Inscrição na biblioteca: Guest
Critical Reviews™ in Biomedical Engineering

Publicou 6 edições por ano

ISSN Imprimir: 0278-940X

ISSN On-line: 1943-619X

SJR: 0.262 SNIP: 0.372 CiteScore™:: 2.2 H-Index: 56

Indexed in

INITIALIZATION, CONCEPTUALIZATION, AND APPLICATION IN THE GENERALIZED (FRACTIONAL) CALCULUS

Volume 35, Edição 6, 2007, pp. 447-553
DOI: 10.1615/CritRevBiomedEng.v35.i6.10
Get accessGet access

RESUMO

This paper provides a formalized basis for initialization in the fractional calculus. The intent is to make the fractional calculus readily accessible to engineering and the sciences. A modified set of definitions for the fractional calculus is provided which formally include the effects of initialization. Conceptualizations of fractional derivatives and integrals are shown. Physical examples of the basic elements from electronics are presented along with examples from dynamics, material science, viscoelasticity, filtering, instrumentation, and electrochemistry to indicate the broad application of the theory and to demonstrate the use of the mathematics. The fundamental criteria for a generalized calculus established by Ross (1974) are shown to hold for the generalized fractional calculus under appropriate conditions. A new generalized form for the Laplace transform of the generalized differintegral is derived. The concept of a variable structure (order) differintegral is presented along with initial efforts toward meaningful definitions.

CITADO POR
  1. Chen Chang-Ming, Liu F., Anh V., Turner I., Numerical Schemes with High Spatial Accuracy for a Variable-Order Anomalous Subdiffusion Equation, SIAM Journal on Scientific Computing, 32, 4, 2010. Crossref

  2. Popović Jovan K., Pilipović Stevan, Atanacković Teodor M., Two compartmental fractional derivative model with fractional derivatives of different order, Communications in Nonlinear Science and Numerical Simulation, 18, 9, 2013. Crossref

  3. Podlubny Igor, Skovranek Tomas, Vinagre Jara Blas M., Petras Ivo, Verbitsky Viktor, Chen YangQuan, Matrix approach to discrete fractional calculus III: non-equidistant grids, variable step length and distributed orders, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 371, 1990, 2013. Crossref

  4. Yin Deshun, Wang Yixin, Li Yanqing, Cheng Chen, Variable-order fractional mean square displacement function with evolution of diffusibility, Physica A: Statistical Mechanics and its Applications, 392, 19, 2013. Crossref

  5. Popović Jovan K., Spasić Dragan T., Tošić Jela, Kolarović Jovanka L., Malti Rachid, Mitić Igor M., Pilipović Stevan, Atanacković Teodor M., Fractional model for pharmacokinetics of high dose methotrexate in children with acute lymphoblastic leukaemia, Communications in Nonlinear Science and Numerical Simulation, 22, 1-3, 2015. Crossref

  6. Shen S., Liu F., Liu Q., Anh V., Numerical simulation of anomalous infiltration in porous media, Numerical Algorithms, 68, 3, 2015. Crossref

  7. Zhuang Jiao , YangQuan Chen , Yisheng Zhong , Stability condition of linear time-invariant distributed-order dynamic systems, 2012 American Control Conference (ACC), 2012. Crossref

  8. Li Yan, Chen Yang Quan, Theory and implementation of weighted distributed order integrator, Proceedings of 2012 IEEE/ASME 8th IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications, 2012. Crossref

  9. dos Santos José Paulo Carvalho, Cardoso Lislaine Cristina, Monteiro Evandro, Lemes Nelson H. T., A Fractional-Order Epidemic Model for Bovine Babesiosis Disease and Tick Populations, Abstract and Applied Analysis, 2015, 2015. Crossref

  10. Ewing Robert L., Abdel-Aty-Zohdy Hoda S., Hollenbeck Michael C., Stevens Kenneth S., Fractional-order signal processing using a polymer-electrolyte transistor, 2008 51st Midwest Symposium on Circuits and Systems, 2008. Crossref

  11. Liu Jianping, Li Xia, Wu Limeng, An Operational Matrix of Fractional Differentiation of the Second Kind of Chebyshev Polynomial for Solving Multiterm Variable Order Fractional Differential Equation, Mathematical Problems in Engineering, 2016, 2016. Crossref

  12. Chen Yi-ming, Liu Li-qing, Liu Dayan, Boutat Driss, Numerical study of a class of variable order nonlinear fractional differential equation in terms of Bernstein polynomials, Ain Shams Engineering Journal, 9, 4, 2018. Crossref

  13. Zhang Li, Yu Chenglong, Liu Tao, Control of finite-time anti-synchronization for variable-order fractional chaotic systems with unknown parameters, Nonlinear Dynamics, 86, 3, 2016. Crossref

  14. Liu Zhengguang, Cheng Aijie, Li Xiaoli, A second order Crank–Nicolson scheme for fractional Cattaneo equation based on new fractional derivative, Applied Mathematics and Computation, 311, 2017. Crossref

