2015**, 107, 312–318. [Google Scholar] [CrossRef] - Garrappa, R. Grünwald-Letnikov Operators for Fractional Relaxation in Havriliak-Negami Models. Commun. Nonlinear Sci. Numer. Simul.
**2016**, 38, 178–191. [Google Scholar] [CrossRef] - Scarpi, G. Sulla Possibilità Di Un Modello Reologico Intermediodi Tipo Evolutivo. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat.
**1972**, 52, 912–917. [Google Scholar] - Garrappa, R.; Giusti, A.; Mainardi, F. Variable-Order Fractional Calculus: A Change of Perspective. arXiv
**2021**, arXiv:2102.09932. [Google Scholar]

**Figure 1.**Limits for eigenvalues of the Hermitian Matrix $H={A}_{m}^{*}{A}_{m}$. Left panel shows the minimal and maximal eigenvalues for a Sinc basis and the right panel for a Sinc-Gaussian basis (log-log plot). The limits follow a relation ${\lambda}_{n,M}\sim {\nu}^{\pm \gamma}$ with $\nu =m\times m$ the size of H, and $\gamma \approx 1/2$ for the upper limits and $\gamma \approx 7/100$ for the lower limit of a Sinc approximation. The lower limit for Sinc-Gaussian follows the upper limit with the opposite sign.

**Figure 2.**Approximation of the transfer function $G\left(s\right)$ for the Debye relaxation equation given by (84). The left panel shows the exact transfer function $G\left(s\right)$ with ${\chi}_{0}=1$ and $\tau =1$ in connection with the approximation $\tilde{G}\left(s\right)$ using Thiele’s algorithm (dashed line). The right panel shows the local error between the exact transfer function and its approximation. The total number of points used in the approximation is $m=2N+1=35$.

**Figure 3.**Solution of the Debye relaxation Equation (84) generated by the inverse Laplace transform of $\tilde{G}\left(s\right)$. The left panel also includes the exact solution of the relaxation equation given by $\chi \left(t\right)={\chi}_{0}exp(-(t/\tau ))$ (dashed) and the approximation (solid line). The right panel shows the local error of the exact solution and the approximation. The number of approximation points used in the calculation was $m=2N+1=35$. The parameters to generate the plot are ${\chi}_{0}=1$ and $\tau =1$.

**Figure 4.**Solution of Debye’s relaxation Equation (84) generated by the Sinc inverse Laplace transform based on indefinite integrals. The top left panel includes the exact solution of the relaxation equation given by $\chi \left(t\right)={\chi}_{0}exp(-(t/\tau ))$ (dashed) and the approximation (solid line). The top right panel shows the absolute local error of the difference between the exact solution and the approximation. The number of approximation points used in the calculation was $m=2N+1=769$. The parameters to generate the plot are ${\chi}_{0}=1$ and $\tau =1$. The bottom panels (left) show the error decay as a function of Sinc points N, ${E}_{N}\sim \sqrt{N}exp\left(\right)open="("\; close=")">-{k}_{1}{N}^{1/2}$. Dots represents numerically determined ${L}^{2}$ norms, and the solid line represents Equation (57) with $c=0$, where ${K}_{1}$ and ${k}_{1}$ are adapted accordingly. The right panel shows the structure of the transfer function $G\left(s\right)$ on the complex plane $\mathbb{C}$.

**Figure 5.**Approximation of the transfer function $G\left(s\right)$ for a fractional relaxation Equation (88). The left panel shows the exact transfer function $G\left(s\right)$ with ${\chi}_{0}=1$, $\tau =1$, and $\alpha =2/3$ in connection with the approximation $\tilde{G}\left(s\right)$ using Thiele’s algorithm. The right panel shows the local error between the exact function $G\left(s\right)$ and its approximation. The total number of points used in the approximation is $m=2N+1=35$.

**Figure 6.**Solution of the fractional relaxation Equation (88) generated by the inverse Laplace transform of $\tilde{G}\left(s\right)$. The left panel also includes the exact solution of the fractional relaxation equation given by $\chi \left(t\right)={\chi}_{0}{E}_{\alpha}\left(\right)open="("\; close=")">-{(t/\tau )}^{\alpha}$ where ${E}_{\alpha}\left(t\right)$ is the Mittag-Leffler function. The right panel shows the local error between the exact solution and the approximation. The number of approximation points used in the calculation was $m=2N+1=35$. The parameters to generate the plot are ${\chi}_{0}=1$, $\tau =1$, $\alpha =2/3$.

