Research article Special Issues

Application of Digital Imaging for Cytopathology under Conditions of Georgia

  • Digital imaging in cytopathology can be defined as a series of procedures, those contributing to the quality of the displayed on the computer monitor final image. The procedures include sample preparation and staining, optical image formation by the microscope and afterwards the digital image sampling by the camera sensor; which means digital image post-processing and compression, transmission across the network and display on the monitor. A large amount of data about digital imaging exist. However, there are existing the problems with standardization and understanding of the digital imaging complete process. The field of digital imaging is rapidly evolving. The new models and protocols of the digital imaging are developed around the world, but in Georgia this field is still at evolving stages and revolves around static telecytology. It has been revealed, that the application of easy available and adaptable technology together with the improvement of infrastructure conditions is the essential basis for digital imaging. This tool is very useful for implementation of second opinion consultations on difficult cases. Digital imaging significantly increases knowledge exchange and thereby ensured a better medical service. The article aims description of digital imaging application for cytopathology under conditions of Georgia.

    Citation: Ekaterina Kldiashvili, Archil Burduli, Gocha Ghortlishvili. Application of Digital Imaging for Cytopathology under Conditions of Georgia[J]. AIMS Medical Science, 2015, 2(3): 186-199. doi: 10.3934/medsci.2015.3.186

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  • Digital imaging in cytopathology can be defined as a series of procedures, those contributing to the quality of the displayed on the computer monitor final image. The procedures include sample preparation and staining, optical image formation by the microscope and afterwards the digital image sampling by the camera sensor; which means digital image post-processing and compression, transmission across the network and display on the monitor. A large amount of data about digital imaging exist. However, there are existing the problems with standardization and understanding of the digital imaging complete process. The field of digital imaging is rapidly evolving. The new models and protocols of the digital imaging are developed around the world, but in Georgia this field is still at evolving stages and revolves around static telecytology. It has been revealed, that the application of easy available and adaptable technology together with the improvement of infrastructure conditions is the essential basis for digital imaging. This tool is very useful for implementation of second opinion consultations on difficult cases. Digital imaging significantly increases knowledge exchange and thereby ensured a better medical service. The article aims description of digital imaging application for cytopathology under conditions of Georgia.


    Abbreviations: FFIDDCs: fuzzy fractional implicit delay differential coupled systems; TFFs: triangular fuzzy functions; MDs: membership degrees; IVP: initial value problem; FDDEs: fuzzy delay differential equations; FDEs: fractional differential equations; DFDEs: fractional differential equations with time delay; FFDEs: fuzzy fractional differential equations; FCFDIVP: fuzzy coupled fractional initial value problem with delays; MF: membership function; TFN: triangular fuzzy number

    In an ecosystem, there exist universally competitions between two species, which is very important. And they tend to interact with each other so that there are some relationships such as predation or symbiosis among them (i.e. coupled systems) (see [9,39]). In addition, the delay and ambiguity may appear because of different reasons such as physical properties of equipment used in the system, signal transmission or measurement of system variables (see [14]), so we need to take them into account. Thus, in this paper, motivated by the work of Dong et al. [13] and Fatullayev et al. [14], we come up with the following problem of fuzzy fractional implicit delay differential coupled systems (FFIDDCs):

    $ {cDαx(t)=m1(t)y(tτ)+~F1(t,y(t),cDαx(t)),t>0,cDβy(t)=m2(t)x(tτ)+~F2(t,x(t),cDβy(t)),t>0,x(t)=~Φ1(t),τt0,y(t)=~Φ2(t),τt0,
    $
    (1.1)

    and explore existence and uniqueness result of solution for the FFIDDC (1.1), where $ ^{c}D^{\alpha}f(t) $ denotes $ \alpha $-order Caputo fractional derivative of function $ f(t)\in C_\alpha^1 $ for $ 0 < \alpha < 1 $ (see [22]), $ \tau $ is the value of time delay, $ \widetilde{\Phi_{1}}(t) $ and $ \widetilde{\Phi_{2}}(t) $ are triangular fuzzy functions (TFFs) defined on $ [-\tau, 0] $, $ \widetilde{F_{1}}(t, y(t), ^{c}D^{\alpha}x(t)) $ and $ \widetilde{F_{2}}(t, x(t), ^{c}D^{\beta}y(t)) $ are also TFFs defined on $ (0, \infty)\times {\mathbb R} \times {\mathbb R} $, and $ m_{1}(t) $ and $ m_{2}(t) $ are continuous crisp functions. The concept of TFFs used in this paper was first proposed by Gasilov et al. [17], then it has been widely employed [7,16,27]. Unlike in the past, Gasilov et al. [17] thought that a fuzzy function is a fuzzy bunch of real functions rather than a fuzzy number-valued function. Each of these real functions has a certain membership degree. The new concept of TFFs broadened the way (see [14]), and so we deal with fuzzy functions and fuzzy problems.

    Remark 1.1. In this paper, we consider the problem with Caputo fractional derivative, which is based on Riemann-Liouville fractional derivative [22]. We note that for temporal variable, the Caputo fractional derivative has been widely recognized in real application. For example, the Caputo fractional derivative of a constant is $ 0 $, but in the case of finite number at the lower limit of the interval, the Riemann-Liouville fractional derivative of a constant is not equal to $ 0 $. And the initial conditions given by Riemann-Liouville method cannot be explained physically. For more related work, see [28,34]. However, can one discuss the existence and uniqueness results for the solutions of the FFIDDC (1.1) with Riemann-Liouville fractional derivative or Hadamard fractional derivative? These are usually presented as important and significative problems in future research.

    Some special cases of the FFIDDC (1.1) are listed as follows:

    (i) If $ \tau = 0, 0\leq t\leq 1 $ and $ \alpha_{j} = 1 $, where $ \alpha_{j} $ is the membership degrees (MDs) of $ \widetilde{F_{j}} $ for $ j = 1, 2 $, the FFIDDC (1.1) can be rewritten as the problem investigated by Dong et al. [13]:

    $ {cDαx(t)=f(t,y(t),cDαx(t)),0t1,cDβy(t)=g(t,x(t),cDβy(t)), 0t1,x(0)=x(0),y(0)=y(0).
    $

    Making full use of fixed point theorem and the vector form of Gronwall inequality, Dong et al. [13] gained existence and uniqueness of solution to the initial value problem (IVP), and discussed the estimation of the solution.

    (ii) If $ \alpha = \beta = 1 $, the FFIDDC (1.1) reduces to the following fuzzy implicit delay differential coupled systems:

    $ {x(t)=m1(t)y(tτ)+~F1(t,y(t),x(t)),t>0,y(t)=m2(t)x(tτ)+~F2(t,x(t),y(t)),t>0,x(t)=~Φ1(t),τt0,y(t)=~Φ2(t),τt0,
    $
    (1.2)

    which is brand new and is worth studying in the future.

    We note that the system (1.2) is an extension of the linear inhomogeneous fuzzy delay differential equations (FDDEs) considered by Fatullayev et al. [14]:

    $ {x(t)=n(t)x(t)+m(t)x(tτ)+˜F(t),t>0,x(t)=˜Φ(t),τt0.
    $

    The authors presented a method to represent the solution of FDDE as a fuzzy set of real functions, and proved existence and uniqueness of solution for FDDE involving TFFs. Furthermore, Fatullayev et al. [14] showed clearly that the proposed method can be extended to the system of FDDEs by using the research results of Gasilov and Amrahov [15]. The purpose of this paper is to discuss existence and uniqueness of solution for the FFIDDC (1.1).

    Comparing with the simpler integer-order differential equations, fractional differential equations (FDEs) have more wide applications and play an important role in engineering, physics, finance and other fields. Nowadays, FDEs have been used to more accurately describe the dynamics of many systems when several complex phenomena in numerous seemingly diverse and widespread fields of science and engineering is modeled and investigated. See, for example, [3,6,22,28,34] and references therein. In addition, to study ecosystem problems in biology, one often discovers that an ecosystem does not have only a single species. That is to say, there must be multiple species, so there exists competition among them and coupling is taken into account (see [9,39]). Coupling relationship refers to the interaction and mutual influence between two or more objects. In fact, the coupled systems of FDEs have been studied by multitudinous researchers. See, for example, [2,38] and the references cited therein. Recently, by applying standard fixed point theorems for multivalued maps, existence of solutions for coupled systems of fractional differential inclusions with coupled boundary conditions was obtained by Ahmad et al. [5].

