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doi: 10.3934/dcdss.2021085
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Optimal control problem of variable-order delay system of advertising procedure: Numerical treatment

1. 

Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt

2. 

Department of Mathematics, Faculty of Science, Umm Al-Qura University, Saudi Arabia

3. 

Department of Engineering Mathematics and Physics, Future University in Egypt, Egypt

Received  March 2021 Revised  May 2021 Early access July 2021

This paper presents an optimal control problem of the general variable-order fractional delay model of advertising procedure. The problem describes the flow of the clients from the unaware people group to the conscious or bought band. The new formulation generalizes the model that proposed by Muller. Two control variables are considered to increase the number of customers who purchased the products. An efficient nonstandard difference approach is used to study numerically the behavior of the solution of the mentioned problem. Properties of the proposed system were introduced analytically and numerically. The proposed difference schema maintains the properties of the analytic solutions as boundedness and the positivity. Numerical examples, for testing the applicability of the utilized method and to show the simplicity, accuracy and efficiency of this approximation approach, are presented with some comprising with standard difference methods.

Citation: Nasser H. Sweilam, Taghreed A. Assiri, Muner M. Abou Hasan. Optimal control problem of variable-order delay system of advertising procedure: Numerical treatment. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2021085
References:
[1]

A. I. Abbas, On a Thermoelastic Fractional Order Model, Journal of Physics, 1 (2012), 24-30.   Google Scholar

[2]

O. P. Agrawal, A formulation and numerical scheme for fractional optimal control problems, J. Vibr. Control, 14 (2008), 1291-1299.  doi: 10.1177/1077546307087451.  Google Scholar

[3]

I. AreaJ. J. Nieto and J. Losada, A note on the fractional logistic equation, Physica A, 444 (2016), 182-187.  doi: 10.1016/j.physa.2015.10.037.  Google Scholar

[4]

A. Atangana and A. H. Cloot, Stability and convergence of the space fractional variable-order Schrödinger equation, Adv. Difference Equ., 2013 (2013). doi: 10.1186/1687-1847-2013-80.  Google Scholar

[5]

D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calaulus, Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, 3, Springer Science and Business Media LLC, 2012. doi: 10.1142/9789814355216.  Google Scholar

[6]

D. Baleanu, J. A. T. Machado and A. C. J. Luo, Fractional Dynamics and Control, Springer, New York, 2012. doi: 10.1007/978-1-4614-0457-6.  Google Scholar

[7]

D. A. BensonS. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resour. Res., 36 (2000), 1403-1412.  doi: 10.1029/2000WR900031.  Google Scholar

[8]

A. V. Chechkin, R. Gorenflo and I. M. Sokolov, Fractional diffusion in inhomogeneous media, J. Phys. A: Math. Gen., 38 (2005), L679–L684. doi: 10.1088/0305-4470/38/42/L03.  Google Scholar

[9]

B. Chen-CharpentierG. González-Parra and A. J. Arenas, Fractional order financial models for awareness and trial advertising decisions, Comput. Econ., 48 (2016), 555-568.  doi: 10.1007/s10614-015-9546-z.  Google Scholar

[10]

C. ChenF. LiuK. Burrage and Y. Chen, Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative, IMA J. Appl. Math., 78 (2013), 924-944.  doi: 10.1093/imamat/hxr079.  Google Scholar

[11]

C. M. ChenF. LiuV. Anh and I. Turner, Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term, Appl. Math. Comput., 217 (2011), 5729-5742.  doi: 10.1016/j.amc.2010.12.049.  Google Scholar

[12]

C. F. M. Coimbra, Mechanics with variable-order differential operators, Ann. Phys., 12 (2003), 692-703.  doi: 10.1002/andp.200310032.  Google Scholar

[13]

A. J. Dodson and E. Muller, Models of new product diffusion through advertising and word-of-mouth, Management Science, 24 (1978), 1557-1676.  doi: 10.1287/mnsc.24.15.1568.  Google Scholar

[14]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, New York, NY, USA, 1975.  Google Scholar

[15]

J. HuangM. Leng and L. Liang, Recent developments in dynamic advertising research, European Journal of Operational Research, 220 (2012), 591-609.  doi: 10.1016/j.ejor.2012.02.031.  Google Scholar

[16]

R. C. Koeller, Application of fractional calculus to the theory of viscoelasticity, J. Appl. Mech., 51 (1984), 229-307.  doi: 10.1115/1.3167616.  Google Scholar

[17]

W. Lin, Global existence theory and chaos control of fractional differential equations, J. Math. Anal. Appl., 332 (2007), 709-726.  doi: 10.1016/j.jmaa.2006.10.040.  Google Scholar

[18]

C. F. Lorenzo and T. T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dyn., 29 (2002), 57-98.  doi: 10.1023/A:1016586905654.  Google Scholar

[19]

C. F. Lorenzo and T. T. Hartley, Initialization, conceptualization, and application in the generalized fractional calculus, Critical Reviews in Biomedical Engineering, 5 (2007), 447-553.   Google Scholar

[20]

D. L. Lukes, Differential Equations: Classical to controlled, Mathematics in Science and Engineering, 162, Academic Press, New York, NY, USA, 1982.  Google Scholar

[21]

D. Matignon, Stability result on fractional differential equations with applications to control processing, Computational Engineering in Systems Applications, 2 (1996), 963-968.   Google Scholar

[22]

R. E. Mickens, Nonstandard Finite Difference Model of Differential Equations, World Scientific, Singapore, 1994.  Google Scholar

[23]

R. E. Mickens, Exact solutions to a finite-difference model of a nonlinear reaction-advection equation: Implications for numerical analysis, Numerical Methods for Partial Differential Equations, 5 (1989), 313-325.  doi: 10.1002/num.1690050404.  Google Scholar

[24]

R. E. Mickens, Nonstandard finite difference schemes for differential equations, Journal of Difference Equations and Applications, 8 (2002), 823-847.  doi: 10.1080/1023619021000000807.  Google Scholar

[25]

E. Muller, Trial/awareness advertising decisions: A control problem with phase diagrams with non-stationary boundaries, Journal of Economic Dynamics and Control, 6 (1983), 333-350.   Google Scholar

[26]

Z. M. Odibat and N. T. Shawagfeh, Generalized taylor's formula, Applied Mathematics and Computation, 186 (2007), 286-293.  doi: 10.1016/j.amc.2006.07.102.  Google Scholar

[27] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.   Google Scholar
[28]

Y. Povstenko, Fractional Thermoelasticity, Solid Mechanics and Its Applications, Springer International Publishing Switzerland, 2015. doi: 10.1007/978-3-319-15335-3.  Google Scholar

[29]

F. A. RihanS. LakshmananA. H. HashishR. Rakkiyappan and E. Ahmed, Fractional-order delayed predator-prey systems with Holling type-II functional response, Nonlinear Dyn., 80 (2015), 777-789.  doi: 10.1007/s11071-015-1905-8.  Google Scholar

[30]

S. G. Samko and B. Ross, Integration and differentiation to a variable fractional order, Integral Transform and Special Functions, 1 (1993), 277-300.   Google Scholar

[31]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, New York Gordon and Breach Science Publishers, 1993.  Google Scholar

[32]

R. SchererS. KallaY. Tang and J. Huang, The Grünwald-Letnikov method for fractional differential equations, Comput. Math. Appl., 62 (2011), 902-917.  doi: 10.1016/j.camwa.2011.03.054.  Google Scholar

[33]

S. ShenF. LiuV. AnhI. Turner and J. Chen, A characteristic difference method for the variable-order fractional advection-diffusion equation, J. Appl. Math. Comput., 42 (2013), 371-386.  doi: 10.1007/s12190-012-0642-0.  Google Scholar

[34]

S. ShenF. LiuJ. ChenI. Turner and V. Anh, Numerical techniques for the variable order time fractional diffusion equation, Appl. Math. Comput., 218 (2012), 10861-10870.  doi: 10.1016/j.amc.2012.04.047.  Google Scholar

[35]

H. G. SunW. ChenH. Wei and Y. Q. Chen, A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems, Eur. Phys. J. Spec. Top., 193 (2011), 185-192.  doi: 10.1140/epjst/e2011-01390-6.  Google Scholar

[36]

H. G. SunA. ChangY. Zhang and W. Chen, A review on variable-order fractional differential equations: Mathematical foundations, physical models, numerical methods and applications, Fract. Calc. Appl. Anal., 22 (2019), 27-59.  doi: 10.1515/fca-2019-0003.  Google Scholar

[37]

N. H. Sweilam and S. M. AL-Mekhlafi, Optimal control for a time delay multi-strain tuberculosis fractional model: A numerical approach, IMA Journal of Mathematical Control and Information, 36 (2019), 317-340.  doi: 10.1093/imamci/dnx046.  Google Scholar

[38]

N. H. Sweilam and S. M. AL-Mekhlafi, On the optimal control for fractional multi-strain TB model, Optimal Control Applications and Methods, 37 (2016), 1355-1374.  doi: 10.1002/oca.2247.  Google Scholar

[39]

N. H. Sweilam and S. M. AL-Mekhlafi, Legendre spectral-collocation method for solving fractional optimal control of HIV infection of $Cd4^{+}T$ cells mathematical model, The Journal of Defense Modeling and Simulation, 14 (2017), 273-284.  doi: 10.1177/1548512916677582.  Google Scholar

[40]

N. H. Sweilam and M. M. Abou Hasan, Numerical solutions of a general coupled nonlinear system of parabolic and hyperbolic equations of thermoelasticity, Eur. Phys. J. Plus, 132 (2017). doi: 10.1140/epjp/i2017-11484-x.  Google Scholar

[41]

N. H. Sweilam and M. M. Abou Hasan, Numerical approximation of Lévy-Feller fractional diffusion equation via Chebyshev-Legendre collocation method, Eur. Phys. J. Plus, 131 (2016). doi: 10.1140/epjp/i2016-16251-y.  Google Scholar

[42]

N. H. Sweilam and M. M. Abou Hasan, Numerical simulation for the variable-order fractional Schrödinger equation with the quantum Riesz-Feller derivative, Adv. Appl. Math. Mech., 9 (2017), 990-1011.  doi: 10.4208/aamm.2015.m1312.  Google Scholar

[43]

N. H. SweilamM. M. Abou Hasan and D. Baleanu, New studies for general fractional financial models of awareness and trial advertising decisions, Chaos, Solitons and Fractals, 104 (2017), 772-784.  doi: 10.1016/j.chaos.2017.09.013.  Google Scholar

[44]

N. H. Sweilam and M. M. Abou Hasan, An improved method for nonlinear variable order Lévy-Feller advection-dispersion equation, Bull. Malays. Math. Sci. Soc., 42 (2019), 3021-3046.  doi: 10.1007/s40840-018-0644-7.  Google Scholar

[45]

V. E. Tarasov, Fractional Dynamics Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer Science and Business Media, 2011. doi: 10.1007/s10773-009-0202-z.  Google Scholar

[46]

M. WangQ. GouC. Wu and L. Liang, An aggregate advertising responsemodel based on consumer population dynamics, International Journal of Applied Management Science, 5 (2013), 22-38.   Google Scholar

[47]

P. ZhuangF. LiuV. Anh and I. Turner, Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Numer. Anal., 47 (2009), 1760-1781.  doi: 10.1137/080730597.  Google Scholar

show all references

References:
[1]

A. I. Abbas, On a Thermoelastic Fractional Order Model, Journal of Physics, 1 (2012), 24-30.   Google Scholar

[2]

O. P. Agrawal, A formulation and numerical scheme for fractional optimal control problems, J. Vibr. Control, 14 (2008), 1291-1299.  doi: 10.1177/1077546307087451.  Google Scholar

[3]

I. AreaJ. J. Nieto and J. Losada, A note on the fractional logistic equation, Physica A, 444 (2016), 182-187.  doi: 10.1016/j.physa.2015.10.037.  Google Scholar

[4]

A. Atangana and A. H. Cloot, Stability and convergence of the space fractional variable-order Schrödinger equation, Adv. Difference Equ., 2013 (2013). doi: 10.1186/1687-1847-2013-80.  Google Scholar

[5]

D. Baleanu, K. Diethelm, E. Scalas and J. J. Trujillo, Fractional Calaulus, Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, 3, Springer Science and Business Media LLC, 2012. doi: 10.1142/9789814355216.  Google Scholar

[6]

D. Baleanu, J. A. T. Machado and A. C. J. Luo, Fractional Dynamics and Control, Springer, New York, 2012. doi: 10.1007/978-1-4614-0457-6.  Google Scholar

[7]

D. A. BensonS. W. Wheatcraft and M. M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resour. Res., 36 (2000), 1403-1412.  doi: 10.1029/2000WR900031.  Google Scholar

[8]

A. V. Chechkin, R. Gorenflo and I. M. Sokolov, Fractional diffusion in inhomogeneous media, J. Phys. A: Math. Gen., 38 (2005), L679–L684. doi: 10.1088/0305-4470/38/42/L03.  Google Scholar