**Figure 7.**Solution of the fractional relaxation Equation (90) generated by the inverse Laplace transform of $G\left(s\right)$. The left panel on top also includes the exact solution of the fractional relaxation equation given by $\chi \left(t\right)={\chi}_{0}{E}_{\alpha}\left(\right)open="("\; close=")">-{(t/\tau )}^{\alpha}$, where ${E}_{\alpha}\left(t\right)$ is the one parameter Mittag-Leffler function. The right panel on top shows the local error between the exact solution and the approximation. The number of approximation points used in the calculation was $m=2N+1=1025$. The parameters to generate the plot are ${\chi}_{0}=1$, $\tau =1$, $\alpha =3/4$. The bottom panels (left) show the error decay as a function of Sinc points N, ${E}_{N}\sim \sqrt{N}exp\left(\right)open="("\; close=")">-{k}_{1}{N}^{1/2}$. Dots represent numerically determined ${L}^{2}$ norms, and the solid line represents Equation (57) with $c=0$, where ${K}_{1}$ and ${k}_{1}$ are adapted accordingly. The right panel shows the pole structure with a branch cut along the negative real axis of the transfer function $G\left(s\right)$ on the complex plane $\mathbb{C}$.

**Figure 8.**A variety of solutions of (88) for different values of $\alpha $. From top to bottom on the right end of the graph, the $\alpha $ values vary between 0.05 and 1 in steps of 0.01. The solid line represents the Sinc inverse Laplace result, while the dashed line is the Mathematica implementation of ${E}_{\alpha}\left(\right)open="("\; close=")">-{(t/\tau )}^{\alpha}$. Parameters are ${\chi}_{0}=1$, $\tau =1$, and $N=256$.

**Figure 9.**Approximation of the transfer function $G\left(s\right)$ for a fractional relaxation equation given by (94). The left panel shows the exact transfer function $G\left(s\right)$ with ${\chi}_{0}=1$, $\tau =1$, $\alpha =1/2$, and $\beta =1/3$ in connection with the approximation using Thiele’s algorithm. The right panel shows the local error between the exact function and its approximation. The total number of points used in the approximation is $m=2N+1=35$.

**Figure 10.**Solution of the fractional relaxation Equation (92) generated by the inverse Laplace transform of $\tilde{G}\left(s\right)$. The left panel also includes the exact solution of the fractional relaxation equation given by $\chi \left(t\right)={\chi}_{0}{t}^{\beta -1}{E}_{\alpha ,\beta}\left(\right)open="("\; close=")">-{(t/\tau )}^{\alpha}$, where ${E}_{\alpha ,\beta}\left(z\right)$ is the two parameter Mittag-Leffler function. The right panel shows the local error between the exact solution and the approximation. The number of approximation points used in the calculation was $m=2N+1=35$. The parameters to generate the plot are ${\chi}_{0}=1$, $\tau =1$, $\alpha =1/2$, and $\beta =1/3$.

**Figure 11.**Approximation of the fractional relaxation Equation (92) generated by the SG inverse Laplace transform of $G\left(s\right)$. The left panel on top includes the exact solution of the fractional relaxation equation given by $\chi \left(t\right)={\chi}_{0}{t}^{\beta -1}{E}_{\alpha ,\beta}\left(\right)open="("\; close=")">-{(t/\tau )}^{q}$, where ${E}_{\alpha ,\beta}\left(t\right)$ is the Mittag-Leffler function. The right panel on top shows the local error between the exact solution and the approximation. The number of approximation points used in the calculation was $m=2N+1=2049$. The parameters to generate the plot are ${\chi}_{0}=1$, $\tau =1$, $\alpha =2/3$, and $\beta =3/4$. The bottom panels (left) show the error decay as a function of Sinc points N, ${E}_{N}\sim \sqrt{N}exp\left(\right)open="("\; close=")">-{k}_{1}{N}^{1/2}$. Dots represents numerically determined ${L}^{2}$ norms, and the solid line represents Equation (57) with $c=1/150$, where ${K}_{1}$ and ${k}_{1}$ are adapted accordingly. The right panel shows the pole structure with a branch cut along the negative real axis of the transfer function $G\left(s\right)$ on the complex plane $\mathbb{C}$.

**Figure 12.**A variety of solutions of (94) for different values of $\alpha $. From top to bottom on the right end of the graph the $\alpha $ values vary between 0.1 and 1.5 in steps of 0.1 with $\beta =11/9$. The solid line represents the SG inverse Laplace result, while the overlaid dashed line is the Mathematica implementation of ${t}^{\beta -1}{E}_{\alpha ,\beta}\left(\right)open="("\; close=")">-{(t/\tau )}^{\alpha}$. Parameters are ${\chi}_{0}=1$, $\tau =1$, and $N=96$.

**Figure 13.**An example where the condition $\alpha <\beta $ is violated. Parameters are ${\chi}_{0}=1$, $\tau =1$, $\alpha =1/2$, and $\beta =7/20$. The pole of the transfer function is replaced by a singularity at zero (right panel).