    In science and engineering and other practical applications, time delay usually occurs on account of manual measurement, signal transmission, aging of equipments and so on (see [30]). Thereby, in recent years, the scholars have been interested in solving the fractional differential equations with time delay (DFDEs) (see, for instance, [30,35] and the references therein). Brzdek and Eghbali [10] considered the Ulam's stability of DFDEs and also proved that under some appropriate assumptions, each approximate solution of DFDEs is close to its only exact solution. In order to acquire existence of solutions for a time dependent delay differential equation with constant delay, Tabassum et al. [33] extended the previous approaches about fuzzy weakly contraction mapping principle. Of course, there are not just one type of FDEs, whether they are linear or nonlinear. And the common ones are Caputo type, Riemann-Liouville type and Hadamard type (see [11,21,36]). Moreover, the delay is not necessarily finite, it can be also infinite. Some researchers dedicated themselves to the study of FDEs with infinite delay. Banach fixed point theorem and the nonlinear alternative of Leray-Schauder type initiated by Granas and Dugundji [18], had been utilized by Benchohra et al. [8] to investigate existence results of FDEs with infinite delay.

    Certainly, in actual application, there are not only time delay, but also exists the possibility of ambiguity. Thus, one needs to take the fuzzy into account, and then the fuzzy fractional differential equations (FFDEs) are proposed. Agarwal et al. [1] put forward a notion of fuzzy Riemann-Liouville differentiability based on Hukuhara differentiability, which can be employed to solve the IVPs of FFDEs. After that, many investigators have extended the concepts and expression forms of FFDEs, and bent themselves to obtain existence and uniqueness of solution for FFDEs. These theories are also well applied in real life. See, for example, [24,25,29] and the references therein. As Fatullayev et al. [14] pointed out, "the types of fuzzy items are more important than their numbers when a fuzzy problem needs to be solved". In fact, we can easily deal with $ n $-th order homogeneous linear differential equations with fuzzy initial values (i.e. the problem with $ n $ fuzzy inputs) by using Zadeh's extension principle. But if a fuzzy forcing function appears in the problem, even for the delay differential equation of first order (i.e. with 2 fuzzy inputs), it will be indistinct how to use the principle.

    It is well know that explicit equations are special cases of implicit equations, and the implicit equations are more general (see [12]). Cubiotti and Yao [12] introduced and studied a class of implicit second-order ordinary differential equations with known endpoint values. They also proved existence results under two conditions. However, as Hoa and Vu [21] stated briefly, "to the best of our knowledge, there exists no literature devoted to the uncertain fractional implicit differential equation in the fuzzy setting with the concepts of Riemann-Liouville, Caputo, Hadamard fractional derivatives". And then, Hoa and Vu [21] proposed some existence results for solutions of the fuzzy fractional implicit differential equations associated with the fuzzy Caputo-type, fuzzy Riemann-Liouville-type and fuzzy Hadamard-type concepts of fractional derivative, respectively. Very recently, Son and Dong [31] proposed global existence and some properties of solutions for the nonlocal problem of implicit fuzzy fractional differential systems. For more work on various forms of implicit FDE models, one can refer to [26,19,20] and the references therein.

    The major methods of this paper are as follows:

    (i) We generalize TFF (Definition 2.1 of [17]) to a group of TFFs (see Example 2.1).

    (ii) To solve the FFIDDC (1.1), we use the method of steps proposed by Fatullayev et al. [14]. Firstly, we figure out the solution of the FFIDDC (1.1) on interval $ [0, \tau] $. Afterwards, we similarly find the solution on interval $ [\tau, 2\tau] $. And as we go along, eventually, the solution on interval $ [0, +\infty) $ can be obtained.

    (iii) Because of the fuzzy coupled fractional initial value problem with delays (FCFDIVP) (3.1), which is the identical transformation of the FFIDDC (1.1), is linear, the superposition principle can be used to work out it. As a consequence, we can decompose the FCFDIVP (3.1) into three subproblems (3.4)-(3.6) and solve them separately.

    The remainder of this paper is organized as follows. The necessary preliminaries about the fuzzy theory that we are going to use are listed in Section 2. In Section 3, we introduce the concept of solution for the FFIDDC (1.1) and describe how to get it. Ultimately, a solution algorithm is proposed. In Section 4, we present an example to show that the hypotheses in all theorems can be met. Finally, in Section 5, we summarize the research results of this paper, and provide the content that one can study in the future.

    Based on the work of Zadeh [37], we define a fuzzy set $ \widetilde{A} $ as a pair of the universal set $ U $ and the membership function (MF) $ \mu:U\rightarrow[0, 1] $. The MF of a fuzzy set $ \widetilde{A} $ can be denoted as $ \mu_{\widetilde{A}} $. For each $ x\in U $, the numerical value $ \mu_{\widetilde{A}}(x) $ is called the MD of $ x $ in $ \widetilde{A} $. The crisp set $ supp(\widetilde{A}) = \{x\in U\mid\mu_{\widetilde{A}}(x) > 0\} $ is called the support of $ \widetilde{A} $.

    After that, let $ U $ be the set of real numbers $ {\mathbb R} $, and $ a, c $ and $ b $ be real numbers which meet $ a\leq c\leq b $. Then the set $ \widetilde{u} $ with MF

    $ μ(x)={xaca, a<x<c,1,x=c,bxbc, c<x<b,0,otherwise
    $

    is called a triangular fuzzy number (TFN) and we denote it as $ \widetilde{u} = (a, c, b) $. On the grounds of the geometric interpretation, the number $ c $ is called the vertex of $ \widetilde{u} $, we denote $ \underline{u} = a $ and $ \overline{u} = b $ to represent the left and the right end-points of $ \widetilde{u} $, respectively. Frequently, we express $ \widetilde{u} = (a, c, b) $ as $ \widetilde{u} = u_{cr}+\widetilde{u}_{un} $. Here, $ u_{cr} = c $ is the crisp part and $ \widetilde{u}_{un} = (a-c, 0, b-c) $ is the uncertain part of $ \widetilde{u} $.

    It is also useful to represent the fuzzy sets though their $ \alpha $-$ cuts $. For each $ \alpha\in(0, 1] $, the crisp set $ A_{\alpha} = \{x\in U\mid \mu_{\widetilde{A}}(x)\geq \alpha\} $ is called the $ \alpha $-$ cut $ of $ \widetilde{A} $. For $ \alpha = 0 $, the $ A_{0} = closure(supp(\widetilde{A})) $.

    For the TFN $ \widetilde{u} = (a, c, b) $, the $ \alpha $-$ cuts $ are intervals $ u_{\alpha} = [\underline{u_{\alpha}}, \overline{u_{\alpha}}] $, where $ \underline{u_{\alpha}} = a+\alpha(c-a) $ and $ \overline{u_{\alpha}} = b+\alpha(c-b) $. These formulas can be rewritten as $ \underline{u_{\alpha}} = c+(1-\alpha)(a-c) $ and $ \overline{u_{\alpha}} = c+(1-\alpha)(b-c) $. Therefore, $ u_{\alpha} = [\underline{u_{\alpha}}, \overline{u_{\alpha}}] = c+(1-\alpha)[a-c, b-c] $. From here we can see that an $ \alpha $-$ cut $ is homothetic to $ [a, b] $ (which is the $ 0 $-$ cut $) with center $ c $ and with ratio $ (1-\alpha) $.

    There are different notions about the fuzzy functions. In this study, we use the concept of the fuzzy function which was brought forward by Gasilov et al. [17], namely, fuzzy function is a bunch of fuzzy real functions. As a value $ \widetilde{F}(t) $ of a fuzzy bunch $ \widetilde{F} $ at time $ t $, we understand the fuzzy set, which elements are the values of the real functions at $ t $, with the higher MD of the corresponding functions. Mathematically,

    $ μ˜F(t)(x)=α  y():(μ˜F(y)=αy(t)=x)z():(μ˜F(z)>αz(t)x),
    $

    where "$ \wedge $" and "$ \rightarrow $" are the logical conjunction and implication symbols, respectively.

    Definition 2.1. ([17]) Let U be set of continuous functions defined on an interval I, and $ F_{a}(\cdot), F_{c}(\cdot), F_{b}(\cdot)\in U $. We call the fuzzy subset $ \widetilde{F} $ of U, determined by the MF as follows:

    $ μ˜F(y())={α,y=Fa+α(FcFa)and0<α1,α,y=Fb+α(FcFb)and0<α1,0,otherwise,
    $

    as TFF and denote it as $ \widetilde{F} = \langle F_{a}, F_{c}, F_{b}\rangle $.

    According to this definition, a TFF is a fuzzy set (or, fuzzy bunch) of real functions. Among them only two functions have the MD $ \alpha $: the functions $ y_{1} = F_{a}+\alpha(F_{c}-F_{a}) $ and $ y_{2} = F_{b}+\alpha(F_{c}-F_{b}) $.

    Referring to the example of TFF given by Gasilov et al. [16], we give the following example of a group of TFFs.

    Example 2.1. In Figure 1, we depict a group of TFFs as

    Figure 1.  A group of TFFs.
    $ {~F1=F1a,F1c,F1b~F2=F2a,F2c,F2b
    $

    where

    $ F_{1a}(t) = -t^{2}+5\, t-4 $ $($MD is $ 0 $, the black curve that is at bottom on $ [0, 1] $ and at upper on $ [1,2] )$;

    $ F_{1c}(t) = 0.5\, t^{2}-1.5\, t+1 $ $($MD is $ 1 $, the black dashed line);

    $ F_{1b}(t) = t^{2}-5\, t+4 $ $($MD is $ 0 $, the black curve that is at upper on $ [0, 1] $ and at bottom on $ [1,2]) $;

    $ F_{2a}(t) = -2\, t^{2}+4\, t-2 $ $($MD is $ 0 $, the blue curve that is at bottom on $ [0, 2]) $;

    $ F_{2c}(t) = t^{2}-2\, t+1 $ $($MD is $ 1 $, the blue dashed line$)$;

    $ F_{2b}(t) = 2\, t^{2}-4\, t+2 $ $($MD is $ 0 $, the blue curve that is at upper on $ [0, 2]) $.

    The TFF $ \widetilde{F_{1}} $ with MDs $ 0.7 $ and $ 0.3 $ are depicted by red dotted and dashed-dotted lines, respectively. And the TFF $ \widetilde{F_{2}} $ with MDs $ 0.7 $ and $ 0.3 $ are described by cyan dotted and dashed-dotted lines which are marked, respectively.

    The value of a TFF at a time $ t\in I $ can be expressed by the following formula:

    $ ˜F(t)=(min{Fa(t),Fc(t),Fb(t)},Fc(t),max{Fa(t),Fc(t),Fb(t)}).
    $

    We can easily find out that this value is a TFN, and a TFF $ \widetilde{F} = \langle F_{a}, F_{b}, F_{c}\rangle $ is not a fuzzy number-valued function. Actually, it is a fuzzy subset of the universe of continuous functions. Each element of this fuzzy subset is a real function with a certain MD.

    If $ F_{a}(t)\leq F_{c}(t)\leq F_{b}(t) $ for all $ t\in I $, then the TFF $ \widetilde{F} = \langle F_{a}, F_{c}, F_{b}\rangle $ is regular TFF on $ I $, and we have $ \widetilde{F}(t) = (F_{a}(t), F_{c}(t), F_{b}(t)) $. Further, when a TFF $ \widetilde{F} $ is not regular, we call it as non-regular TFF. It means that for a non-regular TFF, in general, $ \widetilde{F}(t)\neq (F_{a}(t), F_{c}(t), F_{b}(t)) $. And the graphs of functions $ F_{a}, F_{c} $ and $ F_{b} $ can interchanged as $ t $ goes. In Figure 1, the TFFs $ \widetilde{F_{1}} $ and $ \widetilde{F_{2}} $ are non-regular and regular on $ [0, 2] $, respectively. Without loss of generality, no matter regular TFFs or non-regular TFFs, main results and the algorithm in this paper are all valid. In the sequel, we assume that TTFs are regular.

    Now, we will consider the FFIDDC (1.1) in this section. With respect to the FFIDDC (1.1), we can solve it by using the method of steps introduced by Fatullayev et al. [14].

    First of all, we deal with the FFIDDC (1.1) on interval $ [0, \tau] $. Because $ x(t-\tau) = \widetilde{\Phi_{1}}(t-\tau), y(t-\tau) = \widetilde{\Phi_{2}}(t-\tau) $ for $ t\in[0, \tau] $, the FFIDDC (1.1) is equivalent to FCFDIVP as follows:

    $ {cDαx(t)=~G1(t,y(t),cDαx(t)),cDβy(t)=~G2(t,x(t),cDβy(t)),x(0)=~Φ1(0),y(0)=~Φ2(0),
    $
    (3.1)

    where

    $ ~G1(t,y(t),cDαx(t))=m1(t)~Φ2(tτ)+~F1(t,y(t),cDαx(t)),~G2(t,x(t),cDβy(t))=m2(t)~Φ1(tτ)+~F2(t,x(t),cDβy(t)).
    $

    Thus, we transform the FCFDIVP (3.1) in matrix form:

    $ {DZ(t)=˜G(),Z(0)=˜Φ(0).
    $
    (3.2)

    Here, $ Z(t) = \left(x(t)y(t)

    \right) $, $ D = \left(cDα00cDβ
    \right) $, $ \widetilde{G}(\cdot) = \left(~G1(t,y(t),cDαx(t))~G2(t,x(t),cDβy(t))
    \right) $ and $ \widetilde{\Phi}(0) = \left(~Φ1(0)~Φ2(0)
    \right). $

    Definition 3.1. For the problem $(3.2)$, the fuzzy set $ \widetilde{Z} $ with MF

    $ μ˜Z(Z())=min{μ˜Φ(ϕ(0)),μ˜G(DZ(t))}
    $
    (3.3)

    is called to be a solution of the problem $(3.2)$, which is also a solution to the FCFDIVP $(3.1)$, where $ \phi(0) = Z(0) $.

    As for the formula (3.3), let $ Z(t) $ be a functional matrix. We determine $ \phi(0) = Z(0) $ for $ t\in[0, \tau] $ and calculate $ \mu_{1}\triangleq \mu_{\widetilde{\Phi}}(\phi(0)) $. After that, we compute $ g(\cdot) = D\cdot Z(t) $ on interval $ [0, \tau] $ and determine $ \mu_{2}\triangleq \mu_{\widetilde{G}}(g(\cdot)) $. Finally, we calculate the MD $ \mu $ as $ \mu = \min\{\mu_{1}, \mu_{2}\} $. We assign the number $ \mu $ as the MD of $ Z(t) $. We define the set of all functional matrices such as $ Z(t) $ with their MDs $ \mu $ as the fuzzy solution $ \widetilde{Z} $.

    According to Definition 3.1., the solution $ \widetilde{Z} $ is a fuzzy bunch of real functional matrices, which consists of functional matrices such as $ Z(t) $. If a functional matrix $ Z(t) $ satisfies

    $ {DZ(t)=g(),Z(0)=ϕ(0)
    $

    for some functional matrices $ g\in supp(\widetilde{G}) $ and $ \phi(0)\in supp(\widetilde{\Phi}(0)) $, then it has a positive MD.

    Let us represent $ \widetilde{G_{1}} = g_{1cr}+\widetilde{g_{1}} $ (crisp part + uncertainty), where $ g_{1cr} = G_{1c} $ and $ \widetilde{g_{1}} = \langle g_{1a}, 0, g_{1b}\rangle = \langle G_{1a}-G_{1c}, 0, G_{1b}-G_{1c}\rangle $. Similarly, $ \widetilde{G_{2}} = g_{2cr}+\widetilde{g_{2}} $, here $ g_{2cr} = G_{2c} $ and $ \widetilde{g_{2}} = \langle g_{2a}, 0, g_{2b}\rangle $.

    Assume that $ \widetilde{\Phi_{1}} $ and $ \widetilde{\Phi_{2}} $ are regular TFFs. Then $ \widetilde{\Phi_{1}}(0) $ and $ \widetilde{\Phi_{2}}(0) $ are TFNs. Further, $ \widetilde{\Phi_{1}}(0) = \phi_{1cr}(0)+\widetilde{\phi_{1}}(0) $, here $ \phi_{1cr}(0) = \Phi_{1c}(0) $ and $ \widetilde{\phi_{1}}(0) = (\phi_{1a}(0), 0, \phi_{1b}(0)) = (\Phi_{1a}(0)-\Phi_{1c}(0), 0, \Phi_{1b}(0)-\Phi_{1c}(0)) $. In the same way, $ \widetilde{\Phi_{2}}(0) = \phi_{2cr}(0)+\widetilde{\phi_{2}}(0) $, here $ \phi_{2cr}(0) = \Phi_{2c}(0) $ and $ \widetilde{\phi_{2}}(0) = (\phi_{2a}(0), 0, \phi_{2b}(0)) $.

    Since the FCFDIVP (3.1) is linear, we can solve it using the superposition principle. This follows that in order to find the solution of the FCFDIVP (3.1), we can separately consider the following subproblems:

    (1) The associated crisp problem:

    $ {cDαx(t)=g1cr(t,y(t),cDαx(t)),cDβy(t)=g2cr(t,x(t),cDβy(t)),x(0)=ϕ1cr(0),y(0)=ϕ2cr(0).
    $
    (3.4)

    (2) The problem with initial TFFs:

    $ {cDαx(t)=0,cDβy(t)=0,x(0)=~ϕ1(0),y(0)=~ϕ2(0).
    $
    (3.5)

    (3) The problem with fuzzy source functions and zero initial functions:

    $ {cDαx(t)=~g1(t,y(t),cDαx(t)),cDβy(t)=~g2(t,x(t),cDβy(t)),x(0)=0,y(0)=0.
    $
    (3.6)

    By solving the above three subproblems (3.4)-(3.6), the solution of the FFIDDC (1.1) on interval $ [0, \tau] $ can be obtained. Similarly, at the next step, we can find the solution on interval $ [\tau, 2\tau] $. Hence, we can conclude that the solution of the FFIDDC (1.1) exists for $ t\in[0, \infty) $ using the method of steps [14].