  15. Zafar Zain Ul Abadin, Rehan Kashif, Mushtaq M, RETRACTED ARTICLE: Fractional-order scheme for bovine babesiosis disease and tick populations, Advances in Difference Equations, 2017, 1, 2017. Crossref

  16. Liu Zhengguang, Cheng Aijie, Li Xiaoli, A second-order finite difference scheme for quasilinear time fractional parabolic equation based on new fractional derivative, International Journal of Computer Mathematics, 95, 2, 2018. Crossref

  17. Belkhatir Zehor, N’Doye Ibrahima, Laleg-Kirati Taous Meriem, Estimation Methods for Fractional-Order Systems, in Fractional Order Systems, 2018. Crossref

  18. Salehi Farideh, Saeedi Habibollah, Mohseni Moghadam Mahmood, A Hahn computational operational method for variable order fractional mobile–immobile advection–dispersion equation, Mathematical Sciences, 12, 2, 2018. Crossref

  19. Wang Shengnan, Wang Zhendong, Li Gongsheng, Wang Yingmei, A Simultaneous Inversion Problem for the Variable-Order Time Fractional Differential Equation with Variable Coefficient, Mathematical Problems in Engineering, 2019, 2019. Crossref

  20. Beltempo Angela, Bonelli Alessio, Bursi Oreste S., Zingales Massimiliano, A numerical integration approach for fractional‐order viscoelastic analysis of hereditary‐aging structures, International Journal for Numerical Methods in Engineering, 121, 6, 2020. Crossref

  21. Soradi-Zeid Samaneh, Jahanshahi Hadi, Yousefpour Amin, Bekiros Stelios, King algorithm: A novel optimization approach based on variable-order fractional calculus with application in chaotic financial systems, Chaos, Solitons & Fractals, 132, 2020. Crossref

  22. Zeid Samaneh Soradi, Approximation methods for solving fractional equations, Chaos, Solitons & Fractals, 125, 2019. Crossref

  23. Wang Li-ye, Zhou Feng-xi, Analysis of elastic-viscoplastic creep model based on variable-order differential operator, Applied Mathematical Modelling, 81, 2020. Crossref

  24. Ul Abadin Zafar Zain, Ali Nigar, Zaman Gul, Thounthong Phatiphat, Tunç Cemil, Analysis and numerical simulations of fractional order Vallis system, Alexandria Engineering Journal, 59, 4, 2020. Crossref

  25. Patnaik Sansit, Semperlotti Fabio, Application of variable- and distributed-order fractional operators to the dynamic analysis of nonlinear oscillators, Nonlinear Dynamics, 100, 1, 2020. Crossref

  26. Patnaik Sansit, Semperlotti Fabio, Variable-order particle dynamics: formulation and application to the simulation of edge dislocations, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 378, 2172, 2020. Crossref

  27. Das Souvik, Sen Siddhartha, Effect of past History on Transient Response of a Closed Loop System with Fractional PI Controller, 2018 15th IEEE India Council International Conference (INDICON), 2018. Crossref

  28. Du Ruilian, Alikhanov Anatoly A., Sun Zhi-Zhong, Temporal second order difference schemes for the multi-dimensional variable-order time fractional sub-diffusion equations, Computers & Mathematics with Applications, 79, 10, 2020. Crossref

  29. Patnaik Sansit, Hollkamp John P., Semperlotti Fabio, Applications of variable-order fractional operators: a review, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 476, 2234, 2020. Crossref

  30. Wang Lei, Chen Yi-Ming, Shifted-Chebyshev-polynomial-based numerical algorithm for fractional order polymer visco-elastic rotating beam, Chaos, Solitons & Fractals, 132, 2020. Crossref

  31. Patnaik Sansit, Semperlotti Fabio, Modeling Contacts and Hysteretic Behavior in Discrete Systems Via Variable-Order Fractional Operators, Journal of Computational and Nonlinear Dynamics, 15, 9, 2020. Crossref

  32. dos Santos Maike A.F., Nobre Fernando D., Curado Evaldo M.F., Monitoring Lévy-process crossovers, Communications in Nonlinear Science and Numerical Simulation, 92, 2021. Crossref

  33. Shirkhorshidi S. M. R., Othman W. A. M., Awang M. A. Omar, Rostamy D., Shirkhorshidi A. S., The analytical interface coupling of arbitrary-order fractional nonlinear hyperbolic scalar conservation laws, Advances in Difference Equations, 2020, 1, 2020. Crossref

  34. Kupper Martin, Funk Christopher, Eckert Marius, Hohmann Soren, Fractional Extended and Unscented Kalman Filtering for State of Charge Estimation of Lithium-Ion Batteries, 2018 Annual American Control Conference (ACC), 2018. Crossref

  35. Habibirad Ali, Hesameddini Esmail, Heydari Mohammad Hossein, Roohi Reza, An efficient meshless method based on the moving Kriging interpolation for two‐dimensional variable‐order time fractional mobile/immobile advection‐diffusion model, Mathematical Methods in the Applied Sciences, 44, 4, 2021. Crossref

  36. Stark Oliver, Karg Philipp, Hohmann Soren, Iterative Method for Online Fractional Order and Parameter Identification, 2020 59th IEEE Conference on Decision and Control (CDC), 2020. Crossref

  37. Rezapour Shahram, Souid Mohammed Said, Etemad Sina, Bouazza Zoubida, Ntouyas Sotiris K., Asawasamrit Suphawat, Tariboon Jessada, Mawhin’s Continuation Technique for a Nonlinear BVP of Variable Order at Resonance via Piecewise Constant Functions, Fractal and Fractional, 5, 4, 2021. Crossref

  38. Ding Wei, Patnaik Sansit, Sidhardh Sai, Semperlotti Fabio, Applications of Distributed-Order Fractional Operators: A Review, Entropy, 23, 1, 2021. Crossref

  39. Ahmed Hoda F, Melad Marina B, Efficient method for solving variable-order pantograph models, Pramana, 95, 4, 2021. Crossref

  40. El-Gazar E.F., Zahra W.K., Hassan Hamdy, Rabia Sherif I., Fractional modeling for enhancing the thermal performance of conventional solar still using hybrid nanofluid: Energy and exergy analysis, Desalination, 503, 2021. Crossref

  41. Rezapour Shahram, Souid Mohammed Said, Bouazza Zoubida, Hussain Azhar, Etemad Sina, Scapellato Andrea, On the Fractional Variable Order Thermostat Model: Existence Theory on Cones via Piece-Wise Constant Functions, Journal of Function Spaces, 2022, 2022. Crossref

  42. Sun HongGuang, Chang Ailian, Zhang Yong, Chen Wen, A Review on Variable-Order Fractional Differential Equations: Mathematical Foundations, Physical Models, Numerical Methods and Applications, Fractional Calculus and Applied Analysis, 22, 1, 2019. Crossref

  43. Lenka Bichitra Kumar, Bora Swaroop Nandan, New global asymptotic stability conditions for a class of nonlinear time-varying fractional systems, European Journal of Control, 63, 2022. Crossref

  44. Ahmed Hoda F., Melad Marina B., A new numerical strategy for solving nonlinear singular Emden-Fowler delay differential models with variable order, Mathematical Sciences, 2022. Crossref

  45. Zerari Amina, Odibat Zaid, Shawagfeh Nabil, Numerical schemes for variable exponent fractional‐type integral equations, Mathematical Methods in the Applied Sciences, 2022. Crossref

  46. Liu Ri, Wang Zhe, Zhang Xuefeng, Ren Jianxu, Gui Qinglong, Robust Control for Variable-Order Fractional Interval Systems Subject to Actuator Saturation, Fractal and Fractional, 6, 3, 2022. Crossref

  47. Stark Oliver, Strehle Felix, Hohmann Soren, Convergence Analysis of the Short-Memory Principle for Fractional Models, 2021 European Control Conference (ECC), 2021. Crossref

  48. Belkhatir Zehor, Alhazmi Fahd, Bahloul Mohamed A., Laleg-Kirati Taous-Meriem, Parameter Sensitivity and Experimental Validation for Fractional-Order Dynamical Modeling of Neurovascular Coupling, IEEE Open Journal of Engineering in Medicine and Biology, 3, 2022. Crossref

  49. Sweilam Nasser H., Assiri Taghreed A., Hasan Muner M. Abou, Optimal control problem of variable-order delay system of advertising procedure: Numerical treatment, Discrete & Continuous Dynamical Systems - S, 15, 5, 2022. Crossref

  50. Duc Tran Minh, Van Hoa Ngo, Stabilization of impulsive fractional-order dynamic systems involving the Caputo fractional derivative of variable-order via a linear feedback controller, Chaos, Solitons & Fractals, 153, 2021. Crossref

  51. Zidan Ahmed M., Khan Adnan, Shah Rasool, Alaoui Mohammed Kbiri, Weera Wajaree, Evaluation of time-fractional Fisher's equations with the help of analytical methods, AIMS Mathematics, 7, 10, 2022. Crossref

  52. Bahloul Mohamed A., Belkhatir Zehor, Kirati Taous-Meriem Laleg, Initialization of fractional order systems for the joint estimation of parameters and fractional differentiation orders, 2022 American Control Conference (ACC), 2022. Crossref

  53. Wu Chunxue, Yuan Jian, Shi Bao, Stability of initialization response of fractional oscillators, Journal of Vibroengineering, 18, 6, 2016. Crossref

Portal Digital Begell Biblioteca digital da Begell eBooks Diários Referências e Anais Coleções de pesquisa Políticas de preços e assinaturas Begell House Contato Language English 中文 Русский Português German French Spain