/ / Article

Research Article | Open Access

Volume 2020 |Article ID 1274251 | https://doi.org/10.1155/2020/1274251

Xin Liu, Kamran, Yukun Yao, "Numerical Approximation of Riccati Fractional Differential Equation in the Sense of Caputo-Type Fractional Derivative", Journal of Mathematics, vol. 2020, Article ID 1274251, 12 pages, 2020. https://doi.org/10.1155/2020/1274251

# Numerical Approximation of Riccati Fractional Differential Equation in the Sense of Caputo-Type Fractional Derivative

Revised06 Aug 2020
Accepted11 Aug 2020
Published07 Sep 2020

#### Abstract

The Riccati differential equation is a well-known nonlinear differential equation and has different applications in engineering and science domains, such as robust stabilization, stochastic realization theory, network synthesis, and optimal control, and in financial mathematics. In this study, we aim to approximate the solution of a fractional Riccati equation of order with Atangana–Baleanu derivative (ABC). Our numerical scheme is based on Laplace transform (LT) and quadrature rule. We apply LT to the given fractional differential equation, which reduces it to an algebraic equation. The reduced equation is solved for the unknown in LT space. The solution of the original problem is retrieved by representing it as a Bromwich integral in the complex plane along a smooth curve. The Bromwich integral is approximated using the trapezoidal rule. Some numerical experiments are performed to validate our numerical scheme.

#### 1. Introduction

In applied mathematics, the fractional differential equation (FDE) is an equation which contains derivatives of arbitrary order. Fractional derivatives first appeared in 1695 . Recently, the research community took more interest in fractional calculus because of its applications in engineering and other sciences . Many physical systems show fractional order behavior that may vary with time or space. Fractional derivatives have many kinds. The three important and most commonly used fractional derivatives are Grnwald–Letnikov derivative, Riemann–Liouville fractional derivative, and Caputo derivative . However, these classical fractional derivatives have a singular kernel, and hence, they may face difficulties in describing the nonlocality of real-world dynamics. In order to handle the nonlocal systems in a better way, recently, new fractional derivatives with nonsingular kernels are defined such as the Caputo–Fabrizio (CF) derivative and Atangana–Baleanu (ABC) derivative [6, 7]. Fractional derivatives with nonsingular kernels become more valuable due to the fact that numerous phenomena cannot properly be modeled by fractional derivatives with singular kernels .

In this work, we aim to approximate a Riccati differential equation (RDE) with ABC derivative. RDEs have many applications such as random processes, optimal control, and diffusion process . The RDE of fractional order has been studied by many authors; for example, in , the authors developed the Adomain decomposition method for the solution of RDE of fractional order. In , some analytic techniques are presented for the solution of RDE. The authors  obtained the solution of RDE using the differential transform method. In , the authors have developed a Laplace-transform Adomain decomposition method for the solution of RDE. Other works on the analytic solution of RDE can be found in [14, 15] and the references therein.

Most of the time, the exact/analytical solution of FDEs cannot be found, so numerical approximations must be utilized . Numerous numerical methods have been developed for numerical approximation of FDEs such as the Chebyshev collocation method , variation iteration method , reproducing kernel Hilbert space method [18, 19], and homotopy perturbation method ( and references therein). The numerical solutions of fractional-order RDE have been studied by a large number of researchers. For example, the authors  studied fractional-order RDE with ABC derivatives, and they established the existence and uniqueness results using Banach fixed-point theorem. A reproducing kernel Hilbert space method  for approximating RDEs and Bernoulli differential equations of fractional order with ABC derivative has been presented. In , the authors developed an iterative reproducing kernel Hilbert space method for numerical approximation of fractional RDE. The authors  studied the numerical solution of fractional-order RDE using the modified homotopy perturbation method. Numerical solution of fractional-order RDEs using Bernstein polynomials is considered . A fractional Chebyshev finite difference method  for numerical investigation of RDE of fractional order is proposed. Jafari and Tajadodi  proposed a variational iteration method for solving the RDE of fractional order. The authors in  proposed a hybrid method based on the quasilinearization technique and reproducing kernel method for RDE of fractional order. In , the solution of fractional-order RDE is approximated using fractional polynomial approximations. The authors in  developed a method based on finite difference and Padé-variational iteration method for solving the RDE of noninteger order. A numerical method based on the path following methods and the Tau Legendre is presented  for the solution of fractional-order RDE. The authors of  proposed a modified variational iteration method based on Adomain polynomials for the solution of RDE. Khashan et al.  have utilized the Haar wavelets for the approximation of fractional-order RDE. The authors of  utilized the finite difference scheme for the approximation of RDE. Other works on the numerical approximation of the solution of fractional-order RDE can be found in  and references therein. In this work, we approximate the solution of fractional-order RDE with ABC derivative of the following form:here, denotes Atangana–Baleanu fractional derivative of order .

