Correcting the initialization of models with fractional derivatives via history-dependent conditions

Du, Maolin; Wang, Zaihua

doi:10.1007/s10409-015-0469-7

Cited by 23 publications

(8 citation statements)

References 43 publications

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“…A third possibility is to try to adjust the decay function from the data actually measured by separating it from the response to the post-initial current. Du et al mentioned the difficulties of obtaining the initialization function in practical situations [15], although an algorithm to accomplish this task has been recently proposed [36]. In fact, this is not an easy task, as we show in the example below.…”

Section: Discussionmentioning

confidence: 94%

“…Equation (10) only uses integer derivatives at the initial time, so it is easier to apply in practice. However, the inability of the Caputo definition to provide a satisfactory solution to the initial condition problem has been pointed out [13,15], and we comment on it below.…”

Section: Fractional Derivativesmentioning

confidence: 99%

“…Some other authors also indicated that initial histories rather than initial values at a point should be considered [12][13][14]. The term "aberration" has been introduced in relation to this phenomenon [15] and has been used by other authors [16]. One procedure to deal with this problem is to define an initialized operator, related to the history, to be imposed in the fractional derivative at the initial point [17,18].…”

Section: Introductionmentioning

confidence: 99%

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Constant Phase Element in the Time Domain: The Problem of Initialization

López‐Villanueva

Rodríguez‐Bolívar

2022

Energies

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The constant phase element (CPE) is found in most battery and supercapacitor equivalent circuit models proposed to interpret data in the frequency domain. When these models are used in the time domain, the initial conditions in the fractional differential equations must be correctly imposed. The initial state problem remains controversial and has been analyzed by various authors in the last two decades. This article attempts to clarify this problem by proposing a procedure to prepare the initial state and defining a decay function that reveals the effect of the initial state in several illustrative examples. This decay function depends on the previous history, which is reflected in the time needed to prepare the initial state and on the current profile assumed for this purpose. This effect of the initial state is difficult to separate and can lead to the misinterpretation of the CPE parameter values.

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Section: Discussionmentioning

confidence: 94%

Section: Fractional Derivativesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Constant Phase Element in the Time Domain: The Problem of Initialization

López‐Villanueva

Rodríguez‐Bolívar

2022

Energies

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“…Utilizing the one of most common numerical methods, the L1 scheme [78][79][80][81], the numerical approximation of the FOD of (t) is…”

Section: Memory Trace and Hereditary Traitsmentioning

confidence: 99%

A Fractional Modeling of Tumor–Immune System Interaction Related to Lung Cancer with Real Data

et al. 2021

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In this study, we investigate a new fractional-order mathematical model which considers population dynamics among tumor cells-macrophage cells-active macrophage cells, and host cells involving the Caputo fractional derivative. Firstly, the stability of the positive steady state of the model is studied. Subsequently, the conditions for existence and uniqueness of the solutions are examined. Then, the least squares curve fitting method (LSCFM) which is one of the prominent methods for parameter estimation is used to fit the parameters of the model. It is aimed to fit the relevant parameters with the help of the tumor tissue samples which were collected from the patient with non-small cell lung cancer who had chemotherapy-naive hospitalized at Kayseri Erciyes University hospital in Turkey. A total of 12 parameters in the model are estimated using the data of lung tumor cells of this patient for 14 days. Moreover, the numerical simulations are given by considering the different fractional orders and different parameters for the model. So, it is achieved how the change in $$\alpha $$ α affects the dynamic behavior of the system. In the sequel, to point out the advantages of the fractional-order modeling, the memory trace and hereditary traits are taken into consideration. Finally, the interpretations in terms of biological science are provided in conclusion. We believe that this interdisciplinary study will open new doors for other similar studies and will shed light on the studies to be developed on the use of real data in the mathematical modeling of cancer.

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“…Utilizing the one of most common numerical methods, the L1 scheme [22,23,24,21], the numerical approximation of the FOD of Φ (t) is…”

Section: Memory Trace and Hereditary Traitsmentioning

confidence: 99%

Fractional-order mathematical modelling of cancer cells-cancer stem cells-immune system interaction with chemotherapy

Özköse¹,

Şenel²,

Habbireeh³

2021

mmnsa

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In this paper, we present a mathematical model of stem cells and chemotherapy for cancer treatment, in which the model is represented by fractional order differential equations. Local stability of equilibrium points is discussed. Then, the existence and uniqueness of the solution are studied. In addition, in order to point out the advantages of the fractional order modeling, the memory trace and hereditary traits are taken into consideration. Numerical simulations have been used to investigate how the fractional order derivative and different parameters affect the population dynamics, the graphs have been illustrated according to different values of fractional order α and different parameter values. Moreover, we have examined the effect of chemotherapy on tumor cells and stem cells over time. Furthermore, we concluded that the memory effect occurs as the α decreases from 1 and the chemotherapy drug is quite effective on the populations. We hope that this work will contribute to helping medical scientists take the necessary measures during the screening process and treatment.

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Correcting the initialization of models with fractional derivatives via history-dependent conditions

Cited by 23 publications

References 43 publications

Constant Phase Element in the Time Domain: The Problem of Initialization

Constant Phase Element in the Time Domain: The Problem of Initialization

A Fractional Modeling of Tumor–Immune System Interaction Related to Lung Cancer with Real Data

Fractional-order mathematical modelling of cancer cells-cancer stem cells-immune system interaction with chemotherapy

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