Scopus İndeksli Yayınlar Koleksiyonu

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Author: search.filters.author.Baleanu, D.Subject: search.filters.subject.Fractional Calculus

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  • Article
    Citation - Scopus: 2
    Non-Integer Variable Order Dynamic Equations on Time Scales Involving Caputo-Fabrizio Type Differential Operator
    (Eudoxus Press, LLC, 2018) Baleanu, D.; Baleanu, Dumitru; Nategh, M.; Matematik
    This work deals with the conecept of a Caputo-Fabrizio type non-integer variable order differential opertor on time scales that involves a non-singular kernel. A measure theoretic discussion on the limit cases for the order of differentiation is presented. Then, corresponding to the fractional derivative, we discuss on an integral for constant and variable orders. Beside the obtaining solutions to some dynamic problems on time scales involving the proposed derivative, a fractional folrmulation for the viscoelastic oscillation problem is studied and its conversion into a third order dynamic equation is presented. © 2018 by Eudoxus Press, LLC. All rights reserved.
  • Article
    Recent Advances in Special Functions, Fractional Operators and Their Real World Applications
    (Cambridge Scientific Publishers, 2021) Singh, J.; Baleanu, Dumitru; Baleanu, D.; Kumar, D.; Hammouch, Z.; Matematik
    This special issue ”Recent Advances in Special Functions, Fractional Operators and their Real World Applications” of the journal Mathematics in Engineering, Science and Aerospace (MESA) is mainly collection of the research articles presented in 3rd International Conference on Mathematical Mod-elling, Applied Analysis and Computation (ICMMAAC-20) organized by the Department of Mathe-matics, JECRC University, Jaipur, India during August 7-9, 2020. This collection of articles is mainly concerned to address a wide range of special functions, operators of fractional order and their uses in mathematical modelling and computation of distinct problems of physical sciences, chemical sci-ences, biological sciences, engineering sciences, social science and economics. In the this special is-sue, expository and original research papers associated with the new trends and challenges in special functions and fractional order calculus and as well as their uses in real world problems are collected. Some are invited papers. © CSP - Cambridge, UK; I&S - Florida, USA, 2021
  • Article
    Citation - Scopus: 3
    On Some Impulsive Fractional Neutral Differential Systems With Nonlocal Condition Through Fractional Operators
    (Cambridge Scientific Publishers, 2017) Anuradha, A.; Baleanu, Dumitru; Baleanu, D.; Suganya, S.; Arjunan, M.M.; Matematik
    According to semigroup theories, fractional calculus, Banach contraction principle and Schaefer's fixed point theorem, this paper is fundamentally involved with the existence of mild solutions for an impulsive fractional neutral differential systems (abbreviated, IFNDS) with nonlocal conditions (abbreviated, NLC) in Banach space X. At last, an illustration is also presented to exhibit the use of our theoretical results. © CSP - Cambridge, UK; I & S - Florida, USA, 2017.
  • Article
    Citation - WoS: 20
    Citation - Scopus: 22
    Analytical Treatment of System of Abel Integral Equations by Homotopy Analysis Method
    (Editura Acad Romane, 2014) Jafarian, A.; Baleanu, Dumitru; Ghaderi, P.; Golmankhaneh, Alireza K.; Baleanu, D.; Matematik
    Abel equation has important applications in describing the least time for an object which is sliding on surface without friction in uniform gravity, and the classical theory of elasticity of materials is modeled by a system of Abel integral equations. In this manuscript, the homotopy analysis method is presented for obtaining analytical solutions of a system of Abel integral equations as fractional equations. The applied method has lessened the size of calculation and improved the accuracy of solution in the case of the singular Abel integral equation. The illustrated examples and numerical results have proved the assertion.
  • Article
    Citation - WoS: 1
    Citation - Scopus: 1
    Fractional Systems With Multi-Parameters Fractional Derivatives
    (Springer, 2025) Muslih, S.I.; Agrawal, O.P.; Baleanu, D.
    Recently, a generalization of fractional variational formulations in terms of multiparameter fractional derivatives was introduced by Agrawal and Muslih. This treatment can be used to obtain the Lagrangian and Hamiltonian equations of motion. In this paper, we also extend our work to introduce the generalization of the formulation for constrained mechanical systems containing multi-parameter fractional derivatives. Three examples for regular and constrained fractional systems are analyzed. © The Author(s) 2025.
  • Article
    Citation - Scopus: 14
    The Effect of Deformation of Special Relativity by Conformable Derivative
    (Sociedad Mexicana de Fisica, 2022) Al-Masaeed, M.; Rabei, E.M.; Baleanu, D.; Al-Jamel, A.
    In this paper, the deformation of special relativity within the frame of conformable derivative is formulated. Within this context, the two postulates of the theory are re-stated. Then, the addition of velocity laws are derived and used to verify the constancy of the speed of light. The invariance principle of the laws of physics is demonstrated for some typical illustrative examples, namely, the conformable wave equation, the conformable Schrodinger equation, the conformable Klein-Gordon equation, and conformable Dirac equation. The current formalism may be applicable when using special relativity in a nonlinear or dispersive medium. © 2022, Revista Mexicana de Fisica. All Rights Reserved.
  • Article
    Citation - WoS: 9
    Citation - Scopus: 11
    Correcting 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: 4
    Mittag-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: 8
    Computational 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: 6
    Artificial 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.