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Advances in Mathematical Physics
Volume 2018, Article ID 6162043, 11 pages
https://doi.org/10.1155/2018/6162043
Research Article

Diffusion Process and Reaction on a Surface

1Departamento de Engenharia Química, Universidade Tecnologica Federal do Paraná, Ponta Grossa, PR 84016-210, Brazil
2Departamento de Engenharia Química, Universidade Federal do Paraná, Curitiba, PR 81531-990, Brazil
3Departamento de Física, Universidade Estadual de Ponta Grossa, Ponta Grossa, PR 87030-900, Brazil
4National Institute of Science and Technology for Complex Systems, Centro Brasileiro de Pesquisas Física, Rio de Janeiro, RJ 22290-180, Brazil

Correspondence should be addressed to E. K. Lenzi; rb.gpeu@iznelke

Received 3 May 2018; Revised 16 July 2018; Accepted 30 July 2018; Published 9 August 2018

Academic Editor: Antonio Scarfone

Copyright © 2018 M. E. K. Fuziki et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We investigate the influence of the surface effects on a diffusive process by considering that the particles may be sorbed or desorbed or undergo a reaction process on the surface with the production of a different substance. Our analysis considers a semi-infinite medium, where the particles may diffuse in contact with a surface with active sites. For the surface effects, we consider integrodifferential boundary conditions coupled with a kinetic equation which takes non-Debye relation process into account, allowing the analysis of a broad class of processes. We also consider the presence of the fractional derivatives in the bulk equations. In this scenario, we obtain solutions for the particles in the bulk and on the surface.

1. Introduction

The description of the dynamic processes present on the surface such as adsorption-desorption or reactions and their influence on the diffusion is very important due to the broadness of applications in several fields such as optimization of industrial processes, electronic devices, the action of pharmaceuticals in the organism, and materials science [1]. Particularly in chemical engineering applications of reactions that occur on solid surfaces are of the great interest and advances have recently been achieved by using the heterogeneous catalysis in the pollutants removal [2], with emphasis on oxidative pathways involving the formation of radicals [35], and in the biodiesel production [68]. In this sense, it is worth not only studying chemical reactions on substrates, but also analyzing different phenomena occurring in a given surface, for example, the adsorption of proteins [912], which can be controlled by nanoroughness on the surface of the material [13, 14] or hybridization of DNA [1517], among others. In these systems, different species may diffuse and react [1821], which implies considering suitable changes in the diffusion equation or in the boundary conditions to account for the processes of interest [18, 19, 22, 23]. These contexts usually are analyzed by using the standard approaches related to Markovian processes. Therefore, extensions of these approaches become very important in order to consider non-Markovian processes where anomalous diffusion or non-Debye relaxations on the kinetic processes are present.

Here, we analyze the surface effects on a diffusion process by considering one-dimensional semi-infinite media in contact with a surface, which may adsorb (and desorb) particles or absorb the particles to perform a reaction process with the formation of another kind of particle. Figure 1 illustrates the scenario analyzed in this manuscript, where two species (particles or substances) and diffuse in the bulk which is in contact with a surface. Species may be adsorbed or sorbed by the surface. For the first process, species is removed from the bulk by the surface and after some time is desorbed with a characteristic time. In the other case, when species is sorbed, a reaction process may occur with the formation of species . In this sense, the reaction only occurs on the surface, for instance, in the presence of an specific catalyst. Thus, species reacts promoting the formation of species , in other words, following a first-order irreversible reaction . This reaction can be described by a kinetic equation [21, 24, 25]. In order to describe these processes, the model considered here is a set of coupled equations that may represent reaction, adsorption, and desorption processes of a substrate on a surface and also the diffusion process in the bulk. These developments are performed in Section 2 by considering generalized diffusion equations [2640] for the species ( and ) in the bulk coupled with kinetic equations as boundary conditions. The discussion and conclusions are presented in Section 3.

Figure 1: Illustration of the interactions occurring between particles and, for example, a catalyst surface. Species in contact with the surface can either be adsorbed or sorbed and by a reaction process produce species , i.e., .