[9]

B. Chen-CharpentierG. González-Parra and A. J. Arenas, Fractional order financial models for awareness and trial advertising decisions, Comput. Econ., 48 (2016), 555-568.  doi: 10.1007/s10614-015-9546-z.  Google Scholar

[10]

C. ChenF. LiuK. Burrage and Y. Chen, Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative, IMA J. Appl. Math., 78 (2013), 924-944.  doi: 10.1093/imamat/hxr079.  Google Scholar

[11]

C. M. ChenF. LiuV. Anh and I. Turner, Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term, Appl. Math. Comput., 217 (2011), 5729-5742.  doi: 10.1016/j.amc.2010.12.049.  Google Scholar

[12]

C. F. M. Coimbra, Mechanics with variable-order differential operators, Ann. Phys., 12 (2003), 692-703.  doi: 10.1002/andp.200310032.  Google Scholar

[13]

A. J. Dodson and E. Muller, Models of new product diffusion through advertising and word-of-mouth, Management Science, 24 (1978), 1557-1676.  doi: 10.1287/mnsc.24.15.1568.  Google Scholar

[14]

W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, New York, NY, USA, 1975.  Google Scholar

[15]

J. HuangM. Leng and L. Liang, Recent developments in dynamic advertising research, European Journal of Operational Research, 220 (2012), 591-609.  doi: 10.1016/j.ejor.2012.02.031.  Google Scholar

[16]

R. C. Koeller, Application of fractional calculus to the theory of viscoelasticity, J. Appl. Mech., 51 (1984), 229-307.  doi: 10.1115/1.3167616.  Google Scholar

[17]

W. Lin, Global existence theory and chaos control of fractional differential equations, J. Math. Anal. Appl., 332 (2007), 709-726.  doi: 10.1016/j.jmaa.2006.10.040.  Google Scholar

[18]

C. F. Lorenzo and T. T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dyn., 29 (2002), 57-98.  doi: 10.1023/A:1016586905654.  Google Scholar

[19]

C. F. Lorenzo and T. T. Hartley, Initialization, conceptualization, and application in the generalized fractional calculus, Critical Reviews in Biomedical Engineering, 5 (2007), 447-553.   Google Scholar

[20]

D. L. Lukes, Differential Equations: Classical to controlled, Mathematics in Science and Engineering, 162, Academic Press, New York, NY, USA, 1982.  Google Scholar

[21]

D. Matignon, Stability result on fractional differential equations with applications to control processing, Computational Engineering in Systems Applications, 2 (1996), 963-968.   Google Scholar

[22]

R. E. Mickens, Nonstandard Finite Difference Model of Differential Equations, World Scientific, Singapore, 1994.  Google Scholar

[23]

R. E. Mickens, Exact solutions to a finite-difference model of a nonlinear reaction-advection equation: Implications for numerical analysis, Numerical Methods for Partial Differential Equations, 5 (1989), 313-325.  doi: 10.1002/num.1690050404.  Google Scholar

[24]

R. E. Mickens, Nonstandard finite difference schemes for differential equations, Journal of Difference Equations and Applications, 8 (2002), 823-847.  doi: 10.1080/1023619021000000807.  Google Scholar

[25]

E. Muller, Trial/awareness advertising decisions: A control problem with phase diagrams with non-stationary boundaries, Journal of Economic Dynamics and Control, 6 (1983), 333-350.   Google Scholar

[26]

Z. M. Odibat and N. T. Shawagfeh, Generalized taylor's formula, Applied Mathematics and Computation, 186 (2007), 286-293.  doi: 10.1016/j.amc.2006.07.102.  Google Scholar

[27] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.   Google Scholar
[28]

Y. Povstenko, Fractional Thermoelasticity, Solid Mechanics and Its Applications, Springer International Publishing Switzerland, 2015. doi: 10.1007/978-3-319-15335-3.  Google Scholar

[29]

F. A. RihanS. LakshmananA. H. HashishR. Rakkiyappan and E. Ahmed, Fractional-order delayed predator-prey systems with Holling type-II functional response, Nonlinear Dyn., 80 (2015), 777-789.  doi: 10.1007/s11071-015-1905-8.  Google Scholar

[30]

S. G. Samko and B. Ross, Integration and differentiation to a variable fractional order, Integral Transform and Special Functions, 1 (1993), 277-300.   Google Scholar

[31]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, New York Gordon and Breach Science Publishers, 1993.  Google Scholar

[32]

R. SchererS. KallaY. Tang and J. Huang, The Grünwald-Letnikov method for fractional differential equations, Comput. Math. Appl., 62 (2011), 902-917.  doi: 10.1016/j.camwa.2011.03.054.  Google Scholar

[33]

S. ShenF. LiuV. AnhI. Turner and J. Chen, A characteristic difference method for the variable-order fractional advection-diffusion equation, J. Appl. Math. Comput., 42 (2013), 371-386.  doi: 10.1007/s12190-012-0642-0.  Google Scholar

[34]

S. ShenF. LiuJ. ChenI. Turner and V. Anh, Numerical techniques for the variable order time fractional diffusion equation, Appl. Math. Comput., 218 (2012), 10861-10870.  doi: 10.1016/j.amc.2012.04.047.  Google Scholar

[35]

H. G. SunW. ChenH. Wei and Y. Q. Chen, A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems, Eur. Phys. J. Spec. Top., 193 (2011), 185-192.  doi: 10.1140/epjst/e2011-01390-6.  Google Scholar

[36]

H. G. SunA. ChangY. Zhang and W. Chen, A review on variable-order fractional differential equations: Mathematical foundations, physical models, numerical methods and applications, Fract. Calc. Appl. Anal., 22 (2019), 27-59.  doi: 10.1515/fca-2019-0003.  Google Scholar

[37]

N. H. Sweilam and S. M. AL-Mekhlafi, Optimal control for a time delay multi-strain tuberculosis fractional model: A numerical approach, IMA Journal of Mathematical Control and Information, 36 (2019), 317-340.  doi: 10.1093/imamci/dnx046.  Google Scholar

[38]

N. H. Sweilam and S. M. AL-Mekhlafi, On the optimal control for fractional multi-strain TB model, Optimal Control Applications and Methods, 37 (2016), 1355-1374.  doi: 10.1002/oca.2247.  Google Scholar

[39]

N. H. Sweilam and S. M. AL-Mekhlafi, Legendre spectral-collocation method for solving fractional optimal control of HIV infection of $Cd4^{+}T$ cells mathematical model, The Journal of Defense Modeling and Simulation, 14 (2017), 273-284.  doi: 10.1177/1548512916677582.  Google Scholar

[40]

N. H. Sweilam and M. M. Abou Hasan, Numerical solutions of a general coupled nonlinear system of parabolic and hyperbolic equations of thermoelasticity, Eur. Phys. J. Plus, 132 (2017). doi: 10.1140/epjp/i2017-11484-x.  Google Scholar

[41]

N. H. Sweilam and M. M. Abou Hasan, Numerical approximation of Lévy-Feller fractional diffusion equation via Chebyshev-Legendre collocation method, Eur. Phys. J. Plus, 131 (2016). doi: 10.1140/epjp/i2016-16251-y.  Google Scholar

[42]

N. H. Sweilam and M. M. Abou Hasan, Numerical simulation for the variable-order fractional Schrödinger equation with the quantum Riesz-Feller derivative, Adv. Appl. Math. Mech., 9 (2017), 990-1011.  doi: 10.4208/aamm.2015.m1312.  Google Scholar

[43]

N. H. SweilamM. M. Abou Hasan and D. Baleanu, New studies for general fractional financial models of awareness and trial advertising decisions, Chaos, Solitons and Fractals, 104 (2017), 772-784.  doi: 10.1016/j.chaos.2017.09.013.  Google Scholar

[44]

N. H. Sweilam and M. M. Abou Hasan, An improved method for nonlinear variable order Lévy-Feller advection-dispersion equation, Bull. Malays. Math. Sci. Soc., 42 (2019), 3021-3046.  doi: 10.1007/s40840-018-0644-7.  Google Scholar

[45]

V. E. Tarasov, Fractional Dynamics Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer Science and Business Media, 2011. doi: 10.1007/s10773-009-0202-z.  Google Scholar

[46]

M. WangQ. GouC. Wu and L. Liang, An aggregate advertising responsemodel based on consumer population dynamics, International Journal of Applied Management Science, 5 (2013), 22-38.   Google Scholar

[47]

P. ZhuangF. LiuV. Anh and I. Turner, Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Numer. Anal., 47 (2009), 1760-1781.  doi: 10.1137/080730597.  Google Scholar

