Modified Results by Using New Generalized Definition of Fractional Derivative without Singular Kernel by applying new generalized five parameter Mittag-Leffler function
Modified Results by Using New Generalized Definition of Fractional Derivative
Abstract
In a series of papers [43-46] reviewed all results and generalized the existing results by modifications.In this article, a new approach of the derivative of arbitrary order (FD) with the kernel of the smooth type that gains different depictions for the temporal and spatial variables has been given. It first applies to the time variables and hence it is fit to us transform of Laplace type (LT). Secondly, a definition is linked to the spatial type variables, by a global derivative of arbitrary order (FD), for which we will apply the transform of Fourier type (FT). The courtesy for this new methodology with a kernel of regular type was native from the vision that there is a period of global systems, which can designate the material heterogeneities and the fluctuations of unlike scales, which cannot be well described by traditional local theories or by arbitrary order models with the kernel of singular type.In this endeavour we are introducing a new generalized five parameter Mittag-Lefflerfunction which is used in the definition of fractional derivative.
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References
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[30] M. Caputo, Modeling social and economic cycles, in Alternative Public Economics, Forte F., Navarra P., Mudambi R. (Eds.), Elgar, Cheltenham, 2014.
[31] G. Jumarie, New stochastic fractional models of the Malthusian growth, thePoissonian birth process and optimal management of populations, Math.Comput. Modell. 44, 231-254 (2006).
[32] G. Iaffaldano, M. Caputo and S. Martino, Experimental and theoretical memorydiffusion of water in sand, Hydrology and Earth System Sciences, 10, 93–100(2006).
[33] M. El Shaed, A Fractional Calculus Model of Semilunar Heart Valve Vibrations, 2003 International Mathematica Symposium, London, 2003.
[34] R. L. Magin, Fractional Calculus in Bioengineering, Begell House Inc. Publishers, 2006.
[35] I.M. Gel’fand and G.E. Shilov, Generalized Functions 1–5, Academic Press,1968.
[36] E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag,1963.
[37] G.Z. Voyiadjis and R. K. Abu Al-Rub, Nonlocal gradient-dependent thermodynamics for modeling scale-dependent plasticity, Int. J. Multiscale Comput. 5, 295-323 (2007).
[38] G. Z. Voyiadjis and A. H. Almasri, Variable material length scale associatedwith Nano indentation experiments, J. Eng. Mech. ASCE 135(3), 139-148(2009).
[39] M. Fabrizio, Fractional rheological models for thermomechanical systems.Dissipation and free energies, Fract. Calc. Appl. Anal. 17, 206-223 (2014).
[40] M. Fabrizio and A. Morro, Dissipativity and irreversibility of electromagnetic systems, Math. Mod. Meth. Appl. Sci. 10, 217-246 (2000).
[41] J.H.Williamson, Lebesgue Integration, Holt, Rinehart and Winston Inc. New York,1962.
[42] M. Caputo, Mean fractional-order-derivatives differential equations and filters,Annali Universit`a di Ferrara, Sez. VII, ScienzeMatematiche, 41, 73–83(1995).
[43] M. Caputo and M. Fabrizio, A New Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl. 1, No. 2, 73-85 (2015).
[44] L. Mourya, M. Sharma and R. Mishra: New Results by Using Advanced Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl. 9, No. 4, 589-595 (2023).
[45] M. Al-Refai and D. Baleanu, On an extension of the operator with Mittag – Lefflerkernel, Fractals, Vol. 30, No. 5, 2240129-(1-7) (2022).
[46] A. Atangana, and D. Baleanu, New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model, the Journal Thermal Science (2016)
[2] R. Caponetto, Fractional Order Systems (Modelling and Control Applications), World Scientific, 2010.
[3] M. Caputo, Elasticit`a e Dissipazione, Zanichelli, Bologna, 1965.
[4] K. Diethelm, The Analysis of Fractional Differential Equations, an Application Oriented, Exposition Using Differential Operators of Caputo type, Lecture Notes in Mathematics nr. 2004, Springer, Heidelbereg, 2010.
[5] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing Co, Singapore, 2003.
[6] Z. Jiao, Y. Chen and I. Podlubny, Distributed order Dynamic Systems, Modeling, Analysis and Simulation, Springer, 201).
[7] A.A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications ofFractional Differential Equations, North-Holland Mathematical Studies,Amsterdam, 2006.
[8] V. Kiryakova, Generalized Fractional Calculus and Applications, PitmanResearch Notes in Mathematics N◦ 301, Longman, Harlow, 1994.
[9] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: AnIntroduction to Mathematical Models, World Scientific Publishing, 2010.
[10] A.C. McBride, Fractional Calculus and Integral Transforms of GeneralizedFunctions, Pitman Res. Notes in Mathem., nr. 31, Pitman, London, 1979.
[11] K.S. Miller and B. Ross, An Introduction to Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
[12] I. Petras, Fractional Calculus Non-Linear Systems, Modelling, Analysis andSimulation, Springer, 2011.
[13] S. G. Samko, A. A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Sc. Publ, 1993.
[14] I. Podlubny, Fractional Differential Equations, Academic Press, New York,2009.
[15] J. Sabatier, O.P. Agrawal and J.A. Tenorio Machado, Advances in FractionalCalculus, Springer, 2009.
[16] D. F. M. Torres and A. B. Malinowski, Introduction to the Fractional Calculus of Variations, Imperial College Press, 2012.
[17] L. Ying and Y. Chen, Fractional Order Motion Controls, Wiley, 2012.
[18] A.A. Kilbas and S.A. Marzan, Nonlinear Differential Equations with Caputo Fractional Derivative in the Space of Continuously Differentiable Functions, Differ. Equ. 41, 84-95 (2005).
[19] E. Heinsalu, M. Patriarca, I. Goychuk, G. Schmid and P. H¨anggi , Fractional Fokker Plank dynamics: Numerical algorithm and simulation, Phys. Rev. E 73, 046133 (2006).
[20] Y. Luchko and R. Gorenflo, an operational method for solving fractional differential equations with the Caputo derivative, Acta Math. Vietnamica 24,207-233 (1999).
[21] N. Laskin Fractional Schr¨odinger equation, Phys. Rev. E 66, 056108 (2002).
[22] M. Naber Time fractional Schr¨odinger equation, J. Math. Phys. 45(8), 3339-3352 (2004).
[23] D. Baleanu, A. K. Golmankhaneh and A. K. Golmankhaneh, The dual actionand fractional multi time Hamilton equations, Int. J. Theor. Phys. 48, 2558-2569 (2009). c 2015 NSP Natural Sciences Publishing Cor. Progr. Fract. Differ.Appl. 1, No. 2, 73-85 (2015) / www.naturalspublishing.com/Journals.asp 85
[24] P. Zavada, Relativistic wave equations with fractional derivatives and pseudodifferential operators, J. