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May  2022, 15(5): 1247-1268. doi: 10.3934/dcdss.2021085

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 Published  May 2022 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 and Continuous Dynamical Systems - S, 2022, 15 (5) : 1247-1268. doi: 10.3934/dcdss.2021085
References:
[1]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[14]

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

[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.

[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.

[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.

[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.

[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. 

[20]

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

[21]

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

[22]

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

[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.

[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.

[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. 

[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.

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

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

[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.

[30]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

show all references

References:
[1]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[12]

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

[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.

[14]

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

[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.

[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.

[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.

[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.

[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. 

[20]

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

[21]

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

[22]

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

[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.

[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.

[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. 

[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.

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

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

[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.

[30]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

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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