**Figure 14.**Approximation of the transfer function $G\left(s\right)$ for a three parameter fractional relaxation equation represented by (96). The left panel shows the exact transfer function $G\left(s\right)$ with ${\chi}_{0}=1$, $\tau =1$, $\alpha =1/2$, $\beta =2/3$, and $\gamma =9/10$ in connection with the approximation using Thiele’s algorithm. The right panel shows the local error between the exact function and its approximation. The total number of point used in the approximation is $m=2N+1=35$.

**Figure 15.**Solution of the fractional relaxation Equation (96) generated by the inverse Laplace transform of $\tilde{G}\left(s\right)$. The left panel also includes the two parameter ML function ${t}^{\beta -1}{E}_{\alpha .\beta}\left(\right)open="("\; close=")">-{(t/\tau )}^{\alpha}$ as a reference. The right panel shows the local error between the reference and the approximation. The number of approximation points used in the calculation was $m=2n+1=35$. The parameters to generate the plot are ${\chi}_{0}=1$, $\tau =1$, $\alpha =1/2$, $\beta =2/3$, and $\gamma =9/10$.

**Figure 16.**Three different parameter selections for Prabhakar’s function $\chi \left(t\right)={\chi}_{0}{t}^{\beta -1}{E}_{\alpha ,\beta}^{\gamma}\left(\right)open="("\; close=")">-{(t/\tau )}^{q}$. The left panels include the SG approximation (solid line) and the series representation truncated by 126 terms. The right panels show the relative absolute local errors using the series approximation as reference. The number of approximation points used in the calculation was $N=128$. The parameter sets used from top to bottom are {${\chi}_{0}=1$, $\tau =1$, $\alpha =7/10$, $\beta =3/10$, $\gamma =22/10$}, {${\chi}_{0}=1$, $\tau =1$, $\alpha =7/10$, $\beta =3/10$, $\gamma =12/10$}, {${\chi}_{0}=1$, $\tau =1$, $\alpha =7/10$, $\beta =9/10$, and $\gamma =8/10$}.

**Figure 17.**A variety of solutions of the HN model with $\beta =\alpha \gamma $ for different values of $\gamma $. From top to bottom on the right end of the graph, the $\gamma $ values varied and $\alpha $ kept fixed. The solid line represents the SG inverse Laplace result, while the dashed line is the series implementation of ${t}^{\beta -1}{E}_{\alpha ,\beta}^{\gamma}\left(\right)open="("\; close=")">-{(t/\tau )}^{\alpha}$ using 126 terms. The SG inverse Laplace results are not limited to the shown scale. The limitation results here by using the series representation. Parameters are ${\chi}_{0}=1$, $\tau =1$, $\alpha =1/2$, and $\gamma $ is taken from [3/5,2] in steps of 1/10. The number of Sinc points is $N=128$.

**Figure 18.**A variety of solutions with $\beta =2\alpha \gamma $ for different values of $\gamma $. The solid line represents the SG inverse Laplace result, while the dashed lines are the series approximation of ${t}^{\beta -1}{E}_{\alpha ,\beta}^{\gamma}\left(\right)open="("\; close=")">-{(t/\tau )}^{\alpha}$ using 126 terms. The limitation on the left panel for the range results from using the series representation and the bad convergence for large arguments. The right panel shows the same SG inverse Laplace transform on a wider range. Parameters are ${\chi}_{0}=1$, $\tau =1$, $\alpha =4/5$, and $\gamma $ is taken from [2/5,2] in steps of 1/10. The number of Sinc points is $N=128$.

**Figure 19.**Variable order fractional relaxation with $\alpha \left(t\right)=1-{t}^{k}{e}^{-t}$ and $k=0,1,2,3$. The number of Sinc points is $N=128$.

**Figure 20.**Extensions of Scarpi’s fractional equation. Left panel $\alpha \left(t\right)=1\left(\right)open="/"\; close>cosh{\left(t\right)}^{2}$ and $\beta \left(t\right)=cosh\left(t\right)$ with $N=128$. Right panel the same functions for $\alpha \left(t\right)$ and $\beta \left(t\right)$ as in the left panel, but $\gamma \left(t\right)=sin\left(t\right)$ and $N=256$.

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**MDPI and ACS Style**

Baumann, G.
Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus. *Fractal Fract.* **2021**, *5*, 43.
https://doi.org/10.3390/fractalfract5020043

**AMA Style**

Baumann G.
Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus. *Fractal and Fractional*. 2021; 5(2):43.
https://doi.org/10.3390/fractalfract5020043

**Chicago/Turabian Style**

Baumann, Gerd.
2021. "Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus" *Fractal and Fractional* 5, no. 2: 43.
https://doi.org/10.3390/fractalfract5020043