doi: 10.3934/dcdss.2020173

Fractional diffusion equation described by the Atangana-Baleanu fractional derivative and its approximate solution

Département de Mathématiques de la Décision, Université Cheikh Anta Diop de Dakar, Laboratoire Lmdan, BP 5683 Dakar Fann, Sénégal

* Corresponding author: Ndolane Sene

Received  March 2019 Revised  May 2019 Published  December 2019

In this paper, we propose the approximate solution of the fractional diffusion equation described by a non-singular fractional derivative. We use the Atangana-Baleanu-Caputo fractional derivative in our studies. The integral balance methods as the heat balance integral method introduced by Goodman and the double integral method developed by Hristov have been used for getting the approximate solution. In this paper, the existence and uniqueness of the solution of the fractional diffusion equation have been provided. We analyze the impact of the fractional operator in the diffusion process. We represent graphically the approximate solution of the fractional diffusion equation.

Citation: Ndolane Sene. Fractional diffusion equation described by the Atangana-Baleanu fractional derivative and its approximate solution. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020173
References:
[1]

T. Abdeljawad and D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10 (2017), 1098-1107.  doi: 10.22436/jnsa.010.03.20.  Google Scholar

[2]

T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, J. Ineq. Appli., 2017 (2017), 11 pp. doi: 10.1186/s13660-017-1400-5.  Google Scholar

[3]

T. Abdeljawad, Different type kernel h-fractional differences and their fractional h-sums, Chaos Soli. Fract., 116 (2018), 146-156.  doi: 10.1016/j.chaos.2018.09.022.  Google Scholar

[4]

T. Abdeljawad and Q. M. Al-Mdallal, Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality, J. Comput. Appl. Math., 339 (2018), 218-230.  doi: 10.1016/j.cam.2017.10.021.  Google Scholar

[5]

T. Abdeljawad and D. Baleanu, Discrete fractional differences with non-singular discrete Mittag-Leffler kernels, Adv. Diff. Equat., 2016 (2016), 18 pp. doi: 10.1186/s13662-016-0949-5.  Google Scholar

[6]

T. Abdeljawad and D. Baleanu, On fractional derivatives with generalized Mittag-Leffler kernels, Adv. Diff. Equat., 2018 (2018), 15 pp. doi: 10.1186/s13662-018-1914-2.  Google Scholar

[7]

T. Abdeljawad, Fractional operators with generalized Mittag-Leffler kernels and their differintegrals, Chaos, 29 (2019), 023102, 10 pp. doi: 10.1063/1.5085726.  Google Scholar

[8]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763â€"769, arXiv: 1602.03408. doi: 10.2298/TSCI160111018A.  Google Scholar

[9]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 1-15.   Google Scholar

[10]

J. Fahd and T. Abdeljawad, A modified Laplace transform for certain generalized fractional operators, Results in Nonlinear Analysis, 2 (2018), 88-98.   Google Scholar

[11]

F. JaradT. Abdeljawad and D. Baleanu, On the generalized fractional derivatives and their Caputo modification, J. Nonlinear Sci. Appl., 10 (2017), 2607-2619.  doi: 10.22436/jnsa.010.05.27.  Google Scholar

[12]

J. Hristov, On the Atangana-Baleanu derivative and its relation to the fading memory concept: The diffusion equation formulation, Fractional Derivatives with Mittag-Leffler Kernel, Stud. Syst. Decis. Control, Springer, Cham, 194 (2019), 175-193.   Google Scholar

[13]

J. Hristov, Approximate solutions to fractional subdiffusion equations, The European Physical Journal Special Topics, 193 (2011), 229-243.  doi: 10.1140/epjst/e2011-01394-2.  Google Scholar

[14]

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, 20 (2016), 757-762.  doi: 10.2298/TSCI160112019H.  Google Scholar

[15]

J. Hristov, Multiple integral-balance method: Basic idea and an example with mullin's model of thermal grooving, Thermal Science, 2 (2017), 555-1560.  doi: 10.2298/TSCI170410124H.  Google Scholar

[16]

J. Hristov, The non-linear dodson diffusion equation: Approximate solutions and beyond with formalistic fractionalization, Math. Natur. Sci., 1 (2017), 1-17.  doi: 10.22436/mns.01.01.01.  Google Scholar

[17]