    In the sequel, we make clear how to solve each of these three subproblems (3.4)-(3.6).

    To solve the problem (3.4), we provide the following theorem.

    Theorem 3.1. For the problem $(3.4)$, we can get a unique solution to it, which is denoted as $ Z_{cr}(t) = \left(xcr(t)ycr(t)

    \right) $, when the following conditions fulfill:

    $ (H_{1}) $ $ g_{1cr}, g_{2cr}: [0, \tau]\times {\mathbb R}\times {\mathbb R}\rightarrow {\mathbb R} $ are continuous.

    $ (H_{2}) $ There exist constants $ K_{i}, L_{i}\in (0, 1) $ for $ i = 1, 2 $ such that

    $ 0 < \max\left\{K_{2}+\dfrac{K_{1}}{\Gamma(\beta+1)}, \, L_{2}+\dfrac{L_{1}}{\Gamma(\alpha+1)}\right\} < 1 $

    and for each $ x_{1}, x_{2}, y_{1}, y_{2}\in {\mathbb R} $,

    $ g1cr(t,x1,y1)g1cr(t,x2,y2)∣≤K1x1x2+K2y1y2,g2cr(t,x1,y1)g2cr(t,x2,y2)∣≤L1x1x2+L2y1y2.
    $

    Proof. The proof is similar to Theorem 3.1 of Dong et al. [13] and it is omitted.

    In this subsection, we will give solvability of the problem (3.5) as follows.

    Theorem 3.2. Consider the problem $(3.5)$, where $ \widetilde{\phi_{1}}(0) = (\phi_{1a}(0), 0, \phi_{1b}(0)), \widetilde{\phi_{2}}(0) = (\phi_{2a}(0), 0, \phi_{2b}(0)) $. If $ Z_{a}(t) = \left(xa(t)ya(t)

    \right) $ and $ Z_{b}(t) = \left(xb(t)yb(t)
    \right) $ are solutions of the problem

    $ {cDαx(t)=0,cDβy(t)=0,x(0)=ϕ1(0),y(0)=ϕ2(0)
    $
    (3.7)

    for $ \left(ϕ1(0)ϕ2(0)

    \right) = \left(ϕ1a(0)ϕ2a(0)
    \right) $ and $ \left(ϕ1(0)ϕ2(0)
    \right) = \left(ϕ1b(0)ϕ2b(0)
    \right) $, respectively, then the problem $(3.5)$ has a unique solution $ \widetilde{Z_{\phi}} $, which is a TFF matrix given by

    $ ~Zϕ=Za,0,Zb.
    $
    (3.8)

    Proof. On the one hand, according to Definition 3.1., it is easy for us to know that each $ Z(t) $ with non-zero MD from the bunch $ \widetilde{Z_{\phi}} $ is a solution of the problem (3.7) for some $ \left(ϕ1(0)ϕ2(0)

    \right) $ from $ \left(~ϕ1(0)~ϕ2(0)
    \right) $.

    On the other hand, in terms of Definition 2.1, the bunch $ \widetilde{\phi_{1}} = \langle \phi_{1a}, 0, \phi_{1b}\rangle $ consists of functions $ k\phi_{1a} $ and $ k\phi_{1b} $ $ \left([0, 1]\ni k = 1-\alpha\right) $ owing to $ \phi_{1c} = 0 $. And the bunch $ \widetilde{\phi_{2}} = \langle\phi_{2a}, 0, \phi_{2b}\rangle $ is similar to the former.

    Moreover, if a Z(t) is a solution of the problem (3.7), then $ kZ(t) $ is a solution of the same equation with $ \left(kϕ1(0)kϕ2(0)

    \right) $ taken instead of $ \left(ϕ1(0)ϕ2(0)
    \right) $.

    From the above reasoning we can get the conclusion that the bunch $ \widetilde{Z_{\phi}} $ consists of $ kZ_{a} $ and $ kZ_{b} $. Therefore, the bunch $ \widetilde{Z_{\phi}} $ is a TFF matrix determined to be $ \widetilde{Z_{\phi}} = \langle Z_{a}, 0, Z_{b}\rangle $.

    Actually, we can get the solutions $ Z_{a}(t) = \left(ϕ1a(0)ϕ2a(0)

    \right) $ and $ Z_{b}(t) = \left(ϕ1b(0)ϕ2b(0)
    \right) $.

    We can express the value of the TFF matrix $ \widetilde{Z_{\phi}} $ (3.8) at a time $ t $ by the formula $ \widetilde{Z_{\phi}}(t) = (\min\{Z_{a}(t), 0, Z_{b}(t)\}, 0, \max\{Z_{a}(t), 0, Z_{b}(t)\}) $. Note that this value is a matrix of TFNs.

    As for the solution of the problem (3.6), we give the theorem as follows.

    Theorem 3.3. In the problem $(3.6)$, assume $ \widetilde{g_{1}} = \langle g_{1a}, 0, g_{1b}\rangle, \widetilde{g_{2}} = \langle g_{2a}, 0, g_{2b}\rangle $, where $ \left(g1ag2a

    \right) $ and $ \left(g1bg2b
    \right) $ separately
    satisfy the conditions $ (H_{1}) $ and $ (H_{2}) $ of Theorem 3.1. If $ Z_{u}(t) = \left(xu(t)yu(t)
    \right) $ and $ Z_{v}(t) = \left(xv(t)yv(t)
    \right) $ are solutions of the following problem:

    $ {cDαx(t)=g1(t,y(t),cDαx(t)),cDβy(t)=g2(t,x(t),cDβy(t)),x(0)=0,y(0)=0
    $
    (3.9)

    for $ \left(g1g2

    \right) = \left(g1ag2a
    \right) $ and $ \left(g1g2
    \right) = \left(g1bg2b
    \right) $ in several, then the problem $(3.6)$ has a unique solution $ \widetilde{Z_{g}} $, which is a TFF matrix given by

    $ ~Zg=Zu,0,Zv.
    $
    (3.10)

    Proof. Since $ g_{1a} $ and $ g_{2a} $ satisfy $ (H_{1}) $ and $ (H_{2}) $, it follows from Theorem 3.1 that there will be a unique solution when $ \left(g1g2

    \right) = \left(g1ag2a
    \right) $ in (3.9). By the same token, from the assumption for $ \left(g1g2
    \right) = \left(g1bg2b
    \right) $ in the problem (3.9), only one solution will exist in this case. Thus, the proof can be done by the same way as in Theorem 3.2.

    Thus, we can also express the value of the TFF matrix $ \widetilde{Z_{g}} $ (3.10) at a time $ t $ by $ \widetilde{Z_{g}}(t) = (\min\{Z_{u}(t), 0, Z_{v}(t)\}, 0, \max\{Z_{u}(t), 0, Z_{v}(t)\}) $, which is a matrix of TFNs.

    By integrating Theorems 3.1-3.3 above, we can obtain the following solution algorithm for solving the FFIDDC (1.1).

    Algorithm 3.1. Step 1. Using the method of steps and dealing with the FFIDDC $(1.1)$ on interval $ [0, \tau] $, transform the FFIDDC $(1.1)$ into the FCFDIVP $(3.1)$.

    Step 2. Represent the initial values and source functions as

    $ ~G1=g1cr+g1a,0,g1b,~G2=g2cr+g2a,0,g2b,~Φ1(0)=ϕ1cr(0)+(ϕ1a(0),0,ϕ1b(0)),~Φ2(0)=ϕ2cr(0)+(ϕ2a(0),0,ϕ2b(0)).
    $

    Step 3. Find the solution $ Z_{cr}(t) $ of the problem $(3.4)$.

    Step 4. Seek the solutions $ Z_{a}(t) $ and $ Z_{b}(t) $ of the problem $(3.7)$ in regard to $ \left(ϕ1(0)ϕ2(0)

    \right) = \left(ϕ1a(0)ϕ2a(0)
    \right) $ and $ \left(ϕ1(0)ϕ2(0)
    \right) = \left(ϕ1b(0)ϕ2b(0)
    \right) $, respectively, and define

    $ ~Zϕ(t)=(min{Za(t),0,Zb(t)},0,max{Za(t),0,Zb(t)}).
    $

    Step 5. Solve the problem $(3.9)$ and denote the solutions by $ Z_{u}(t) $ and $ Z_{v}(t) $, corresponding to $ \left(g1g2

    \right) = \left(g1ag2a
    \right) $ and $ \left(g1g2
    \right) = \left(g1bg2b
    \right) $, respectively, and let

    $ ~Zg(t)=(min{Zu(t),0,Zv(t)},0,max{Zu(t),0,Zv(t)}).
    $

    Step 6. Construct the unique solution of the FFIDDC $(1.1)$ on interval $ [0, \tau] $ as follows:

    $ ˜Z(t)=Zcr(t)+~Zϕ(t)+~Zg(t).
    $

    Similarly, we can find the unique solution on interval $ [\tau, 2\tau] $, $ \cdots $. In consequence, we obtain a unique solution of the FFIDDC (1.1) which exists at $ t\geq 0 $.

