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Optimal control problem of variable-order delay system of advertising procedure: Numerical treatment

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  • 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.

    Mathematics Subject Classification: 26A33, 49M25, 65L03.


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  • 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.
     | Show Table
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    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
     | Show Table
    DownLoad: CSV
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