2. Diffusion and Surface Effects

Let us start our analysis about the surface effects on the diffusion processes by considering the system governed in the bulk by the following generalized diffusion equationsandwith , where and are the generalized diffusion coefficients related to particles and , respectively. The quantities and represent the density of each particle present in the bulk and , in (1) and (2), is the integrodifferential operator:where defines how the past history of each system influences the time evolution of and depending on the choice of different fractional differential operators may be obtained. For example, the casecorresponds to the well-known Riemann-Liouville fractional operator [41] for , which has been applied in several situations related to anomalous diffusion [4245]. The exponential kernelwith and being a normalization factor, corresponds to the fractional operator of Caputo-Fabrizio [46]. Further possibilities for describing the kernel are discussed by Gómez-Aguilar et al. [47], in particular the kernel , where is the Mittag-Leffler function [41], which leads us to Atangana-Baleanu fractional operator [4850]. It has been considered in the Bloch system [51], by resulting in a different behavior from the one obtained in the usual context where differential operators of integer order are considered. These operators, by taking a variable order into account, have been used to extend the Gray-Scott reaction-diffusion model which describe irreversible reaction between two species. In this scenario, the Atangana-Baleanu-Caputo fractional differential operator has shown a faster stabilization behavior and Liouville-Caputo fractional differential operator presented a stronger memory effect [52]. From these choices for , we observe that the Riemann-Liouville operator has a singularity at the origin (), while the recently proposed Caputo-Fabrizio and Atangana-Baleanu operators are nonsingular [4648, 5355], and the last one may manifest different regimes of diffusion.

On the surface, we consider that the processes are governed by the following equations:andHere is related to the rate of particle sorption by the surface which by a reaction process produces substance . In addition to (6) and (7), we also have the boundary conditions and to solve (1) and (2). In (6), represents the density of particles which are adsorbed by the surface. For the adsorption and desorption processes on the surface, we assume that they may be modeled by the following kinetic equation [56]:In (8), is the bulk density just in front of the surface, which may be adsorbed by the surface. The parameter is connected to the adsorption phenomenon, being related to a characteristic adsorption time , and is a kernel that governs the desorption phenomenon. Thus, the surface density of adsorbed particles depends on the bulk density of particles just in front of the membrane, and on the surface density of particles already sorbed [56]. Equation (8) also extends the usual kinetic equations of first order to situations characterized by unusual relaxations, i.e., non-Debye relaxations for which a nonexponential behavior of the densities can be obtained, depending on the choice of the kernels [56, 57].

From (1), (2), (6), and (7), it is possible to show thatandIn (9) the term implies the removal of the particles from the bulk by the surface to promote the production of species by a reaction process. Thus, (11) shows that the mass (number of particles) variation on species is connected to the variations on species . In particular, the negative sign shows that the variation of particles of one species produces an opposite variation on the other. The particles produced by a reaction process on the surface are being released from the surface to the bulk. Equations (9) and (10) imply which consequently yieldsi.e., a direct consequence of the conservation of the total number of particles present in the system.

These systems of coupled equations can be solved by using the Laplace transform and the Green function approach. In fact, by applying the Laplace transform ( and ) it is possible to simplify the previous equations and obtain thatandwhich are subjected, in the Laplace domain, to the boundary conditions:with and , where . By using the Green function approach, the solutions for and can found by considering the previous equations and they are given byandwhere and correspond to the Green function for each species and represents the quantity of particles which may initially be present on the surface. These Green functions are obtained by solving the following equation:by taking into account the suitable boundary conditions for each species related to (17) and (18). Thus, the solutions given by (17) and (18) are obtained from the combination of (15), (16), and (19) by taking into account their boundary conditions, following the procedure employed in [58]. It is also interesting to note that the boundary conditions chosen for the Green function were performed in order to evidence the adsorption-desorption process manifested by the surface on the distribution . For species , by performing some calculations, it possible to show that the Green function is given bywhen the boundary conditions, which are consistent with (17),and are considered to solve (21). Note that the boundary condition used to solve the equation for the Green function incorporates the reaction on the surface. The term related to the adsorption-desorption process was not incorporated in (21) to be evident on the solution as an additional term. In (20), the first term corresponds to the spreading of the initial condition and the second term shows the influence of the surface on the spreading of the system, which in this case corresponds to a reaction sorption process where species is removed from the system. It is worth mentioning that depending on the choice of (or ) the previous Green function may present a stationary solution. In particular, for case with , for (i.e., for long times), we have, for the Green function, the following stationary behavior:which in this case is independent of , where . This feature is interesting and shows that the surface effects with this characteristic play an important role at intermediate times. For the case (or ), we also have an stationary solution and it is given bywhich is different from (22) due to the boundary conditions used to obtain the Green function. The scenario is typical, for example, of the Caputo-Fabrizio operator, which is characterized by an exponential kernel as mentioned before.

