Scopus İndeksli Yayınlar Koleksiyonu
Permanent URI for this collectionhttps://hdl.handle.net/20.500.12416/8651
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Article Citation - WoS: 9Citation - Scopus: 11Correcting Dimensional Mismatch in Fractional Models With Power, Exponential and Proportional Kernel: Application To Electrical Systems(Elsevier, 2022) Correa-Escudero, I. L.; Gomez-Aguilar, J. F.; Lopez-Lopez, M. G.; Alvarado-Martinez, V. M.; Baleanu, D.Fractional calculus is a powerful tool for describing diffusion phenomena, anomalous behaviors, and in general, systems with highly complex dynamics. However, the application of fractional operators for modeling purposes, produces a dimensional problem. In this paper, the fractional models of the RC, RL, RLC electrical circuits, a supercapacitor, a bank of supercapacitors, a LiFePO4 battery and a direct current motor are presented. A correction parameter is included in their formulation in order to preserve dimensionality in the physical equations. The optimal value of this parameter was determined via particle swarm optimization algorithm using numerical simulations and experimental data. Thus, a direct and effective approach for the construction of dimensionally corrected fractional models with power, exponential-decay and constant proportional Caputo hybrid derivative is presented. To show the effectiveness of the procedure, the time-response of the models is compared with experimental data and the modeling error is computed. The numerical solutions of the models were obtained using a numerical method based on the Adams methods.Book Part Citation - Scopus: 4Mittag-Leffler Functions With Heavy-Tailed Distributions' Algorithm Based on Different Biology Datasets To Be Fit for Optimum Mathematical Models' Strategies(Elsevier, 2022) Karaca, Y.; Baleanu, D.Complexity of living organisms owing to their inherent functional properties points toward a systems biology approach due to the fact that structural and topological uncertainties exist along with abrupt transitions characterized by unknown inputs, time-varying parameters and unpredictable observation states. The related uncertain, emergent and evolving qualities of organisms along with their varying quantities and states present in the related complex system need to be identified in biological datasets based on mathematical models in a way that enables the structural identification analysis in a reasonable time frame, the detection of nonlinear dependencies among the many parameters involved and practical analysis for the identification of data at stake. Superstatistics, which is concerned with the study of nonlinear systems, has proven to be a significant tool to examine the dynamic aspects of organisms, substances, particles and other biological elements. Superstatistics is characterized by the superposition of varying statistical models to achieve the desired nonlinearity. The challenge of integrating fractional calculus in cases of complexity requires an effective use of empirical, numerical, experimental and analytical methods to tackle complexity. One of the most noteworthy tools in the fractional calculus context is the Mittag-Leffler (ML) functions. Mittag-Leffler distributions have extensive application domains when dealing with irregular and nonhomogeneous environments for dynamic problems' solutions. These distributions can be used in reliability modeling as an alternative for exponential distribution; and thus, the proposed integrated approach in this study addresses the Mittag-Leffler (ML) function with two parameters (α,β) in order to investigate the dynamics of diseases related to biological elements. Arising in the different solutions of varying complex biological systems, ML function generalizes the exponential function; and to this end, firstly, we applied the ML function with two parameters to biological datasets (cancer cell dataset and diabetes dataset, namely raw datasets) in order to obtain the new datasets (ml_cancer cell dataset and ml_diabetes dataset) with significant attributes for diagnosis, prognosis and classification of diseases. Secondly, heavy-tailed distributions (The Mittag-Leffler distribution, Pareto distribution, Cauchy distribution and Weibull distribution) were applied to the new datasets obtained, and their comparison was made with regard to the performances, by employing the log likelihood value (MLE) and the Akaike Information Criterion (AIC). Fitting algorithm Mittag-Leffler function is based on heavy-tailed distributions. Subsequently, the ML functions that represent the cancer cell and diabetes data were identified so that the two parameters Eα,β(z) yielding the optimum value based on the distributions fit could be found. By finding the most significant attributes with heavy-tailed distributions (The Mittag-Leffler distribution, Pareto distribution, Cauchy distribution and Weibull distribution) based on Mittag-Leffler function with two parameters (α,β) the diagnosis, prognosis and classification of the diseases has been enabled in our study. In this way, through this proposed integrative scheme, optimal strategical means have been obtained for accurate and robust mathematical models' strategies concerning the diagnosis and progress of the diseases. The results obtained by the current study for diseases on biological datasets based on mathematical models demonstrate that the integrative approach with Mittag-Leffler with heavy-tailed distributions algorithm is applicable and fits very well to the related data with the robust parameters' values observed and estimated in transient chaotic and unpredictable settings. The analysis results obtained by the data fitting algorithm scheme proposed have demonstrated its criticality for understanding the dynamics of transmission and prevalence operating in the complex biological