#### 2. Preliminaries

In this section, we present some basic results and definitions. For details about fractional calculus, we refer , and for details about modeling, we refer [50, 51] and references therein.

Definition 1. The Mittag–Leffler (ML) function with one parameter is defined as [52, 53]

Definition 2. The two-parameter Mittag–Leffler (ML) function is defined as [53, 54]

Definition 3. The ABC fractional derivative of order of with base point at is defined as where is a first-order Sobolev space equipped with -norm over the region , which is defined asand the term is given as

Definition 4. The (ABC) fractional integral of order of with base point at is defined as 

Definition 5. The Laplace transform of a piecewise continuous function is defined as

Definition 6. The LT of function with one parameter is defined as 

Definition 7. The LT of two-parameter function is defined as 

Definition 8. If , then the LT of the ABC derivative is defined by 

#### 3. Laplace Transform Method for Fractional Riccati Equation with ABC Derivative

In this section, we give a detailed description of our proposed numerical scheme for approximating the inverse Laplace transform. First, we will apply the Laplace transform to the given fractional problem, which will transform it to an algebraic equation. After solving the reduced equation, the solution of the original problem can be obtained by representing it as a contour integral in the left half of the complex plane. The trapezoidal rule is then utilized to approximate the contour integral. Applying the Laplace transform to equation (1), we obtainwhich can be written in simplified form aswhere

In our method, first we represent the solution of original problem (1) as a contour integral:where, for , is appropriately large and is an initially appropriately chosen line perpendicular to the real axis in the complex plane, with . Integral (15) is just the inverse transform of , with the condition that it must be analytic to the right of . To make sure the contour of integration remains in the domain of analyticity of , we select as a deformed contour in the set , which behaves as a pair of asymptotes in the left half plane, with when , which force to decay towards both ends of . In our work, we choose aswhere

By writing , we notice that (16) is the left branch of the following hyperbola:The asymptotes for (18) are and x-intercept at . Condition (17) confirms that lies in the sector and grows into the left half plane. From (16) and (15), we obtain

The trapezoidal rule is used for the approximation of equation (19) with step as follows:where .

#### 4. Error Analysis

In the process of obtaining the solution of problem equations (1)–(13), the fractional integrodifferential equation is first transformed to an algebraic equation using Laplace transform, and this causes no error. The transformed equation is then solved for the unknown in the Laplace space. Finally, the solution is obtained using inverse Laplace transform via integral representation (19). The integral is then approximated using quadrature rule. In the process of approximating integral (19), convergence is achieved at different rates depending on the path . In approximating integral (19), the convergence order relies on the step of the quadrature rule and the time domain . The proof for the order of quadrature error is given in the next theorem.

Theorem 1 ( Theorem 2.1). Let be the solution of (1) with being analytic in . Let , and define by , where , , and and let . Then, for equation (20), with , we have , for , , , , , and . Hence, the error estimate for the proposed scheme is

#### 5. Results and Discussion

Most of the time analytical methods cannot be applied to handle a real-world problem. So we need numerical methods to approximate the solutions of the problems. In this section, we consider fractional-order Riccati equations to validate our method.

Problem 1. Here, we consider the fractional Riccati equation as follows:with .
The problem has exact solution . In this experiment, the optimal parameters utilized are . The quadrature nodes are generated using the MATLAB commands . The values of approximate solution for different fractional orders and are depicted in Table 1. The absolute errors for various quadrature nodes and fractional order are depicted in Table 2. The method produced almost exact values for different values of fractional order . The plots of absolute error for fractional orders are displayed in Figure 1(a), where Figure 1(b) shows the comparison between the absolute error and error estimate for the fractional order . Figures 2(a) and 2(b) show the error functions for fractional orders and , respectively. It can be seen that the method can solve fractional Riccati equation with ABC derivative efficiently.

 0.1 1.414213562373095 1.414213562373095 1.414213562373095 1.414213562373095 1.414213562373095 0.2 1.414213562373094 1.414213562373094 1.414213562373094 1.414213562373094 1.414213562373094 0.3 1.414213562373096 1.414213562373095 1.414213562373096 1.414213562373095 1.414213562373096 0.4 1.414213562373100 1.414213562373100 1.414213562373100 1.414213562373100 1.414213562373100 0.5 1.414213562373101 1.414213562373101 1.414213562373101 1.414213562373101 1.414213562373101 0.6 1.414213562373092 1.414213562373093 1.414213562373092 1.414213562373093 1.414213562373093 0.7 1.414213562373096 1.414213562373096 1.414213562373097 1.414213562373096 1.414213562373097 0.8 1.414213562373100 1.414213562373100 1.414213562373101 1.414213562373100 1.414213562373101 0.9 1.414213562373099 1.414213562373100 1.414213562373099 1.414213562373100 1.414213562373100 1 1.414213562373098 1.414213562373098 1.414213562373097 1.414213562373097 1.414213562373097
 30 60 90 120 150 180 210 250 

Problem 2. Here, we consider the fractional Riccati equation as follows:with .
The problem has exact solution . In this experiment, the optimal parameters utilized are . The quadrature nodes are generated using the MATLAB commands . The values of approximate solution for different fractional orders and are depicted in Table 3. The method produced exact values for . The plots of numerical numerical solutions for different fractional orders are displayed in Figure 3, where Figure 4(a) shows the absolute error for the fractional order and Figure 4(b) shows the error function for .

 0.1 0.099999999999997 0.231168929042977 0.378890254407207 0.540603871314406 0.2 0.199999999999935 0.339457009420931 0.492611583158604 0.654981683985162 0.3 0.299999999999814 0.442211492048623 0.594981117995457 0.752630554959894 0.4 0.400000000000399 0.541529712645641 0.690611916885552 0.840788989032917 0.5 0.500000000000783 0.638368560911693 0.781525188669415 0.922501829604263 0.6 0.600000000000226 0.733276357205215 0.868847629153083 0.999420047678484 0.7 0.700000000001987 0.826607814998613 0.953292789025402 1.072567200353862 0.8 0.799999999997327 0.918610710161770 1.035350520844109 1.142633846360632 0.9 0.899999999996349 1.009467601183178 1.115376469957896 1.210114016809452 1 0.999999999994301 1.099318390756997 1.193639720785398 1.275376747522151

Problem 3. Here, we consider the fractional Riccati equation as follows:withwhere,The problem has exact solution . In this experiment, the same set of optimal parameters is utilized. The values of approximate solution for different fractional orders and are depicted in Table 4. The absolute errors for various values of and fractional order are shown in Table 5. Figure 5 shows the comparison between the absolute error and error_est, and a good agreement between them is observed. The plots of numerical and exact solutions for different fractional orders are displayed in Figures 69. It can be seen that the proposed method has produced good results and the exact and numerical solutions are in good agreement. This shows that this method can solve fractional Riccati equation with ABC derivative efficiently.

 0.1 1.010000000081090 1.013000508297990 1.016475393193733 1.020156483509860 0.2 1.039999999904037 1.048519612196495 1.057370650922991 1.065488607772575 0.3 1.090000000107037 1.104831237979493 1.119029345287758 1.130473075366759 0.4 1.160000000118295 1.181081596140041 1.199776210363150 1.212773114819303 0.5 1.250000000104207 1.276696291556891 1.298525747082721 1.310930820826873 0.6 1.359999999746213 1.391244013660800 1.414484253729630 1.423908271904983 0.7 1.489999999970782 1.524380587892006 1.547031402579680 1.550910987123471 0.8 1.639999999836653 1.675820412830372 1.695661169283945 1.691301889020929 0.9 1.810000000220293 1.845319910066929 1.859948222231272 1.844553239275344 1 1.999999999918102 2.032667074163037 2.039527028680209 2.010217216849950
 30 60 90 120 150 180 210 250 252 

Problem 4. Here, we consider the fractional Riccati equation as follows:with
The problem has exact solution . In this experiment, the same set of optimal parameters is utilized. The values of approximate solution for different fractional orders and are depicted in Table 6. Figure 10(a) displays the comparison between the absolute error and error estimate. Figure 10(b) shows the error function for .

 0.1 0.904837418035961 0.878343285218955 0.851704707572839 0.824831134877866 0.2 0.818730753077981 0.795360306677717 0.773306387788940 0.752558353585973 0.3 0.740818220681718 0.722946506074075 0.707343502809177 0.693926556750935 0.4 0.670320046035642 0.659000108719911 0.650476193469615 0.644550914567384 0.5 0.606530659712630 0.602175814084098 0.600817516047575 0.602146031627945 0.6 0.548811636094031 0.551473582617326 0.557090577010358 0.565261898262492 0.7 0.496585303791411 0.506098747938276 0.518352782107219 0.532883565755115 0.8 0.449328964117216 0.465396594634668 0.483872422015733 0.504257605945836 0.9 0.406569659740603 0.428815700079375 0.453062839223986 0.478802532603470 1 0.367879441171453 0.395884829945778 0.425443065181985 0.456057014564435

Problem 5. Here, we consider the fractional Riccati equation as follows:with
The problem has exact solution . In this experiment, the same set of optimal parameters is utilized. The values of approximate solution for different fractional orders and are displayed in Table 7. The comparison between absolute error and error estimate is shown in Figure 11(a), and Figure 11(b) shows the error function for fractional order .

 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

#### 6. Conclusion

In this work, we developed a numerical scheme based on LT and inverse LT for approximation of the solution of fractional Riccati equations with ABC derivative. The inverse LT is approximated using the quadrature rule. The proposed method approximated the fractional Riccati equation with ABC derivative accurately and efficiently. From the results, it can be seen that this method is an excellent alternative for approximation of such type of equations.

#### Data Availability

Data are included within this paper.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Authors’ Contributions

Xin Liu improved literature review and gave numerical applications of results and present examples. Kamran wrote the paper, and Yukun Yao gave error analysis and plotted error functions.

#### Acknowledgments

This research was partitively supported by the funds of HEC Pakistan.