For species , we have thatwhen the boundary conditions, which are consistent with Eq. (18),are employed to solve (21). Equation (24) may also present a stationary solution depending on the choice of , similar to (20).

In order to perform the inverse of Laplace transform and obtain the time dependent behavior of (20) and (24), we first consider the Riemann-Liouville fractional time derivative and after that the Caputo-Fabrizio fractional time derivative. We also consider that ; i.e., the sorption of species by the surface occurs in a constant rate. For the Riemann-Liouville fractional time derivative, we have that and, consequently,with , where is the Fox function [59] and is the generalized Mittag-Leffler function [59] (see Figure 2). For the Caputo-Fabrizio fractional time operator, we have thatwith . By substituting the previous equation in (20) and performing some calculations with , we obtain thatfor the Caputo-Fabrizio fractional time operator with :where is the complementary error function and . Figure 3 shows the behavior of (29) for different values of . In Figure 4, we present the behavior of (29) for different times in order to show that for long times a stationary state is reached. For (24), we havefor the Riemann-Liouville operator (see Figure 5) andfor the Caputo-Fabrizio fractional time operator (see Figure 6).

Figure 2: This figure illustrates the behavior of (27) by considering different values of . We consider, for simplicity, , , and , in arbitrary unities.
Figure 3: This figure illustrates the behavior of (29) by considering different values of . We consider, for simplicity, , , , and , in arbitrary unities.
Figure 4: This figure illustrates the behavior of (29) by considering different times. We consider, for simplicity, , , and , in arbitrary unities.
Figure 5: This figure illustrates the behavior of (31) by considering different values of . We consider, for simplicity, , , and , in arbitrary unities.
Figure 6: This figure illustrates the behavior of (32) by considering different values of . We consider, for simplicity, , , , and , in arbitrary unities.

In (17) and (18), the first term promotes the spreading of the initial condition and the other terms represent the influence of the surface on the diffusive process in the bulk. Thus, we need to quantify these terms, in order to obtain the solution for each species. From (17), after some calculations, it is also possible to show thatwhere . By using the previous result, it is possible to show that the quantity of adsorbed particles by the surface is given by

By using (33) and (34) and by applying the inverse Laplace transform in (13) and (14), it is possible to obtain and . We consider, as performed for the Green functions, the fractional operators characterized by (4) and (5) in order to obtain the inverse Laplace transform of (33) and (34). For the first case, i.e., the Riemann-Liouville fractional time derivative, we obtain for (33) and (34) thatFor the case characterized by (4), i.e., the Caputo-Fabrizio fractional time operator, we obtain thatThe quantities and presented in previous equations are defined as follows:with

By using the previous equations, the inverse of Laplace transform of (17) and (18)andEquations (41) and (42) are the solutions for the scenario analyzed here, which is characterized by a diffusion with processes occurring on a surface. By using these results, it is possible to obtain the survival probability, which is defined as and it is related to the quantity of particles each species present in the bulk. In Figure 7, we illustrate the time behavior of the survival probability for species and by considering different fractional time operators. We observe that for the fractional operator related to the exponential kernel a stationary state is reached. Another interesting quantity to be analyzed is the mean square displacement, i.e., , related to the spreading of the distributions of each species. For example, in absence of adsorption-desorption process ( and ) with , species manifests two different regimes, when the Riemann-Liouville fractional time derivative is considered. In particular, for small times and for long times . In this context, for the Caputo-Fabrizio fractional time operator, we have for small times and for long times. For species , the behavior of the mean square displacement is illustrated in Figure 8 and, similar to species , the Riemann-Liouville fractional time derivative presents two different regimes. For the Caputo-Fabrizio fractional time operator, we observe a superdiffusion (see Figure 8(a)) or a transient (see Figure 8(b)) before reaching a stationary state. The initial behavior observed for species is a consequence of the initial condition considered in this application.