and epidemiological systems along the Mittag-Leffler function based on distribution scale, with temporal and spatial attributes, to improve applicability and accuracy constituting optimal mathematical models' strategies. © 2022 Elsevier Inc. All rights reserved.Book Part Citation - Scopus: 8Computational Fractional-Order Calculus and Classical Calculus Ai for Comparative Differentiability Prediction Analyses of Complex-Systems Paradigm(Elsevier, 2022) Baleanu, D.; Karaca, Y.Modern science having embarked on the thorough and accurate interpretation of natural and physical phenomena has proven to provide successful models for the analysis of complex systems and harnessing of control over the various processes therein. Computational complexity, in this regard, comes to the foreground by providing applicable sets of ideas or integrative paradigms to recognize and understand the complex systems' intricate properties. Thus, while making the appropriate, adaptable and evolutive decisions in complex dynamic systems, it is essential to acknowledge different degrees of acceptance of the problems and construct the model it to account for its inherent constraints or limits. In this respect, while hypothesis-driven research has its inherent limitations regarding the investigation of multifactorial and heterogeneous diseases, a data-driven approach enables the examination of the way variables impact one another, which paves the way for the interpretation of dynamic and heterogeneous mechanisms of diseases. Fractional Calculus (FC), in this scope characterized by complexity, provides the applicable means and methods to solve integral, differential and integro-differential equations so FC enables the generalization of integration and differentiation possible in a flexible and consistent manner owing to its capability of reflecting the systems' actual state properties, which exhibit unpredictable variations. The fractional integration and differentiation of fractional-order is capable of providing better characterization of nonstationary and locally self-similar attributes in contrast to constant-order fractional calculus. It becomes possible to model many complex systems by fractional-order derivatives based on fractional calculus so that related syntheses can be realized in a robust and effective way. To this end, our study aims at providing an intermediary facilitating function both for the physicians and individuals by establishing accurate and robust model based on the integration of fractional-order calculus and Artificial Neural Network (ANN) for the diagnostic and differentiability predictive purposes with the diseases which display highly complex properties. The integrative approach we have proposed in this study has a multistage quality the steps of which are stated as follows: first of all, the Caputo fractional-order derivative, one of the fractional-order derivatives, has been used with two-parametric Mittag-Leffler function on the stroke dataset and cancer cell dataset, manifesting biological and neurological attributes. In this way, new fractional models with varying degrees have been established. Mittag-Leffler function, with its distributions of extensive application domains, can address irregular and heterogeneous environments for the solution of dynamic problems; thus, Mittag-Leffler function has been opted for accordingly. Following this application, the new datasets (mlf_stroke dataset and mlf_cancer cell dataset) have been obtained by employing Caputo fractional-order derivative with the two-parametric Mittag-Leffler function (α,β). In addition, classical derivative (calculus) was applied to the raw datasets; and cd_stroke dataset and cd_cancer cell dataset were obtained. Secondly, the performance of the new datasets as obtained from the Caputo fractional derivative with the two-parametric Mittag-Leffler function, the datasets obtained from the classical derivative application and the raw datasets have been compared by using feed forward back propagation (FFBP) algorithm, one of the algorithms of ANN (along with accuracy rate, sensitivity, precision, specificity, F1-score, multiclass classification (MCC), ROC curve). Based on the accuracy rate results obtained from the application with FFBP, the Caputo fractional-order derivative model that is most suitable for the diseases has been generated. The experimental results obtained demonstrate the applicability of the complex-systems-grounded paradigm scheme as proposed through this study, which has no existing counterpart. The integrative multi-stage method based on mathematical-informed framework with comparative differentiability prediction analyses can point toward a new direction in the various areas of applied sciences to address formidable challenges of critical decision making and management of chaotic processes in different complex dynamic systems. © 2022 Elsevier Inc. All rights reserved.Book Part Citation - Scopus: 6Artificial Neural Network Modeling of Systems Biology Datasets Fit Based on Mittag-Leffler Functions With Heavy-Tailed Distributions for Diagnostic and Predictive Precision Medicine(Elsevier, 2022) Baleanu, D.; Karaca, Y.Being the most complex physical system in the universe, life, at all scales requires the understanding of the massive complexity including its origin, structure, dynamic, adaptation and organization. Both the number of substructures and interacting pathways of each substructure along with other ones and neurons determine the degree of complexity. Neural networks, as descriptive models, in systems biology setting, provide the means to gather, store and use experiential knowledge; and are designed in a way to emulate different operations of the human brain. One of the major ongoing challenges of integrating fractional calculus in cases of complexity requires an effective use of