J. Hristov, Space-fractional diffusion with a potential power-law coefficient: Transient approximate solution, Progr. Fract. Differ. Appl., 3 (2017), 19-39.   Google Scholar

[18]

J. Hristov, Fourth-order fractional diffusion model of thermal grooving: Integral approach to approximate closed form solution of the Mullins model, Mathematical Modelling of Natural Phenomena, 13 (2018), Art. 6, 14 pp. doi: 10.1051/mmnp/2017080.  Google Scholar

[19]

J. Hristov, Integral-balance solution to nonlinear subdiffusion equation, Frontiers in Fractional Calculus, Curr. Dev. Math. Sci., Bentham Sci. Publ., Sharjah, 1 (2018), 70-105.   Google Scholar

[20]

J. Hristov, The heat radiation diffusion equation: Explicit analytical solutions by improved integral-balance method, Thermal Science, 22 (2018), 777-778.  doi: 10.2298/TSCI171011308H.  Google Scholar

[21]

J. Hristov, Integral balance approach to 1-D space-fractional diffusion models, Mathematical Methods in Engineering, Nonlinear Syst. Complex., Springer, Cham, 23 (2019), 111-131.   Google Scholar

[22]

J. Hristov, A transient flow of a non-newtonian fluid modelled by a mixed time-space derivative: An improved integral-balance approach, Mathematical Methods in Engineering, Nonlinear Syst. Complex., Springer, Cham, 24 (2019), 153-174.   Google Scholar

[23]

F. JaradT. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative, Chaos Soli. Fract., 117 (2018), 16-20.  doi: 10.1016/j.chaos.2018.10.006.  Google Scholar

[24]

U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865.  doi: 10.1016/j.amc.2011.03.062.  Google Scholar

[25]

R. KhalilM. Al HoraniA. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comp. Appli. Math., 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002.  Google Scholar

[26]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.  Google Scholar

[27]

S. J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput., 147 (2004), 499-513.  doi: 10.1016/S0096-3003(02)00790-7.  Google Scholar

[28]

S. L. Mitchell, Applying the combined integral method to two-phase Stefan problems with delayed onset of phase change, Journal of computational and Applied Mathematics, 281 (2015), 58-73.  doi: 10.1016/j.cam.2014.11.051.  Google Scholar

[29]

S. L. Mitchell and T. G. Myers, Improving the accuracy of heat balance integral methods applied to thermal problems with time dependent boundary conditions, International Journal of Heat and Mass Transfer, 53 (2010), 3540-3551.  doi: 10.1016/j.ijheatmasstransfer.2010.04.015.  Google Scholar

[30]

S. L. Mitchell and T. G. Myers, Heat balance integral method for one-dimensional finite ablation, Journal of Thermophysics and Heat Transfer, 22 (2008), 508-514.  doi: 10.2514/1.31755.  Google Scholar

[31]

T. G. Myers, Optimal exponent heat balance and refined integral methods applied to Stefan problems, International Journal of Heat and Mass Transfer, 53 (2010), 1119-1127.  doi: 10.1016/j.ijheatmasstransfer.2009.10.045.  Google Scholar

[32]

N. Sene, Analytical solutions of Hristov diffusion equations with non-singular fractional derivatives, Choas, 29 (2019), 023112, 11 pp. doi: 10.1063/1.5082645.  Google Scholar

[33]

N. Sene, Solutions of fractional diffusion equations and Cattaneo-Hristov diffusion models, Int. J. Appli. Anal., 17 (2019), 191-207.   Google Scholar

[34]

N. Sene, Lyapunov characterization of the fractional nonlinear systems with exogenous input, Fractal and Fractional, 2 (2018). doi: 10.3390/fractalfract2020017.  Google Scholar

[35]

N. Sene, Solutions for some conformable differential equations, Progr. Fract. Differ. Appl., 4 (2018), 493-501.   Google Scholar

[36]

N. Sene, Stokes' first problem for heated flat plate with Atangana-Baleanu fractional derivative, Chaos, Solitons & Fractals, 117 (2018), 68-75.  doi: 10.1016/j.chaos.2018.10.014.  Google Scholar

[37]

N. Sene, Generalized mittag-leffler input stability of the fractional differential equations, Symmetry, 11 (2019), 608. doi: 10.3390/sym11050608.  Google Scholar