Figure 7: (a) illustrates the survival probability for species and by considering the Riemann-Liouville fractional time operator. We consider, for simplicity, , , , , and , in arbitrary unities. (b) illustrates the survival probability for species and by considering the Caputo-Fabrizio fractional time operator. We consider, for simplicity, , , , , , and , in arbitrary unities.
Figure 8: (a) illustrates the mean square displacement for species for different values of by taking into account the Riemann-Liouville fractional time operator. We consider, for simplicity, , and , , , and , in arbitrary unities. (b) illustrates the mean square displacement for species for different values of for species by taking into account the Caputo-Fabrizio fractional time operator. We consider, for simplicity, , and , , , and , in arbitrary unities.

3. Discussions and Conclusions

We have investigated a diffusion process in semi-infinite media in contact with a surface, where the particles may be adsorbed (and desorbed) or sorbed by the surface in order to promote though a reaction process the formation of another species of particle. For particles in the bulk, we have considered generalized diffusion equations, which may be related to a different types of fractional differential operator. In particular, we have considered the Riemann-Liouville and Caputo-Fabrizio fractional time derivative. For each one of these operators, we have found the solutions and obtained the quantity of particles adsorbed by the surface as a function of time. The reaction process considered here occurs only on the surface and it is irreversible, i.e., . The particles obtained from the reaction process are released from the surface to the bulk. In the bulk, we have considered that reaction between the particles is absent; i.e., only the diffusion governs the dynamics of the particles in the bulk. The solutions for these cases have been obtained in terms of the Fox H-functions and generalized Mittag-Leffler functions which are usually related to anomalous diffusion. They have exhibited different diffusive behaviors depending on the conditions employed to analyze the system. In particular, for the situation discussed in Figure 8(a), we have verified that species manifests a superdiffusive behavior for small times when the Riemann-Liouville fractional time operator is considered. For long times, the behavior is essentially governed by the diffusion equations and, consequently, subdiffusive behaviors were obtained. In Figure 8(b), we have analyzed the behavior of the mean square displacement for the Caputo-Fabrizio fractional time operator. For small times, we have a transient until the system reaches a stationary solution, i.e., a time independent solution, for long times. The stationary solution manifested by the system, when the Caputo-Fabrizio is employed, may be related to the resetting process connected to this operator as pointed out in [60]. Finally, we hope that the results presented here may be useful to discussion of situations related to anomalous diffusion.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflicts of interest.

Acknowledgments

E. K. Lenzi acknowledges CNPq for partial financial support.

References

  1. G. Hariharan and K. Kannan, “Review of wavelet methods for the solution of reaction-diffusion problems in science and engineering,” Applied Mathematical Modelling, vol. 38, no. 3, pp. 799–813, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. K. Pirkanniemi and M. Sillanpää, “Heterogeneous water phase catalysis as an environmental application: a review,” Chemosphere, vol. 48, no. 10, pp. 1047–1060, 2002. View at Publisher · View at Google Scholar · View at Scopus
  3. D. Wan, W. Li, G. Wang, K. Chen, L. Lu, and Q. Hu, “Adsorption and heterogeneous degradation of rhodamine B on the surface of magnetic bentonite material,” Applied Surface Science, vol. 349, pp. 988–996, 2015. View at Publisher · View at Google Scholar · View at Scopus
  4. W.-D. Oh, Z. Dong, and T.-T. Lim, “Generation of sulfate radical through heterogeneous catalysis for organic contaminants removal: Current development, challenges and prospects,” Applied Catalysis B: Environmental, vol. 194, pp. 169–201, 2016. View at Publisher · View at Google Scholar · View at Scopus
  5. S. Rahim Pouran, A. A. Abdul Raman, and W. M. A. Wan Daud, “Review on the application of modified iron oxides as heterogeneous catalysts in Fenton reactions,” Journal of Cleaner Production, vol. 64, pp. 24–35, 2014. View at Publisher · View at Google Scholar · View at Scopus
  6. S. Semwal, A. K. Arora, R. P. Badoni, and D. K. Tuli, “Biodiesel production using heterogeneous catalysts,” Bioresource Technology, vol. 102, no. 3, pp. 2151–2161, 2011. View at Publisher · View at Google Scholar · View at Scopus
  7. H.-J. Kim, B.-S. Kang, M.-J. Kim et al., “Transesterification of vegetable oil to biodiesel using heterogeneous base catalyst,” Catalysis Today, vol. 93–95, pp. 315–320, 2004. View at Publisher · View at Google Scholar · View at Scopus
  8. A. F. Lee, J. A. Bennett, J. C. Manayil, and K. Wilson, “Heterogeneous catalysis for sustainable biodiesel production via esterification and transesterification,” Chemical Society Reviews, vol. 43, no. 22, pp. 7887–7916, 2014. View at Publisher · View at Google Scholar · View at Scopus
  9. M. Malmsten, Biopolymers at Interfaces, vol. 110, CRC Press, 2003.
  10. D. S. Vieira, P. R. G. Fernandes, H. Mukai, R. S. Zola, G. G. Lenzi, and E. K. Lenzi, “Surface roughness influence on CPE parameters in electrolytic cells,” International Journal of Electrochemical Science, vol. 11, no. 9, pp. 7775–7784, 2016. View at Publisher · View at Google Scholar · View at Scopus
  11. V. Hlady and J. Buijs, “Protein adsorption on solid surfaces,” Current Opinion in Biotechnology, vol. 7, no. 1, pp. 72–77, 1996. View at Publisher · View at Google Scholar · View at Scopus
  12. K. Ishihara, H. Nomura, T. Mihara, K. Kurita, Y. Iwasaki, and N. Nakabayashi, “Why do phospholipid polymers reduce protein adsorption?” Journal of Biomedical Materials Research Part B: Applied Biomaterials, vol. 39, no. 2, pp. 323–330, 1998. View at Publisher · View at Google Scholar · View at Scopus
  13. K. Rechendorff, M. B. Hovgaard, M. Foss, V. P. Zhdanov, and F. Besenbacher, “Enhancement of protein adsorption induced by surface roughness,” Langmuir, vol. 22, no. 26, pp. 10885–10888, 2006. View at Publisher · View at Google Scholar · View at Scopus
  14. M. B. Hovgaard, K. Rechendorff, J. Chevallier, M. Foss, and F. Besenbacher, “Fibronectin adsorption on tantalum: the influence of nanoroughness,” The Journal of Physical Chemistry B, vol. 112, no. 28, pp. 8241–8249, 2008. View at Publisher · View at Google Scholar · View at Scopus
  15. A. Dolatshahi-Pirouz, K. Rechendorff, M. B. Hovgaard, M. Foss, J. Chevallier, and F. Besenbacher, “Bovine serum albumin adsorption on nano-rough platinum surfaces studied by QCM-D,” Colloids and Surfaces B: Biointerfaces, vol. 66, no. 1, pp. 53–59, 2008. View at Publisher · View at Google Scholar · View at Scopus
  16. Y.-P. Ho, M. C. Kung, S. Yang, and T.-H. Wang, “Multiplexed hybridization detection with multicolor colocalization of quantum dot nanoprobes,” Nano Letters, vol. 5, no. 9, pp. 1693–1697, 2005. View at Publisher · View at Google Scholar · View at Scopus
  17. L. C. Brousseau, “Label-Free “Digital Detection” of Single-Molecule DNA Hybridization with a Single Electron Transistor,” Journal of the American Chemical Society, vol. 128, no. 35, pp. 11346-11347, 2006. View at Publisher · View at Google Scholar