empirical, numerical, experimental and analytical methods to tackle complexity. In that regard, Artificial Neural Networks (ANNs), including a family of nonlinear computational methods, are employed to handle experimental data in differing domains owing to their capability of tackling complex computations so that their progressive application can solve practical problems. One of the other most noteworthy tools which arises in the fractional calculus context is the Mittag-Leffler (ML) functions. Mittag-Leffler distributions have extensive application domains when dealing with irregular and nonhomogeneous environments for dynamic problems' solutions. They can be used in reliability modeling as an alternative for exponential distribution, particularly this provides upper hand for diagnostic and predictive purposes in precision medicine through novel algorithmic models. To address this, the proposed method in the current study has obtained the generation of optimum model strategies for different biology datasets along with Mittag-Leffler functions with heavy-tailed distributions (see Part I). Within this framework, the proposed integrated approach in this study investigates the dynamics of diseases related to biological elements; and arising in the different solutions of varying complex biological systems, ML function generalizes the exponential function. To this end, firstly, the two-parametric Mittag-Leffler function was applied to biological datasets (cancer cell dataset and diabetes dataset, namely raw datasets), namely cancer cell and diabetes in order to obtain the new datasets (ml_cancer cell dataset and ml_diabetes dataset). Heavy-tailed distributions (The Mittag-Leffler distribution, Pareto distribution, Cauchy distribution and Weibull distribution) were applied to the new datasets obtained with their comparison performed in relation to the performances (by employing the log likelihood value and the Akaike Information Criterion (AIC)). ML functions that represent the cancer cell and diabetes data were identified so that the two parameters Eα,β(z) yielding the optimum value based on the distributions fit could be found. Secondly, one of the ANN algorithms, namely Multi-layer Perceptron (MLP) (along with the accuracy, sensitivity, precision, specificity, F1-score, multi-class classification (MCC), ROC curve), was applied for the diagnosis and prediction of the disease course regarding the optimum ML functions that represent the cancer cell and diabetes datasets obtained and the performances of the ML functions with heavy-tailed distributions were compared with ANN training functions (Levenberg-Marquart, Bayes Regularization and BFGS-Quasi-Newton) accordingly. The integrative modeling scheme proposed herein, which has not been addressed through this sort of approach before, is concerned with the applicability and reliability of the solutions obtained by Mittag-Leffler functions with heavy-tailed distributions. The results obtained by the current study for diseases related to biological datasets based on mathematical models demonstrate that the integrative approach with Mittag-Leffler function and ANN applications is applicable and fits very well to the related data with the robust parameters' values observed and estimated. When the fact that complex biological phenomena involve various intrinsic and extrinsic aspects is considered, it becomes a major difficulty to make identifications and recognition on the basis of a single type of data merely. Thus, the proposed approach of our study corroborates its applicability for diagnostic and predictive purposes in precision medicine through the novel algorithmic model, which plays a significant role in the effective and timely management of unpredictable phenomena in dynamic and nonlinear complex situations. © 2022 Elsevier Inc. All rights reserved.Article Citation - WoS: 29Citation - Scopus: 33A New Fractional Derivative Operator With Generalized Cardinal Sine Kernel: Numerical Simulation(Elsevier, 2023) Baleanu, Dumitru; Odibat, ZaidIn this paper, we proposed a new fractional derivative operator in which the generalized cardinal sine function is used as a non-singular analytic kernel. In addition, we provided the corresponding fractional integral operator. We expressed the new fractional derivative and integral operators as sums in terms of the Riemann-Liouville fractional integral operator. Next, we introduced an efficient extension of the new fractional operator that includes integrable singular kernel to overcome the initialization problem for related differential equations. We also proposed a numerical approach for the numerical simulation of IVPs incorporating the proposed extended fractional derivatives. The proposed fractional operators, the developed relations and the presented numerical method are expected to be employed in the field of fractional calculus.(c) 2023 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.Article Citation - WoS: 38Citation - Scopus: 41Sir Epidemic Model of Childhood Diseases Through Fractional Operators With Mittag-Leffler and Exponential Kernels(Elsevier, 2021) Chakraverty, Snehashish; Baleanu, Dumitru; Jena, Rajarama MohanVaccination programs for infants have significantly affected childhood morbidity and mortality. The primary goal of health administrators is to protect children against diseases that can be prevented by vaccination. In this manuscript, we have applied the homotopy perturbation Elzaki transform method to obtain the solutions of the epidemic model of childhood diseases involving time-fractional order Atangana-Baleanu and Caputo-Fabrizio derivatives. The present method is the combination of the classical homotopy perturbation method and the Elzaki transform. Although Elzaki transform is an effective method for solving fractional differential equations, this method sometimes