1. U. N. Katugampola, “A new approach to generalized fractional derivatives,” Bulletin of Mathematical Analysis and Applications, vol. 6, no. 4, pp. 1–15, 2014. View at: Google Scholar
2. D. Baleanu, A. Jajarmi, S. S. Sajjadi, and D. Mozyrska, “A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 29, no. 8, Article ID 083127, 2019. View at: Publisher Site | Google Scholar
3. A. Atangana and J. F. Gómez-Aguilar, “Decolonisation of fractional calculus rules: breaking commutativity and associativity to capture more natural phenomena,” The European Physical Journal Plus, vol. 133, no. 4, p. 166, 2018. View at: Publisher Site | Google Scholar
4. I. Podlubny, Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198, Elsevier, Amsterdam, Netherlands, 1998.
5. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science Limited, Amsterdam, Netherlands, 2006.
6. M. Caputo and M. Fabrizio, “A new definition of fractional derivative without singular kernel,” Progress in Fractional Differentiation and Applications, vol. 1, no. 2, pp. 1–13, 2015. View at: Google Scholar
7. 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
8. J. Hristov, “On the Atangana-Baleanu derivative and its relation to the fading memory concept: the diffusion equation formulation,” in Fractional Derivatives with Mittag-Leffler Kernel. Studies in Systems, Decision and Control, Springer, Cham, Switzerland, 2019. View at: Publisher Site | Google Scholar
9. W. T. Reid, Riccati Differential Equations, Elsevier, Amsterdam, Netherlands, 1972.
10. S. Momani and N. Shawagfeh, “Decomposition method for solving fractional Riccati differential equations,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1083–1092, 2006. View at: Publisher Site | Google Scholar
11. Y. Pala and M. O. Ertas, “An analytical method for solving general riccati equation,” International Journal of Mathematical and Computational Sciences, vol. 11, no. 3, pp. 125–130, 2017. View at: Google Scholar
12. J. Biazar and M. Eslami, “Differential transform method for quadratic riccati differential equation,” International Journal of Nonlinear Science, vol. 9, no. 4, pp. 444–447, 2010. View at: Google Scholar
13. P.-Y. Tsai and C. o.-K. Chen, “An approximate analytic solution of the nonlinear riccati differential equation,” Journal of the Franklin Institute, vol. 347, no. 10, pp. 1850–1862, 2010. View at: Publisher Site | Google Scholar
14. M. A. Z. Raja, I. M. Qureshi, and J. A. Khan, “A new stochastic approach for solution of riccati differential equation of fractional order,” Annals of Mathematics and Artificial Intelligence, vol. 60, no. 3-4, pp. 229–250, 2010. View at: Publisher Site | Google Scholar
15. H. Aminikhah, A. H. R. Sheikhani, and H. Rezazadeh, “Approximate analytical solutions of distributed order fractional riccati differential equation,” Ain Shams Engineering Journal, vol. 9, no. 4, pp. 581–588, 2018. View at: Publisher Site | Google Scholar
16. N. H. Sweilam, M. M. Khader, and R. F. Al-Bar, “Numerical studies for a multi-order fractional differential equation,” Physics Letters A, vol. 371, no. 1-2, pp. 26–33, 2007. View at: Publisher Site | Google Scholar
17. M. A. Snyder, Chebyshev Methods in Numerical Approximation, vol. 2, Prentice-Hall, Upper Saddle River, NJ, USA, 1966.
18. M. G. Sakar, A. Akgül, and D. Baleanu, “On solutions of fractional Riccati differential equations,” Advances in Difference Equations, vol. 2017, p. 39, 2017. View at: Publisher Site | Google Scholar
19. O. A. Arqub and B. Maayah, “Modulation of reproducing kernel Hilbert space method for numerical solutions of Riccati and Bernoulli equations in the Atangana-Baleanu fractional sense,” Chaos, Solitons & Fractals, vol. 125, pp. 163–170, 2019. View at: Google Scholar
20. M. I. Syam and M. Al-Refai, “Fractional differential equations with Atangana-Baleanu fractional derivative: analysis and applications,” Chaos, Solitons & Fractals: X, vol. 2, Article ID 100013, 2019. View at: Publisher Site | Google Scholar