    In this section, referring to Example 1 in Fatullayev et al. [14], we take an example to clarify that the hypotheses in Theorems 3.1-3.3 can be satisfied, which can be also employed to verify the main results presented in Dong et al. [13].

    Example 4.1. Let us solve the problem

    $ {cD12x(t)=costy(tπ2)+~F1(t,y(t),cD12x(t)),t>0,cD12y(t)=sintx(tπ2)+~F2(t,x(t),cD12y(t)),t>0,x(t)=~Φ1(t),π2t0,y(t)=~Φ2(t),π2t0,
    $
    (4.1)

    where

    $ ~F1=f1cr+f1a,0,f1b,~F2=f2cr+f2a,0,f2b,~Φ1=ϕ1cr+ϕ1a,0,ϕ1b,~Φ2=ϕ2cr+ϕ2a,0,ϕ2b
    $

    with

    $ f1cr=0.5siny(t)+103sincD12x(t),f2cr=0.7sinx(t)103sincD12y(t),f1a=0.15sintcost, f1b=0.15sintcost,f2a=0.15sintcost,f2b=0.15sintcost,ϕ1cr=cost, ϕ1a=0.15sint0.25cost, ϕ1b=0.15sint+0.25cost,ϕ2cr=sint, ϕ2a=0.15cost0.25sint, ϕ2b=0.15cost+0.25sint.
    $

    In the sequel, we only deal with the problem (4.1) on interval $ \left[0, \dfrac{\pi}{2}\right] $. The unique solution on interval $ [\tau, 2\tau] $, $ \cdots $ can be likewise found.

    Firstly, the problem (4.1) can be transformed as the problem

    $ {cD12x(t)=~G1(t,y(t),cD12x(t)),cD12y(t)=~G2(t,x(t),cD12y(t)),x(0)=~Φ1(0),y(0)=~Φ2(0),
    $
    (4.2)

    where

    $ ~G1(t,y(t),cD12x(t))=cost~Φ2(tπ2)+~F1(t,y(t),cD12x(t)),~G2(t,x(t),cD12y(t))=sint~Φ1(tπ2)+~F2(t,x(t),cD12y(t)).
    $

    Then, after performing some arithmetics, one can get

    $ ~G1=g1cr+g1a,0,g1b,~G2=g2cr+g2a,0,g2b,~Φ1(0)=ϕ1cr(0)+~ϕ1(0),~Φ2(0)=ϕ2cr(0)+~ϕ2(0),
    $

    where

    $ g1cr=cos2t0.5siny(t)+103sincD12x(t),g2cr=sin2t+0.7sinx(t)103sincD12y(t),g1a=0.25cos2t,g1b=0.25cos2t,g2a=0.25sin2t,g2b=0.25sin2t,ϕ1cr(0)=1,~ϕ1(0)=(0.25,0,0.25),ϕ2cr(0)=0,~ϕ2(0)=(0.15,0,0.15).
    $

    Next, we respectively solve the three subproblems of the problem (4.2) on $ \left[0, \dfrac{\pi}{2}\right] $ one by one.

    (i) With respect to $ g_{1cr} $ and $ g_{2cr} $, we can rewrite them in another forms:

    $ g1cr(t,u,v)=cos2t0.5sinu+103sinv,g2cr(t,m,n)=sin2t+0.7sinm103sinn,
    $

    where

    $ u=y(t),v=cD12x(t),m=x(t),n=cD12y(t).
    $

    We can easily see that $ g_{1cr}, g_{2cr}:[0, \dfrac{\pi}{2}]\times {\mathbb R}\times {\mathbb R}\rightarrow {\mathbb R} $ are continuous and satisfy $ (H_{1}) $.

    Moreover, taking $ K_{1} = 0.5 $, $ K_{2} = 10^{-3} $, $ L_{1} = 0.7 $ and $ L_{2} = 10^{-3} $, then we have

    $ max{K2+K1Γ(12+1),L2+L1Γ(12+1)}=max{103+0.5Γ(32),103+0.7Γ(32)}=max{0.5718,0.8000}=0.8000(0,1),
    $

    and

    $ g1cr(t,u1,v1)g1cr(t,u2,v2)∣≤K1u1u2+K2v1v2,g2cr(t,m1,n1)g2cr(t,m2,n2)∣≤L1m1m2+L2n1n2
    $

    for $ i = 1, 2 $ and any $ u_{i} $, $ v_{i} $, $ m_{i} $, $ n_{i}\in {\mathbb R} $. Hence, it is easy to see that $ (H_{2}) $ holds.

    To sum up, all the hypotheses $ (H_{1}) $ and $ (H_{2}) $ in Theorem 3.1 are satisfied and we can obtain a unique solution $ Z_{cr}(t) $, which is illustrated in Figure 2 to the following associated crisp subproblem via using the "L1 method" due to Li and Zeng [23]:

    $ {cD12x(t)=g1cr(t,y(t),cD12x(t))=cos2t0.5siny(t)+103sincD12x(t),cD12y(t)=g2cr(t,x(t),cD12y(t))=sin2t+0.7sinx(t)103sincD12y(t),x(0)=ϕ1cr(0)=1,y(0)=ϕ2cr(0)=0,
    $
    (4.3)
    Figure 2.  The crisp solutions (global).

    which is the first subproblem of the problem (4.2).

    From Figure 2, one can easily see that the order of magnitude of the ordinate value in Figure 2 is $ 10^8 $, and $ x_{cr} $ and $ y_{cr} $ established by relying on the "L1 method" discretization [23] of the Caputo fractional derivatives in (4.3), are growing very fast, and after iterating 43 times ($ t\approx 0.66 $), the values of $ x_{cr} $ and $ y_{cr} $ approach $ 1 $ and $ 27.88 $, respectively. Since the ordinate value of Figure 2 is too large to observe between $ t = 0.6 $ and $ t = 0.7 $, we give the local graph Figure 3 for the crisp discretized solutions to the problem (4.3) on the ordinate value interval $ \left[0, 100\right] $, which right endpoint is far less than that of the ordinate value interval $ [0, 4\times 10^8] $ in Figure 2. We note that Figure 3 locally shows the unique solution $ Z_{cr}(t) = \left(xcr(t)ycr(t)

    \right) = \left(127.88
    \right) $ of the problem (4.3) at $ t\approx 0.66\in \left[0, \dfrac{\pi}{2}\right] $.

    Figure 3.  The crisp solutions (ordinate local interval $ [0, 100] $).

    (ii) Solve the following crisp problem:

    $ {cD12x(t)=0,cD12y(t)=0,x(0)=ϕ1(0),y(0)=ϕ2(0)
    $

    for $ \left(ϕ1(0)ϕ2(0)

    \right) = \left(0.250.15
    \right) $ and $ \left(ϕ1(0)ϕ2(0)
    \right) = \left(0.250.15
    \right) $, and find the solutions $ Z_{a}(t) = \left(0.250.15
    \right) $ and $ Z_{b}(t) = \left(0.250.15
    \right) $, respectively.

    Letting $ \widetilde{\phi_{1}}(0) = (-0.25, 0, 0.25) $ and $ \widetilde{\phi_{2}}(0) = (-0.15, 0, 0.15) $, then it follows that the solution to the second subproblem of the problem (4.2) as hereunder mentioned:

    $ {cD12x(t)=0,cD12y(t)=0,x(0)=~ϕ1(0),y(0)=~ϕ2(0)
    $
    (4.4)

    is the TFF matrix $ \widetilde{Z_{\phi}} = \langle Z_{a}, 0, Z_{b}\rangle $, which is graphed in Figure 4, and one knows that

    $ \widetilde{Z_{\phi}}(t) = \left(\min\{Z_{a}(t), 0, Z_{b}(t)\}, 0, \max\{Z_{a}(t), 0, Z_{b}(t)\}\right). $
    Figure 4.  Uncertainty of the solutions with initial TFFs.

    Hence, Theorem 3.2 holds and there is one and only solution to the problem (4.4). Indeed, since the solutions $ Z_{a}(t) $ and $ Z_{b}(t) $ are all constant matrices in Figure 4, it is easy to see that the values of $ \widetilde{x_{\phi_{1}}} $ and $ \widetilde{y_{\phi_{2}}} $ are all constants when MD $ \alpha\in [0, 1] $ is determined. That is, the values of $ \widetilde{x_{\phi_{1}}} $ and $ \widetilde{y_{\phi_{2}}} $ remain constant as $ t $ increases. For example, $ \widetilde{x_{\phi_{1a}}} = 0.25 $, $ \widetilde{x_{\phi_{1b}}} = -0.25 $, $ \widetilde{y_{\phi_{2a}}} = 0.15 $ and $ \widetilde{y_{\phi_{2b}}} = -0.15 $ when $ \alpha = 0 $.

    (iii) Consider the following crisp problem:

    $ {cD12x(t)=g1(t,y(t),cD12x(t)),cD12y(t)=g2(t,x(t),cD12y(t)),x(0)=0,y(0)=0
    $

    for $ \left(g1g2

    \right) = \left(g1ag2a
    \right) $ and $ \left(g1g2
    \right) = \left(g1bg2b
    \right) $, and find the solutions $ Z_{u}(t) = \left(xu(t)yu(t)
    \right) $ and $ Z_{v}(t) = \left(xv(t)yv(t)
    \right) $, respectively. That is, one solves the following crisp problems:

    $ {cD12x(t)=g1a(t,y(t),cD12x(t))=0.25cos2t,cD12y(t)=g2a(t,x(t),cD12y(t))=0.25sin2t,x(0)=0,y(0)=0
    $
    (4.5)

    and

    $ {cD12x(t)=g1b(t,y(t),cD12x(t))=0.25cos2t,cD12y(t)=g2b(t,x(t),cD12y(t))=0.25sin2t,x(0)=0,y(0)=0.
    $
    (4.6)

    As for the problem (4.5), $ g_{1a} $ and $ g_{2a} $ can be written in following forms:

    $ g1a(t,u,v)=0.25cos2t,g2a(t,m,n)=0.25sin2t,
    $

    where

    $ u=y(t),v=cD12x(t),m=x(t),n=cD12y(t).
    $

    It is clear to find that $ g_{1a}, g_{2a}:[0, \dfrac{\pi}{2}]\times {\mathbb R}\times {\mathbb R}\rightarrow {\mathbb R} $ are continuous.

    On the other hand, let $ K_{1} = 0.4 $, $ K_{2} = 0.1 $, $ L_{1} = 0.6 $ and $ L_{2} = 0.2 $. Then

    $ max{K2+K1Γ(12+1),L2+L1Γ(12+1)}=max{0.1+0.4Γ(32),0.2+0.6Γ(32)}=max{0.5514,0.8770}=0.8770(0,1),
    $

    and for each $ u_{1} $, $ u_{2} $, $ v_{1} $, $ v_{2} $, $ m_{1} $, $ m_{2} $, $ n_{1} $, $ n_{2}\in {\mathbb R} $,

    $ g1a(t,u1,v1)g1a(t,u2,v2)∣≤K1u1u2+K2v1v2,g2a(t,m1,n1)g2a(t,m2,n2)∣≤L1m1m2+L2n1n2.
    $

    According to the above, all the hypotheses $ (H_{1}) $ and $ (H_{2}) $ hold. And that on the basis of Theorem 3.1, we can obtain the unique solution $ Z_{u}(t) $ to the last subproblem of the problem (4.2) involving fuzzy source functions and zero initial functions as follows:

    $ {cD12x(t)=g1a,0,g1b,cD12y(t)=g2a,0,g2b,x(0)=0,y(0)=0.
    $
    (4.7)

    Similarly, we can conclude that the problem (4.6) also has a unique solution $ Z_{v}(t) $. It follows that the solution to the problem (4.7) is the TFF matrix $ \widetilde{Z_{g}} = \langle Z_{u}, 0, Z_{v}\rangle $, which is depicted in Figure 5, and we have

    $ ~Zg(t)=(min{Zu(t),0,Zv(t)},0,max{Zu(t),0,Zv(t)}).
    $
    Figure 5.  Uncertainty of the solutions owing to fuzzy source functions.

    As a consequence, Theorem 3.3 holds and there is a unique solution to the problem (4.7). Moreover, it can be seen that, in Figure 5, the images of $ \widetilde{x_{g_{1}}} $ and $ \widetilde{y_{g_{2}}} $ are respectively symmetric when the corresponding MDs are chosen. For MD $ \alpha\in [0, 1) $, as $ t $ increases, the upper branch of $ \widetilde{x_{g_{1}}} $ firstly increases and then decreases ($ t\approx 0.69 $ is the turning point), and the upper branch of $ \widetilde{y_{g_{2}}} $ gets bigger and bigger. Especially, the values of $ \widetilde{x_{g_{1}}} $ and $ \widetilde{y_{g_{2}}} $ are constant $ 0 $ when the MD $ \alpha = 1 $. However, the ordinate value of Figure 5 is of order of magnitude $ 10^{-1} $ and very small compared with that in Figure 2, so it has little effect on the final solution $ \widetilde{Z}(t) $.

    In conclusion, by solving the problems (4.3), (4.4) and (4.7), it follows from Algorithm 3.1 that we can find the unique solution of the problem (4.1) on interval $ \left[0, \dfrac{\pi}{2}\right] $ as

    $ ˜Z(t)=Zcr(t)+~Zϕ(t)+~Zg(t),
    $

    which can be found from Figure 6. It is easy to note that the image of fuzzy solution $ \widetilde{Z}(t) $ in Figure 6 has a similar trend to that of the crisp solution $ Z_{cr}(t) $ in Figure 2. That is because when MD is fixed, the unique solution $ \widetilde{Z}(t) $ of the problem (4.1) is composed of three parts and the value of the crisp solution $ Z_{cr}(t) $ for the problem (4.3) is too large (order of magnitude $ 10^8 $ of the ordinate value for Figure 2). Further, since the ordinate value of Figure 6 is very big (order of magnitude $ 10^8 $) and the curves of $ \widetilde{x}, \widetilde{y} $ can not be distinguished obviously, a local graph Figure 7 of Figure 6 is given for the fuzzy solutions of the problem (4.1) when the ordinate value interval is restricted to $ \left[0, 30\right] $. Choosing specially MD $ \alpha = 0 $, we have $ \widetilde{x_{\phi_{1a}}} = 0.93 $, $ \widetilde{x_{\phi_{1b}}} = 1.07 $, $ \widetilde{y_{\phi_{2a}}} = 27.68 $ and $ \widetilde{y_{\phi_{2b}}} = 28.08 $.

    Figure 6.  The fuzzy solutions obtained by the proposed method (global).
    Figure 7.  The fuzzy solutions acquired by the proposed method (ordinate local interval $ [0, 30] $). And the black $ \alpha- $cuts of $ \widetilde{x} $: $ 0- $cut (continues lines), $ 1- $cut (dashed line); the blue $ \alpha- $cuts of $ \widetilde{y} $: $ 0- $cut (continues lines), $ 1- $cut (dashed line).

    Before the end of this section, we remark that the unique solution of the problem (4.1) on interval $ \left[\dfrac{\pi}{2}, \pi\right] $, $ \cdots $ can be found in the same way. Therefore, one can find the unique solution of the problem (4.1) for $ t > 0 $ via Algorithm (3.1).

    In this paper, inspired by Fatullayev et al. [14] and Dong et al. [13], by using the methods of steps and separating fuzzy items, we solved the implicit coupled systems of fuzzy fractional delay differential equations with fuzzy initial values and source functions (FFIDDC) (1.1). Under the conditions $ (H_1) $ and $ (H_2) $, and based on the concept of TFFs and examples in [16,17], we obtained existence and uniqueness of solution for the FFIDDC (1.1). We note that the solution is a triangular fuzzy function matrix which is composed of real functional matrices. Furthermore, an algorithm for solution procedure and an example for graphical confirmation of explicit solution were given.

    As Remark 1.1 mentioned, except for considering the problem (1.1) with Caputo fractional derivative, we will explore the problem of form (1.1) with fractional derivatives of Riemann-Liouville or Hadamard in the future. Recently, there are more and more researchers to explore the relevant (fractional) differentiable set-value problems and impulsive problems, such as Amrahov et al. [4] and Sun et al. [32]. Therefore, for further study, we can introduce the set-valued delay and the impulse into the FFIDDC (1.1), and extend the methods proposed in this paper to solve the new problems associated with set-values or impulsive elements.

    We are grateful to the anonymous referees and editors for their valuable comments and helpful suggestions to improve the quality of this paper.

    This work was partially supported by the Sichuan Science and Technology Program (2019YJ0541) the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (2019QZJ03) and the Innovation Fund of Postgraduate, Sichuan University of Science & Engineering (y2020080).

    All authors declare no conflicts of interest in this paper.

    [1] Banta D (2003) The development of health technology assessment. Health Policy 63: 121-132. doi: 10.1016/S0168-8510(02)00059-3
    [2] Chatman C (2010) How cloud computing is changing the face of health care information technology. J Health Care Compliance 12(3): 37-70.
    [3] Clamp S, Keen J (2007) Electronic health records: Is the evidence base any use? J Med Informatics Int Med 32: 5-10.
    [4] Coleman R (2009) Can histology and pathology be taught without microscopes? The advantages and disadvantages of virtual histology. J Acta Histochemica 111(1): 1-4.
    [5] Detmer D (2000) Information technology for quality health care: a summary of United Kingdom and United States experiences. Quality in Health Care 9: 181-189. doi: 10.1136/qhc.9.3.181
    [6] Detmer D (2001) Transforming health care in the internet era. World Hospital Health Ser 37: 2.
    [7] Dixon R, Stahl J (2008) Virtual visits in a general medicine practice: A pilot study. Telemed e-Health 14(6): 525-530.
    [8] Furness PN (1997) The use of digital images in pathology. J Patholo183: 15-24.
    [9] Haughton J (2011) Year of the underdog: Cloud-based EHRs. Healthcare Management Technolo 32(1): 9
    [10] Hayrinen K, Saranto K, Nykanen P (2008) Definition, structure, content, use and impacts of electronic health records: A review of the research literature. Inter J Med Inform 77: 291-304.
    [11] Horbinski C, Wiley CA (2009) Comparison of telepathology systems in neuropathological intraoperative consultations. J Neuropatholo 19(2): 317-322.
    [12] Hufnagl P, Schluns K (2008) Virtual microscopy and routine diagnostics. A discussion paper. J Patholo 29(2): 250-254.
    [13] Kabachinski J (2011) What's the forecast for cloud computing in healthcare? Biomedical Instrumental Technolo 45(2): 146-150.
    [14] Kayser K, Hoshang SA, Metze K, et al. (2008) Texture- and object-related automated information analysis in histological still images of various organs. J Analytical Quantitative Cytolo Histo 30(6): 323-335.
    [15] Kobb R, Lane R, Stallings D (2008) E-learning and telehealth: Measuring your success. Telemedicine and e-Health 14(6): 576-579.
    [16] Lane K (2006) Telemedicine news. Telemedicine and e-Health 12(5): 507-511.
    [17] Lareng L (2002) Telemedicine in Europe. European J Int Med 13: 1-13.
    [18] Leong FJ, Graham AK, Schwarzmann P (2000) Clinical trial of telepathology as an alternative modality in breast histopathology quality assurance. J Telemed E-Health 6: 373-377.
    [19] Mencarelli R, Marcolongo A, Gasparetto A (2008) Organizational model for a telepathology system. J Diagnostic Patholo 15(3): S7.
    [20] Merell R, Doarn C (2008) Is it time for a telemedicine breakthrough? Telemed e-Health 14(6): 505-506.
    [21] Moore D, Green J, Jay S, et al. (1994). Creating a new paradigm for CME: Seizing opportunities within the health care revolution. J Continuing Education in the Health Professionals 14: 4-31. doi: 10.1002/chp.4750140102
    [22] Moura A, Del Giglio A (2000) Education via internet. Revist da Associacao Medica Brasileira, 46(1): 47-51.
    [23] Nannings B, Abu-Hanna A (2006) Decision support telemedicine systems: A conceptual model and reusable templates. Telemed e-Health 12(6): 644-654.
    [24] Pak H (2007) Telethinking. Telemed e-Health 13(5): 483-486.
    [25] Raab SS, Zaleski MS, Thomas PA (1996) Telecytology: diagnostic accuracy in cervical-vaginal smears. American J Clin Patholo 105: 599-603.
    [26] Riva G (2000) From telehealth to e-health: Internet and distributed virtual reality in health care. J CyberPsychology Behavior 3(6): 989-998.
    [27] Riva G (2002) The emergence of e-health: using virtual reality and the internet for providing advanced healthcare services. Int J Healthcare Technolo Management 4 (1/2): 15-40.
    [28] Rocha R, Vassallo J, Soares F, et al. (2009) Digital slides: present status of a tool for consultation, teaching, and quality control in pathology. J Patholo Res Prac 205(11): 735-741.
    [29] Rosenthal A, Mork P, Li MH, et al. (2010) Cloud computing: a new business paradigm for biomedical information sharing. J Biomed Inform 43(2): 342-353.
    [30] Sloot P, Tirado-Ramos A, Altintas I, et al. (2006) From molecule to man: Decision support in individualized E-Health. Computer 39(11): 40-46.
    [31] Van Ginneken AM (2002) The computerized patient record: Balancing effort and benefit. Int J Med Inform 65: 97-119.
    [32] Weinstein RS, Graham AR, Richte LC, et al. (2009) Overview of telepathology, virtual microscopy, and whole slide imaging: prospects for the future. J Human Patholo 40(8): 1057-1069.
    [33] Bautista PA, Yagi Y (2010) Improving the visualization and detection of tissue folds in whole slide images through color enhancement. J Pathol Inform 1:25.
    [34] Bloom K (2009) The virtual consultative network. Presented at Pathol. Vis. Conf., San Diego.
    [35] Bruch LA, De Young BR, Kreiter CD, et al. (2009) Competency assessment of residents in surgical pathology using virtual microscopy. Hum Pathol 40: 1122–1128.
    [36] Chargari C, Comperat E, Magne N, et al. (2011) Prostate needle biopsy examination by means of virtual microscopy. Pathol Res Pract 207: 366-369.
    [37] Costello SS, Johnston DJ, Dervan PA, et al. (2003) Development and evaluation of the virtual pathology slide: a new tool in telepathology. J Med Int Res 5: e11.
    [38] Daniel C, Rojo MG, Klossa J, et al. (2011) Standardizing the use of whole slide images in digital pathology. Comput Med Imaging Graph 35: 496-505.
    [39] Dee FR (2009) Virtual microscopy in pathology education. Hum Pathol 40: 1112-1121.
    [40] Della Mea V, Bortolotti N, Beltrami CA (2009) eSlide suite: an open source software system for whole slide imaging. J Clin Pathol 62: 749-751.
    [41] DICOM Stand. Comm. Work. Group 26 Pathol. Specimen Module and Revised Pathology SOP Classes, suppl. 122., 2008. Available from ftp://medical.nema.org/medical/dicom/final/sup122_ft2.pdf
    [42] DICOM Stand. Comm. Work. Group 26 Pathol. Whole Slide Microscopic Image IOD and SOP Classes, suppl. 145, 2008. Available from ftp://medical.nema.org/medical/dicom/supps/sup145_ft.pdf.
    [43] Difranco MD, O'Hurley G, Kay EW, et al. (2011) Ensemble based system for whole-slide prostate cancer probability mapping using color texture features. Comput Med Imaging Graph 35: 629-645.
    [44] Evans AJ, Chetty R, Clarke BA, et al. (2009) Primary frozen section diagnosis by robotic microscopy and virtual slide telepathology: the University Health Network experience. Hum Pathol 40: 1070-1081.
    [45] Evans AJ (2011) Re: Barriers and facilitators to adoption of soft copy interpretation from the user perspective: lessons learned from filmless radiology for slideless pathology. J Pathol Inform 2: 8.
    [46] Evans AJ, Sinard JH, Fatheree LA, et al. (2011) Validating whole slide imaging for diagnostic purposes in pathology: recommendations of the College of American Pathologists (CAP) Pathology and Laboratory Quality Center. Anal. Cell Pathol 34: 169-203.
    [47] Evered A, Dudding N (2010) Accuracy and perceptions of virtual microscopy compared with glass slide microscopy in cervical cytology. Cytopatholo 22: 82-87.
    [48] Fallon MA, Wilbur DC, Prasad M (2010) Ovarian frozen section diagnosis: Use of whole-slide imaging shows excellent correlation between virtual slide and original interpretations in a large series of cases. Arch Pathol Lab Med 134: 1020-1023.
    [49] Fine JL, Grzybicki DM, Silowash R, et al. (2008) Evaluation of whole slide image immunohistochemistry interpretation in challenging prostate needle biopsies. Hum Pathol 39: 564–572.
    [50] Fonyad L, Gerely L, Cserneky M, et al. (2010) Shifting gears higher—digital slides in graduate education—4 years experience at Semmelweis University. Diagn Pathol 5: 73.
    [51] Fung KM, Hassell LA, Talbert ML, et al. (2012) Whole slide images and digital media in pathology education, testing, and practice: the Oklahoma experience. Anal Cell Pathol 35: 37-40.