[38]

I. SuwanT. Abdeljawad and F. Jarad, Monotonicity analysis for nabla h-discrete fractional Atangana-Baleanu differences, Chaos Soli. Fract., 117 (2018), 50-59.  doi: 10.1016/j.chaos.2018.10.010.  Google Scholar

show all references

References:
[1]

T. Abdeljawad and D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 10 (2017), 1098-1107.  doi: 10.22436/jnsa.010.03.20.  Google Scholar

[2]

T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, J. Ineq. Appli., 2017 (2017), 11 pp. doi: 10.1186/s13660-017-1400-5.  Google Scholar

[3]

T. Abdeljawad, Different type kernel h-fractional differences and their fractional h-sums, Chaos Soli. Fract., 116 (2018), 146-156.  doi: 10.1016/j.chaos.2018.09.022.  Google Scholar

[4]

T. Abdeljawad and Q. M. Al-Mdallal, Discrete Mittag-Leffler kernel type fractional difference initial value problems and Gronwall's inequality, J. Comput. Appl. Math., 339 (2018), 218-230.  doi: 10.1016/j.cam.2017.10.021.  Google Scholar

[5]

T. Abdeljawad and D. Baleanu, Discrete fractional differences with non-singular discrete Mittag-Leffler kernels, Adv. Diff. Equat., 2016 (2016), 18 pp. doi: 10.1186/s13662-016-0949-5.  Google Scholar

[6]

T. Abdeljawad and D. Baleanu, On fractional derivatives with generalized Mittag-Leffler kernels, Adv. Diff. Equat., 2018 (2018), 15 pp. doi: 10.1186/s13662-018-1914-2.  Google Scholar

[7]

T. Abdeljawad, Fractional operators with generalized Mittag-Leffler kernels and their differintegrals, Chaos, 29 (2019), 023102, 10 pp. doi: 10.1063/1.5085726.  Google Scholar

[8]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763â€"769, arXiv: 1602.03408. doi: 10.2298/TSCI160111018A.  Google Scholar

[9]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 1-15.   Google Scholar

[10]

J. Fahd and T. Abdeljawad, A modified Laplace transform for certain generalized fractional operators, Results in Nonlinear Analysis, 2 (2018), 88-98.   Google Scholar

[11]

F. JaradT. Abdeljawad and D. Baleanu, On the generalized fractional derivatives and their Caputo modification, J. Nonlinear Sci. Appl., 10 (2017), 2607-2619.  doi: 10.22436/jnsa.010.05.27.  Google Scholar

[12]

J. Hristov, On the Atangana-Baleanu derivative and its relation to the fading memory concept: The diffusion equation formulation, Fractional Derivatives with Mittag-Leffler Kernel, Stud. Syst. Decis. Control, Springer, Cham, 194 (2019), 175-193.   Google Scholar

[13]

J. Hristov, Approximate solutions to fractional subdiffusion equations, The European Physical Journal Special Topics, 193 (2011), 229-243.  doi: 10.1140/epjst/e2011-01394-2.  Google Scholar

[14]

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, 20 (2016), 757-762.  doi: 10.2298/TSCI160112019H.  Google Scholar

[15]

J. Hristov, Multiple integral-balance method: Basic idea and an example with mullin's model of thermal grooving, Thermal Science, 2 (2017), 555-1560.  doi: 10.2298/TSCI170410124H.  Google Scholar

[16]

J. Hristov, The non-linear dodson diffusion equation: Approximate solutions and beyond with formalistic fractionalization, Math. Natur. Sci., 1 (2017), 1-17.  doi: 10.22436/mns.01.01.01.  Google Scholar

[17]

J. Hristov, Space-fractional diffusion with a potential power-law coefficient: Transient approximate solution, Progr. Fract. Differ. Appl., 3 (2017), 19-39.   Google Scholar

[18]

J. Hristov, Fourth-order fractional diffusion model of thermal grooving: Integral approach to approximate closed form solution of the Mullins model, Mathematical Modelling of Natural Phenomena, 13 (2018), Art. 6, 14 pp. doi: 10.1051/mmnp/2017080.  Google Scholar

[19]

J. Hristov, Integral-balance solution to nonlinear subdiffusion equation, Frontiers in Fractional Calculus, Curr. Dev. Math. Sci., Bentham Sci. Publ., Sharjah, 1 (2018), 70-105.   Google Scholar