21. M. M. Khader, “Introducing an efficient modification of the homotopy perturbation method by using Chebyshev polynomials,” Arab Journal of Mathematical Sciences, vol. 18, no. 1, pp. 61–71, 2012. View at: Publisher Site | Google Scholar
22. Z. Odibat and S. Momani, “Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order,” Chaos, Solitons & Fractals, vol. 36, no. 1, pp. 167–174, 2008. View at: Publisher Site | Google Scholar
23. Ş. Yüzbaşı, “Numerical solutions of fractional Riccati type differential equations by means of the Bernstein polynomials,” Applied Mathematics and Computation, vol. 219, no. 11, pp. 6328–6343, 2013. View at: Google Scholar
24. M. M. Khader, “Numerical treatment for solving fractional Riccati differential equation,” Journal of the Egyptian Mathematical Society, vol. 21, no. 1, pp. 32–37, 2013. View at: Publisher Site | Google Scholar
25. H. Jafari and H. Tajadodi, “He’s variational iteration method for solving fractional Riccati differential equation,” International Journal of Differential Equations, vol. 2010, Article ID 764738, 8 pages, 2010. View at: Publisher Site | Google Scholar
26. X. Y. Li, B. Y. Wu, and R. T. Wang, “Reproducing kernel method for fractional Riccati differential equations,” Abstract and Applied Analysis, vol. 2014, Article ID 970967, 6 pages, 2014. View at: Publisher Site | Google Scholar
27. M. Izadi, “Fractional polynomial approximations to the solution of fractional Riccati equation,” Punjab University Journal of Mathematics, vol. 51, no. 11, pp. 123–141, 2019. View at: Google Scholar
28. N. H. Sweilam, M. M. Khader, and A. M. S. Mahdy, “Numerical studies for solving fractional Riccati differential equation,” Applications and Applied Mathematics, vol. 7, no. 2, pp. 595–608, 2012. View at: Google Scholar
29. M. I. Syam, H. I. Siyyam, and I. Al-Subaihi, “Tau-Path following method for solving the Riccati equation with fractional order,” Journal of Computational Methods in Physics, vol. 2014, Article ID 207916, 7 pages, 2014. View at: Publisher Site | Google Scholar
30. H. Jafari, H. Tajadodi, and D. Baleanu, “A modified variational iteration method for solving fractional Riccati differential equation by Adomian polynomials,” Fractional Calculus and Applied Analysis, vol. 16, no. 1, pp. 109–122, 2013. View at: Publisher Site | Google Scholar
31. M. M. Khashan, R. Amin, and M. I. Syam, “A new algorithm for fractional riccati type differential equations by using haar wavelet,” Mathematics, vol. 7, no. 6, p. 545, 2019. View at: Publisher Site | Google Scholar
32. B. S. Kashkari and M. I. Syam, “A numerical approach for investigating a special class of fractional riccati equation,” Results in Physics, vol. 17, Article ID 103080, 2020. View at: Publisher Site | Google Scholar
33. W. M. Abd-Elhameed and Y. H. Youssri, “New ultraspherical wavelets spectral solutions for fractional Riccati differential equations,” Abstract and Applied Analysis, vol. 2014, Article ID 626275, 8 pages, 2014. View at: Publisher Site | Google Scholar
34. M. Merdan, “On the solutions fractional Riccati differential equation with modified Riemann-Liouville derivative,” International Journal of Differential Equations, vol. 2012, Article ID 346089, 17 pages, 2012. View at: Publisher Site | Google Scholar
35. H. Yaslan, “Numerical solution of fractional Riccati differential equation via shifted Chebyshev polynomials of the third kind,” Journal of Engineering Technology and Applied Sciences, vol. 2, no. 1, pp. 1–11, 2017. View at: Publisher Site | Google Scholar
36. B. Lu, “Bäcklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations,” Physics Letters A, vol. 376, no. 28-29, pp. 2045–2048, 2012. View at: Google Scholar
37. S. W. Jeng and A. Kilicman, “Fractional riccati equation and its applications to rough heston model using numerical methods,” Symmetry, vol. 12, no. 6, p. 959, 2020. View at: Publisher Site | Google Scholar
38. H. Z. Mjthap and S. N. Al-Azzawi, “Mixing sumudu transform and adomain decomposition method for solving riccati equation of variable fractional order,” Journal of Interdisciplinary Mathematics, vol. 22, no. 8, pp. 1559–1563, 2019. View at: Publisher Site | Google Scholar