  18. M. Sinder, V. Sokolovsky, and J. Pelleg, “Reversible AB reaction–diffusion process with initially mixed reactants: Boundary layer function approach,” Physica B: Condensed Matter, vol. 406, no. 15-16, pp. 3042–3049, 2011. View at Publisher · View at Google Scholar
  19. D. ben-Avraham and S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems, Cambridge University Press, Cambridge, UK, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  20. D. Marin, M. A. Ribeiro, H. V. Ribeiro, and E. . Lenzi, “A nonlinear Fokker--Planck equation approach for interacting systems: Anomalous diffusion and Tsallis statistics,” Physics Letters A, vol. 382, no. 29, pp. 1903–1907, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  21. E. K. Lenzi, M. A. Ribeiro, M. E. Fuziki, M. K. Lenzi, and H. V. Ribeiro, “Nonlinear diffusion equation with reaction terms: analytical and numerical results,” Applied Mathematics and Computation, vol. 330, pp. 254–265, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  22. B. Chopard and M. Droz, Cellular automata modeling of physical systems. collection aléa-saclay: Monographs and texts in statistical physics, 1998.
  23. I. Zeldovich, G. I. Barenblatt, V. Librovich, and G. Makhviladze, Mathematical Theory of Combustion and Explosions, Consultants Bureau, New York, NY, USA, 1985.
  24. L. G. Arnaut, S. J. Formosinho, and H. Burrows, Chemical Kinetics: from Molecular Structure to Chemical Reactivity, Elsevier, 2006.
  25. A. Sapora, P. Cornetti, B. Chiaia, E. K. Lenzi, and L. R. Evangelista, “Nonlocal diffusion in porous media: A spatial fractional approach,” Journal of Engineering Mechanics, vol. 143, no. 5, 2017. View at Google Scholar · View at Scopus
  26. L. R. Evangelista and E. K. Lenzi, Fractional Diffusion Equations and Anomalous Diffusion, Cambridge University Press, 2018. View at Publisher · View at Google Scholar
  27. P. C. Assis, R. P. de Souza, P. C. da Silva, L. R. da Silva, L. S. Lucena, and E. K. Lenzi, “Non-Markovian Fokker-Planck equation: Solutions and first passage time distribution,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 73, no. 3, 2006. View at Publisher · View at Google Scholar
  28. E. K. Lenzi, R. S. Mendes, J. S. Andrade Jr, L. R. da Silva, and L. S. Lucena, “N-dimensional fractional diffusion equation and green function approach: Spatially dependent diffusion coefficient and external force,” Physical Review E, vol. 71, no. 5, 2005. View at Google Scholar
  29. E. K. Lenzi, R. Mendes, K. S. Fa, L. C. Malacarne, and L. Da Silva, “Anomalous diffusion: Fractional Fokker–Planck equation and its solutions,” Journal of Mathematical Physics, vol. 44, pp. 2179–2185, 2003. View at Google Scholar
  30. S. A. El-Wakil and M. A. Zahran, “Fractional representation of Fokker-Planck equation,” Chaos, Solitons & Fractals, vol. 12, no. 10, pp. 1929–1935, 2001. View at Publisher · View at Google Scholar
  31. A. Hanyga, “Multi-dimensional solutions of space-time-fractional diffusion equations,” Proceedings A, vol. 458, no. 2018, pp. 429–450, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  32. O. P. Agrawal, “Solution for a fractional diffusion-wave equation defined in a bounded domain,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 145–155, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  33. F. Ren, J. Liang, W. Qiu, X. Wang, Y. Xu, and R. Nigmatullin, “An anomalous diffusion model in an external force fields on fractals,” Physics Letters A, vol. 312, no. 3-4, pp. 187–197, 2003. View at Publisher · View at Google Scholar
  34. R. Gorenflo, A. Iskenderov, and Y. Luchko, “Mapping between solutions of fractional diffusion-wave equations,” Fractional Calculus & Applied Analysis. An International Journal for Theory and Applications, vol. 3, no. 1, pp. 75–86, 2000. View at Google Scholar · View at MathSciNet
  35. B. N. N. Achar and J. W. Hanneken, “Fractional radial diffusion in a cylinder,” Journal of Molecular Liquids, vol. 114, no. 1-3, pp. 147–151, 2004. View at Publisher · View at Google Scholar · View at Scopus
  36. F. Mainardi and G. Pagnini, “The Wright functions as solutions of the time-fractional diffusion equation,” Applied Mathematics and Computation, vol. 141, no. 1, pp. 51–62, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  37. G. M. Zaslavsky, “Chaos, fractional kinetics, and anomalous transport,” Physics Reports A Review Section of Physics Letters, vol. 371, no. 6, pp. 461–580, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  38. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  39. R. Metzler and J. Klafter, “The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics,” Journal of Physics A: Mathematical and General, vol. 37, no. 31, pp. R161–R208, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  40. M. Ashrafuzzaman, “Aptamers as Both Drugs and Drug-Carriers,” BioMed Research International, vol. 2014, Article ID 697923, 21 pages, 2014. View at Publisher · View at Google Scholar · View at Scopus
  41. 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, 1998.
  42. R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, no. 1, pp. 1–77, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  43. V. R. Voller, “Fractional Stefan problems,” International Journal of Heat and Mass Transfer, vol. 74, pp. 269–277, 2014. View at Publisher · View at Google Scholar · View at Scopus
  44. S. Blasiak, “Time-fractional heat transfer equations in modeling of the non-contacting face seals,” International Journal of Heat and Mass Transfer, vol. 100, pp. 79–88, 2016. View at Publisher · View at Google Scholar · View at Scopus
  45. F. R. G. B. Silva, G. Gonçalves, M. K. Lenzi, and E. K. Lenzi, “An extension of the linear Luikov system equations of heat and mass transfer,” International Journal of Heat and Mass Transfer, vol. 63, pp. 233–238, 2013. View at Publisher · View at Google Scholar · View at Scopus
  46. 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. 73–85, 2015. View at Google Scholar
  47. J. F. Gómez-Aguilar and A. Atangana, “Fractional Hunter-Saxton equation involving partial operators with bi-order in Riemann-Liouville and Liouville-Caputo sense,” The European Physical Journal Plus, vol. 132, no. 2, 2017. View at Google Scholar · View at Scopus
  48. A. Atangana and I. Koca, “Model of thin viscous fluid sheet flow within the scope of fractional calculus: fractional derivative with and no singular kernel,” Fundamenta Informaticae, vol. 151, no. 1-4, pp. 145–159, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  49. 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, 2018. View at Publisher · View at Google Scholar
  50. A. Atangana, “Non validity of index law in fractional calculus: A fractional differential operator with MARkovian and non-MARkovian properties,” Physica A: Statistical Mechanics and its Applications, vol. 505, pp. 688–706, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  51. J. Gómez-Aguilar, “Chaos in a nonlinear Bloch system with Atangana-Baleanu fractional derivatives,” in Numerical Methods for Partial Differential Equations, pp. 1–23, 2017. View at Google Scholar
  52. A. Coronel-Escamilla, J. F. Gómez-Aguilar, L. Torres, and R. F. Escobar-Jiménez, “A numerical solution for a variable-order reaction-diffusion model by using fractional derivatives with non-local and non-singular kernel,” Physica A: Statistical Mechanics and its Applications, vol. 491, pp. 406–424, 2018. View at Publisher · View at Google Scholar · View at MathSciNet
  53. J. Hristov, “Transient heat diffusion with a non-singular fading memory: from the cattaneo constitutive equation with Jeffrey's kernel to the caputo-fabrizio time-fractional derivative,” Thermal Science, vol. 20, no. 2, pp. 757–762, 2016. View at Publisher · View at Google Scholar · View at Scopus
  54. J. F. Gómez-Aguilar, “Space-time fractional diffusion equation using a derivative with nonsingular and regular kernel,” Physica A: Statistical Mechanics and its Applications, vol. 465, pp. 562–572, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  55. T. Abdeljawad and D. Baleanu, “Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel,” The Journal of Nonlinear Science and Applications, vol. 10, no. 03, pp. 1098–1107, 2017. View at Publisher · View at Google Scholar
  56. V. G. Guimarães, H. V. Ribeiro, Q. Li, L. R. Evangelista, E. K. Lenzi, and R. S. Zola, “Unusual diffusing regimes caused by different adsorbing surfaces,” Soft Matter, vol. 11, no. 9, pp. 1658–1666, 2015. View at Publisher · View at Google Scholar · View at Scopus
  57. R. S. Zola, E. K. Lenzi, L. R. Evangelista, and G. Barbero, “Memory effect in the adsorption phenomena of neutral particles,” Physical Review E: Statistical, Nonlinear, and Soft Matter Physics, vol. 75, no. 4, 2007. View at Publisher · View at Google Scholar
  58. H. W. Wyld, Mathematical Methods for Physics, Westview Press, 1999.
  59. A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-function : Theory and Applications, Springer Science & Business Media, New York, NY, USA, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  60. A. A. Tateishi, H. V. Ribeiro, and E. K. Lenzi, “The role of fractional time-derivative operators on anomalous diffusion,” Frontiers of Physics, vol. 5, p. 52, 2017. View at Google Scholar