[20]

J. Hristov, The heat radiation diffusion equation: Explicit analytical solutions by improved integral-balance method, Thermal Science, 22 (2018), 777-778.  doi: 10.2298/TSCI171011308H.  Google Scholar

[21]

J. Hristov, Integral balance approach to 1-D space-fractional diffusion models, Mathematical Methods in Engineering, Nonlinear Syst. Complex., Springer, Cham, 23 (2019), 111-131.   Google Scholar

[22]

J. Hristov, A transient flow of a non-newtonian fluid modelled by a mixed time-space derivative: An improved integral-balance approach, Mathematical Methods in Engineering, Nonlinear Syst. Complex., Springer, Cham, 24 (2019), 153-174.   Google Scholar

[23]

F. JaradT. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative, Chaos Soli. Fract., 117 (2018), 16-20.  doi: 10.1016/j.chaos.2018.10.006.  Google Scholar

[24]

U. N. Katugampola, New approach to a generalized fractional integral, Appl. Math. Comput., 218 (2011), 860-865.  doi: 10.1016/j.amc.2011.03.062.  Google Scholar

[25]

R. KhalilM. Al HoraniA. Yousef and M. Sababheh, A new definition of fractional derivative, J. Comp. Appli. Math., 264 (2014), 65-70.  doi: 10.1016/j.cam.2014.01.002.  Google Scholar

[26]

A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006.  Google Scholar

[27]

S. J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput., 147 (2004), 499-513.  doi: 10.1016/S0096-3003(02)00790-7.  Google Scholar

[28]

S. L. Mitchell, Applying the combined integral method to two-phase Stefan problems with delayed onset of phase change, Journal of computational and Applied Mathematics, 281 (2015), 58-73.  doi: 10.1016/j.cam.2014.11.051.  Google Scholar

[29]

S. L. Mitchell and T. G. Myers, Improving the accuracy of heat balance integral methods applied to thermal problems with time dependent boundary conditions, International Journal of Heat and Mass Transfer, 53 (2010), 3540-3551.  doi: 10.1016/j.ijheatmasstransfer.2010.04.015.  Google Scholar

[30]

S. L. Mitchell and T. G. Myers, Heat balance integral method for one-dimensional finite ablation, Journal of Thermophysics and Heat Transfer, 22 (2008), 508-514.  doi: 10.2514/1.31755.  Google Scholar

[31]

T. G. Myers, Optimal exponent heat balance and refined integral methods applied to Stefan problems, International Journal of Heat and Mass Transfer, 53 (2010), 1119-1127.  doi: 10.1016/j.ijheatmasstransfer.2009.10.045.  Google Scholar

[32]

N. Sene, Analytical solutions of Hristov diffusion equations with non-singular fractional derivatives, Choas, 29 (2019), 023112, 11 pp. doi: 10.1063/1.5082645.  Google Scholar

[33]

N. Sene, Solutions of fractional diffusion equations and Cattaneo-Hristov diffusion models, Int. J. Appli. Anal., 17 (2019), 191-207.   Google Scholar

[34]

N. Sene, Lyapunov characterization of the fractional nonlinear systems with exogenous input, Fractal and Fractional, 2 (2018). doi: 10.3390/fractalfract2020017.  Google Scholar

[35]

N. Sene, Solutions for some conformable differential equations, Progr. Fract. Differ. Appl., 4 (2018), 493-501.   Google Scholar

[36]

N. Sene, Stokes' first problem for heated flat plate with Atangana-Baleanu fractional derivative, Chaos, Solitons & Fractals, 117 (2018), 68-75.  doi: 10.1016/j.chaos.2018.10.014.  Google Scholar

[37]

N. Sene, Generalized mittag-leffler input stability of the fractional differential equations, Symmetry, 11 (2019), 608. doi: 10.3390/sym11050608.  Google Scholar

[38]

I. SuwanT. Abdeljawad and F. Jarad, Monotonicity analysis for nabla h-discrete fractional Atangana-Baleanu differences, Chaos Soli. Fract., 117 (2018), 50-59.  doi: 10.1016/j.chaos.2018.10.010.  Google Scholar