39. A. Turan Dincel, “Solution to fractional-order riccati differential equations using euler wavelet method,” Scientia Iranica, vol. 26, no. 3, pp. 1608–1616, 2019. View at: Google Scholar
40. S. N. Tural-Polat, “Third-kind Chebyshev wavelet method for the solution of fractional order riccati differential equations,” Journal of Circuits, Systems and Computers, vol. 28, no. 14, Article ID 1950247, 2019. View at: Publisher Site | Google Scholar
41. W. M. Abd-Elhameed and Y. H. Youssri, “Explicit shifted second-kind Chebyshev spectral treatment for fractional riccati differential equation,” Computer Modeling in Engineering & Sciences, vol. 121, no. 3, pp. 1029–1049, 2019. View at: Publisher Site | Google Scholar
42. J. Biazar, L. B. Hasani, and Z. Ayati, “Asymptotic decomposition method for fractional order riccati differential equation,” Computational Methods for Differential Equations, 2020. View at: Google Scholar
43. S. Mehmood, G. Farid, and G. Farid, “Fractional integrals inequalities for exponentially \(m\)-convex functions,” Open Journal of Mathematical Sciences, vol. 4, no. 1, pp. 78–85, 2020. View at: Publisher Site | Google Scholar
44. S. Mehmood, G. Farid, K. A. Khan, and M. Yussouf, “New hadamard and fejér–hadamard fractional inequalities for exponentially m-convex function,” Engineering and Applied Science Letter, vol. 3, pp. 45–55, 2020. View at: Publisher Site | Google Scholar
45. S. Mehmood, G. Farid, K. A. Khan, and M. Yussouf, “New fractional hadamard and fejér–hadamard inequalities associated with exponentially (h, m)-convex functions,” Engineering and Applied Science Letter, vol. 3, pp. 9–18, 2020. View at: Google Scholar
46. S. I. Butt, M. Nadeem, and G. Farid, “On caputo fractional derivatives via exponential (s, m)-convex functions,” Engineering and Applied Science Letter, vol. 3, no. 2, pp. 32–39, 2020. View at: Google Scholar
47. Y. C. Kwun, G. Farid, W. Nazeer, S. Ullah, and S. M. Kang, “Generalized riemann-liouville k-fractional integrals associated with ostrowski type inequalities and error bounds of hadamard inequalities,” IEEE Access, vol. 6, pp. 64946–64953, 2018. View at: Publisher Site | Google Scholar
48. Y. C. Kwun, M. S. Saleem, M. Ghafoor, W. Nazeer, and S. M. Kang, “Hermite–hadamard-type inequalities for functions whose derivatives are η-convex via fractional integrals,” Journal of Inequalities and Applications, vol. 2019, p. 44, 2019. View at: Publisher Site | Google Scholar
49. S. Kang, G. Abbas, G. Farid, and W. Nazeer, “A generalized fejér–hadamard inequality for harmonically convex functions via generalized fractional integral operator and related results,” Mathematics, vol. 6, no. 7, p. 122, 2018. View at: Publisher Site | Google Scholar
50. A. Singh and A. Prakash, “Risk evaluation in information systems using continuous and discrete distribution laws,” Engineering and Applied Science Letters, vol. 3, no. 1, pp. 35–44, 2020. View at: Publisher Site | Google Scholar
51. C. M. D. Simarmata, N. Susyanto, I. J. Hammadi, and C. Rahmaditya, “A mathematical model of smoking behaviour in Indonesia with density-dependent death rate,” Open Journal of Mathematical Sciences, vol. 4, no. 1, pp. 118–125, 2020. View at: Google Scholar
52. G. M. Mittag-Leffler, “Sur la nouvelle fonction E(x),” Comptes Rendus Mathématique, vol. 137, no. 2, pp. 554–558, 1903. View at: Google Scholar
53. P. Humbert and R. P. Agarwal, “Sur la fonction de Mittag-Leffler et quelques-unes de ses généralisations,” Bulletin des Sciences Mathématiques, vol. 77, no. 2, pp. 180–185, 1953. View at: Google Scholar
54. W. McLean and V. Thomée, “Numerical solution via Laplace transforms of a fractional order evolution equation,” Journal of Integral Equations and Applications, vol. 22, no. 1, pp. 57–94, 2010. View at: Publisher Site | Google Scholar

#### More related articles

Article of the Year Award: Outstanding research contributions of 2020, as selected by our Chief Editors. Read the winning articles.