    [52] Gabril MY, Yousef GM (2010) Informatics for practicing anatomical pathologists: marking a new era in pathology practice. Mod Pathol 23: 349-358.
    [53] Gilbertson JR, Ho J, Anthony L, et al. (2006) Primary histologic diagnosis using automated whole slide imaging: a validation study. BMC Clin Pathol 6: 4.
    [54] Graham AR, Bhattacharyya AK, Scott KM, et al. (2009) Virtual slide telepathology for an academic teaching hospital surgical pathology quality assurance program. Hum Pathol 40: 1129-1136.
    [55] Hassell LA, Fung KM, Chaser B (2011) Digital slides and ACGME resident competencies in anatomic pathology: an altered paradigm for acquisition and assessment. J Pathol Inform 2: 27.
    [56] Hede K (2008) Breast cancer testing scandal shines spotlight on black box of clinical laboratory testing. J Natl Cancer Inst 100: 836-844.
    [57] Hedvat CV (2008) Digital microscopy: past, present, and future. Arch Pathol Lab Med 134: 1666-1670.
    [58] Ho J, Parwani AV, Jukic DM, et al. (2006) Use of whole slide imaging in surgical pathology quality assurance: design and pilot validation studies. Hum Pathol 37: 322-331.
    [59] Ho J (2009) Validating digital slides for clinical use: When is image quality good enough? Presented at Pathol Vis Conf, San Diego.
    [60] Huisman A, Looijen A, van den Brink SM, et al. (2010) Creation of a fully digital pathology slide archive by high-volume tissue slide scanning. Hum Pathol 41: 751-757.
    [61] Jukic DM, Drogowski LM, Martina J, et al. (2011) Clinical examination and validation of primary diagnosis in anatomic pathology using whole slide digital images. Arch Pathol Lab Med 135: 372-378.
    [62] Isaacs M, Lennerz JK, Yates S, et al. (2011) Implementation of whole slide imaging in surgical pathology: a value added approach. J Pathol Inform 2: 39.
    [63] Kalinski T, Zwonitzer R, Sel S, et al. (2008) Virtual3Dmicroscopy usingmultiplane whole slide images in diagnostic pathology. Am J Clin Pathol 130: 259-264.
    [64] Krupinski EA, Tillack AA, Richter L, et al. (2006) Eye-movement study and human performance using telepathology virtual slides: implications for medical education and differences with experience. Hum Pathol 37: 1543-1556.
    [65] Krupinski EA (2009) Virtual slide telepathology workstation of the future: lessons learned from teleradiology. Hum Pathol 40: 1100-1111.
    [66] Li L, Dangott BJ, Parwani AV (2010) Development and use of a genitourinary pathology digital teaching set for trainee education. J Pathol Inform 1: 2.
    [67] Lopez AM, Graham AR, Barker GP, et al. (2009) Virtual slide telepathology enables an innovative telehealth rapid breast care clinic. Hum Pathol 40: 1082-1091.
    [68] Manion E, Cohen MB, Weydert J (2008) Mandatory second opinion in surgical pathology referral material: clinical consequences of major disagreements. Am J Surg Pathol 32: 732-737.
    [69] Massone C, Brunasso AM, Campbell TM, et al. (2008) State of the art of teledermatopathology. Am J Dermatopathol 30: 446-450.
    [70] Marchevsky AM, Khurana R, Thomas P, et al. (2006) The use of virtual microscopy for proficiency testing in gynecologic cytopathology: a feasibility study using ScanScope. Arch Pathol Lab Med 130: 349-355.
    [71] McClintock DS, Lee RE, Gilbertson JR (2012) Using computerized workflow simulations to assess the feasibility of whole slide imaging full adoption in a high-volume histology laboratory. Anal Cell Pathol 35: 57-64. doi: 10.1155/2012/726526
    [72] Merk M, Knuechel R, Perez-Bouza A (2010) Web-based virtual microscopy at the RWTH Aachen University: didactic concept, methods and analysis of acceptance by the students. Ann Anat 192: 383-387.
    [73] Mooney E, Hood AF, Lampros J, et al. (2011) Comparative diagnostic accuracy in virtual dermatopathology. Skin Res Technol 17: 251-255.
    [74] Nielsen PS, Lindebjerg J, Rasmussen J, et al. (2010). Virtual microscopy: an evaluation of its validity and diagnostic performance in routine histologic diagnosis of skin tumors. Hum Pathol 41: 1770-1776.
    [75] Pantanowitz L (2010) Digital images and the future of digital pathology. J Pathol Inform 1: 15.
    [76] Patterson ES, Rayo M, Gill C, et al. (2011) Barriers and facilitators to adoption of soft copy interpretation from the user perspective: lessons learned from filmless radiology for slideless pathology. J Pathol Inform 2: 9.
    [77] Paulsen FP, Eichhorn M, Brauer L (2010) Virtual microscopy: the future of teaching histology in the medical curriculum? Ann Anat 192: 378-382.
    [78] Raab SS, Nakhleh RE, Ruby SG (2005) Patient safety in anatomic pathology: measuring discrepancy frequencies and causes. Arch Pathol Lab Med 129: 459-466.
    [79] Raab SS, Grzybicki DM, Mahood LK, et al. (2008) Effectiveness of random and focused review in detecting surgical pathology error. Am J Clin Pathol 130: 905-912.
    [80] Ramey JP, Fung K, Hassell LA (2011) Validation of pathologist use of whole slide images for remote frozen section evaluation. Mod Pathol 24: 436 (Abstr.).
    [81] Ramey JP, Fung K, Hassell LA (2011) Use of mobile high resolution device for remote frozen section evaluation of whole slide images. Mod Pathol 24: 341 (Abstr.).
    [82] Reyes C, Ikpatt F, Nadiji M, et al. (2011) Is virtual microscopy ready for the prime time? A comparison with conventional microscopy. Mod Pathol 24: 454 (Abstr.).
    [83] Rodriguez-Urrego PA, Cronin AM, Al-Ahmadie HA, et al. (2011) Interobserver and intraobserver reproducibility in digital and routine microscopic assessment of prostate needle biopsies. Hum Pathol 42: 68-74.
    [84] Sawai T, Uzuki M, Kamataki A, et al. (2010) The state of telepathology in Japan. J Pathol Inform 1: 13. doi: 10.4103/2153-3539.68327
    [85] Schrader T, Niepage S, Leuthold T, et al. (2006) The diagnostic path, a useful visualisation tool in virtual microscopy. Diagn Pathol 1: 40.
    [86] Siegel EL, Kolonder RM (1999) Filmless Radiology. New York: Springer.
    [87] Spinosa JC (2008) Minute by minute, digital a boon to tumor board. Coll Am Pathol Today Arch.: Oct.
    [88] Stewart J, Miyazaki K, Bevans-Wilkins K, et al. (2007) Virtual microscopy for cytology proficiency testing: Are we there yet? Cancer 111: 203-209.
    [89] Stratman C (2009) Digital pathology in the clinical histology lab: a time and motion study. Presented at Pathol Vis Conf, San Diego.
    [90] Thrall M, Pantanowitz L, Khalbuss W (2011) Telecytology: clinical applications, current challenges, and future benefits. J Pathol Inform 2: 51.
    [91] Thorstenson S (2009) From the conventional microscope to the digital slide scanner in routine diagnostic histopathology. Presented at Pathol. Vis. Conf., San Diego.
    [92] Tsuchihashi Y, Takamatsu T, Hashimoto Y, et al. (2008) Use of virtual slide system for quick frozen intra-operative telepathology diagnosis in Kyoto, Japan. Diagn Pathol 3(Suppl. 1): 6.
    [93] van den Tweel JG, Bosman FT (2011) The use of virtual slides in the EUROPALS examination. Diagn Pathol 6(Suppl. 1): 23.
    [94] Velez N, Jukic D, Ho J (2008) Evaluation of two whole-slide imaging applications in dermatopathology. Hum Pathol 39: 1341-1349.
    [95] Weinstein RS, Graham AR, Richter LC, et al. (2009) Overview of telepathology, virtual microscopy, and whole slide imaging: prospects for the future. Hum Pathol 40: 1057-1069.
    [96] Wilbur DC, Madi K, Colvin RB, et al. (2009) Whole-slide imaging digital pathology as a platform for teleconsultation: a pilot study using paired subspecialist correlations. Arch Pathol Lab Med 133: 1949-1953.
    [97] Williams S, Henricks WH, Becich MJ, et al. (2010) Telepathology for patient care: What am I getting myself into? Adv Anat Pathol 17: 130-149.
    [98] Woolgar JA, Ferlito A, Devaney KO, et al. (2011) How trustworthy is a diagnosis in head and neck surgical pathology? A consideration of diagnostic discrepancies (errors). Eur Arch Otorhinolaryngol 268: 643-651. doi: 10.1007/s00405-011-1526-x
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