Figure 1.  Approximate solutions of diffusion equation, $ \alpha = 0.5 $
Figure 2.  Approximate solutions of diffusion equation, different $ \alpha $ and $ t = 0.6 $
[1]

Ilknur Koca. Numerical analysis of coupled fractional differential equations with Atangana-Baleanu fractional derivative. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 475-486. doi: 10.3934/dcdss.2019031

[2]

Saif Ullah, Muhammad Altaf Khan, Muhammad Farooq, Ebraheem O. Alzahrani. A fractional model for the dynamics of tuberculosis (TB) using Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 937-956. doi: 10.3934/dcdss.2020055

[3]

Kashif Ali Abro, Ilyas Khan. MHD flow of fractional Newtonian fluid embedded in a porous medium via Atangana-Baleanu fractional derivatives. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 377-387. doi: 10.3934/dcdss.2020021

[4]

G. M. Bahaa. Generalized variational calculus in terms of multi-parameters involving Atangana-Baleanu's derivatives and application. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 485-501. doi: 10.3934/dcdss.2020027

[5]

Binjie Li, Xiaoping Xie. Regularity of solutions to time fractional diffusion equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3195-3210. doi: 10.3934/dcdsb.2018340

[6]

Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615

[7]

Valentin Keyantuo, Mahamadi Warma. On the interior approximate controllability for fractional wave equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3719-3739. doi: 10.3934/dcds.2016.36.3719

[8]

Begoña Barrios, Leandro Del Pezzo, Jorge García-Melián, Alexander Quaas. A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5731-5746. doi: 10.3934/dcds.2017248

[9]

Chun Wang, Tian-Zhou Xu. Stability of the nonlinear fractional differential equations with the right-sided Riemann-Liouville fractional derivative. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 505-521. doi: 10.3934/dcdss.2017025

[10]

Krunal B. Kachhia. Comparative study of fractional Fokker-Planck equations with various fractional derivative operators. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 741-754. doi: 10.3934/dcdss.2020041

[11]

Fahd Jarad, Sugumaran Harikrishnan, Kamal Shah, Kuppusamy Kanagarajan. Existence and stability results to a class of fractional random implicit differential equations involving a generalized Hilfer fractional derivative. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 723-739. doi: 10.3934/dcdss.2020040

[12]

Vincenzo Ambrosio, Giovanni Molica Bisci. Periodic solutions for nonlocal fractional equations. Communications on Pure & Applied Analysis, 2017, 16 (1) : 331-344. doi: 10.3934/cpaa.2017016

[13]

Zaihui Gan, Fanghua Lin, Jiajun Tong. On the viscous Camassa-Holm equations with fractional diffusion. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3427-3450. doi: 10.3934/dcds.2020029

[14]

Dariusz Idczak, Rafał Kamocki. Existence of optimal solutions to lagrange problem for a fractional nonlinear control system with riemann-liouville derivative. Mathematical Control & Related Fields, 2017, 7 (3) : 449-464. doi: 10.3934/mcrf.2017016

[15]

Abbes Benaissa, Abderrahmane Kasmi. Well-posedeness and energy decay of solutions to a bresse system with a boundary dissipation of fractional derivative type. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4361-4395. doi: 10.3934/dcdsb.2018168

[16]

Wanwan Wang, Hongxia Zhang, Huyuan Chen. Remarks on weak solutions of fractional elliptic equations. Communications on Pure & Applied Analysis, 2016, 15 (2) : 335-340. doi: 10.3934/cpaa.2016.15.335

[17]

Lizhi Zhang. Symmetry of solutions to semilinear equations involving the fractional laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2393-2409. doi: 10.3934/cpaa.2015.14.2393

[18]

Daria Bugajewska, Mirosława Zima. On positive solutions of nonlinear fractional differential equations. Conference Publications, 2003, 2003 (Special) : 141-146. doi: 10.3934/proc.2003.2003.141

[19]

Kolade M. Owolabi, Abdon Atangana. High-order solvers for space-fractional differential equations with Riesz derivative. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 567-590. doi: 10.3934/dcdss.2019037

[20]

Hao Yang, Fuke Wu, Peter E. Kloeden. Existence and approximation of strong solutions of SDEs with fractional diffusion coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5553-5567. doi: 10.3934/dcdsb.2019071

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (147)
  • HTML views (138)
  • Cited by (0)

Other articles
by authors

[Back to Top]