Figure 1.  Approximations of the control variables with different final time
Figure 2.  Comparison between the solutions of $ x,\ z $ with control and without control when $ \alpha(t) $ takes different constant values
Figure 3.  Solutions of $ x $ and $ z $ when $ \tau $ and $ \alpha(t) $ have different values
Figure 4.  Solutions of $ x $ and $ z $ when $ \tau $ and $ \alpha(t) $ have different values
Figure 5.  Solutions of $ x $ and $ z $ when $ \tau $ has different values
Figure 6.  Solutions of $ x $ and $ z $ when $ \tau $ has different values
Figure 7.  Solutions of $ x $ and $ z $ when $ \tau $ has different values
Figure 8.  Solutions of $ x $ and $ z $ when $ \tau $ has different values
Figure 9.  Relation between the variables $ x(t) $ and $ x(t-\tau) $
Table 1.  Notations in the proposed model (1)-(2) with their definition
Symbol Definition
$ N(t) $ The whole number of the population, $ N(t)=x(t)+y(t)+z(t). $ (summation of all unknowns)
$ ^{c}_{0}D^{\alpha(t)}_{t} $ Fractional variable order derivative operator in Caputo sense.
$ \alpha(t) $ The order of variable fractional derivative.
$ t $ $ t\geq 0 $, time.
$ x(t) $ The cardinality of the set of persons who did not realize anything about the goods.
$ u $ Awareness, which switches the persons from $ x(t) $, the group who do not aware, into
the prospective one $ y(t) $ by letting them know the goods.
$ y(t) $ The cardinality of the set of persons who realize the goods but they did not purchase it till now.
$ v $ Trial advertisement, which switches the people from $ y(t) $, the prospective group, into
the bought set $ z(t) $ by encouraging them to buy the goods.
$ z(t) $ The cardinality of the set of individuals who really bought the goods.
$ a $ First purchase, (Trial rate).
$ k $ Contact rate.
$ r $ Discount rate.
$ \delta $ Switching rate.
$ c $ $ c=p(r+\delta+g) $.
$ p $ Net price.
$ g $ Repeat purchase.
$ \mu_{_b} $ Birth rate.
$ \mu_{_d} $ Death rate.
Symbol Definition
$ N(t) $ The whole number of the population, $ N(t)=x(t)+y(t)+z(t). $ (summation of all unknowns)
$ ^{c}_{0}D^{\alpha(t)}_{t} $ Fractional variable order derivative operator in Caputo sense.
$ \alpha(t) $ The order of variable fractional derivative.
$ t $ $ t\geq 0 $, time.
$ x(t) $ The cardinality of the set of persons who did not realize anything about the goods.
$ u $ Awareness, which switches the persons from $ x(t) $, the group who do not aware, into
the prospective one $ y(t) $ by letting them know the goods.
$ y(t) $ The cardinality of the set of persons who realize the goods but they did not purchase it till now.
$ v $ Trial advertisement, which switches the people from $ y(t) $, the prospective group, into
the bought set $ z(t) $ by encouraging them to buy the goods.
$ z(t) $ The cardinality of the set of individuals who really bought the goods.
$ a $ First purchase, (Trial rate).
$ k $ Contact rate.
$ r $ Discount rate.
$ \delta $ Switching rate.
$ c $ $ c=p(r+\delta+g) $.
$ p $ Net price.
$ g $ Repeat purchase.
$ \mu_{_b} $ Birth rate.
$ \mu_{_d} $ Death rate.
Table 2.  Final values of the states variables and the values of objective functional using NSFDM and SFDM when $ t_{final} = 10 $ and different $ \alpha(t) $
$ \alpha(t) $ NSFDM
$ J $ $ x $ $ z $
1 $ 286.14 $ 0 834
0.9 $ 267.09 $ 17 804
$ 0.5+0.5e^{-(t)^2-1} $ $ 248.98 $ 42 762
$ \frac{5+cos^2(t)}{10} $ $ 256.74 $ 77 709
$ \alpha(t) $ NSFDM
$ J $ $ x $ $ z $
1 $ 286.14 $ 0 834
0.9 $ 267.09 $ 17 804
$ 0.5+0.5e^{-(t)^2-1} $ $ 248.98 $ 42 762
$ \frac{5+cos^2(t)}{10} $ $ 256.74 $ 77 709
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