fails to handle nonlinear terms from the fractional differential equations. These difficulties may be overcome by coupling this transform with that of HPM. This method offers a rapidly convergent series solutions. Validation and usefulness of the technique are incorporated with new fractional-order derivatives with exponential decay law and with general Mittag-Leffler law. Obtained results are compared with the established solution defined in the Caputo sense. Further, a comparative study among Caputo, Atangana-Baleanu, and Caputo-Fabrizio derivatives is discussed. (C) 2020 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.Article Citation - WoS: 26Citation - Scopus: 32On a More General Fractional Integration by Parts Formulae and Applications(Elsevier, 2019) Gomez-Aguilar, J. F.; Jarad, Fahd; Abdeljawad, Thabet; Atangana, AbdonThe integration by part comes from the product rule of classical differentiation and integration. The concept was adapted in fractional differential and integration and has several applications in control theory. However, the formulation in fractional calculus is the classical integral of a fractional derivative of a product of a fractional derivative of a given function f and a function g. We argue that, this formulation could be done using only fractional operators: thus, we develop fractional integration by parts for fractional integrals, Riemann-Liouville, Liouville-Caputo, Caputo-Fabrizio and Atangana-Baleanu fractional derivatives. We allow the left and right fractional integrals of order alpha > 0 to act on the integrated terms instead of the usual integral and then make use of the fractional type Leibniz rules to formulate the integration by parts by means of new generalized type fractional operators with binomial coefficients defined for analytic functions. In the case alpha = 1, our formulae of fractional integration by parts results in previously obtained integration by parts in fractional calculus. The two disciplines or branches of mathematics are built differently, while classical differentiation is built with the concept of rate of change of a given function, a fractional differential operator is a convolution. (C) 2019 Elsevier B.V. All rights reserved.Article Citation - WoS: 8Citation - Scopus: 8A New Type of Equation of Motion and Numerical Method for a Harmonic Oscillator With Left and Right Fractional Derivatives(Elsevier, 2020) Baleanu, Dumitru; Ullah, Malik ZakaThe aim of this research is to propose a new fractional Euler-Lagrange equation for a harmonic oscillator. The theoretical analysis is given in order to derive the equation of motion in a fractional framework. The new equation has a complicated structure involving the left and right fractional derivatives of Caputo-Fabrizio type, so a new numerical method is developed in order to solve the above-mentioned equation effectively. As a result, we can see different asymptotic behaviors according to the flexibility provided by the use of the fractional calculus approach, a fact which may be invisible when we use the classical Lagrangian technique. This capability helps us to better understand the complex dynamics associated with natural phenomena.Article Citation - WoS: 148Citation - Scopus: 178A New Comparative Study on the General Fractional Model of Covid-19 With Isolation and Quarantine Effects(Elsevier, 2022) Abadi, M. Hassan; Jajarmi, A.; Vahid, K. Zarghami; Nieto, J. J.; Baleanu, D.; Hassan Abadi, M.; Zarghami Vahid, K.A generalized version of fractional models is introduced for the COVID-19 pandemic, including the effects of isolation and quarantine. First, the general structure of fractional derivatives and integrals is discussed; then the generalized fractional model is defined from which the stability results are derived. Meanwhile, a set of real clinical observations from China is considered to determine the parameters and compute the basic reproduction number, i.e., R-0 approximate to 6.6361. Additionally, an efficient numerical technique is applied to simulate the new model and provide the associated numerical results. Based on these simulations, some figures and tables are presented, and the data of reported cases from China are compared with the numerical findings in both classical and fractional frameworks. Our comparative study indicates that a particular case of general fractional formula provides a better fit to the real data compared to the other classical and fractional models. There are also some other key parameters to be examined that show the health of society when they come to eliminate the disease. (C) 2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University.Article Citation - WoS: 94Citation - Scopus: 111A New and General Fractional Lagrangian Approach: A Capacitor Microphone Case(Elsevier, 2021) Baleanu, D.; Vahid, K. Zarghami; Pirouz, H. Mohammadi; Asad, J. H.; Jajarmi, A.; Mohammadi Pirouz, H.; Zarghami Vahid, K.In this study, a new and general fractional formulation is presented to investigate the complex behaviors of a capacitor microphone dynamical system. Initially, for both displacement and electrical charge, the classical Euler-Lagrange equations are constructed by using the classical Lagrangian approach. Expanding this classical scheme in a general fractional framework provides the new fractional Euler-Lagrange equations in which non-integer order derivatives involve a general function as their kernel. Applying an appropriate matrix approximation technique changes the latter fractional formulation into a nonlinear algebraic system. Finally, the derived system is solved numerically with a discussion on its dynamical behaviors. According to the obtained results, various features of the capacitor microphone under study are discovered due to the flexibility in choosing the kernel, unlike the previous mathematical formalism.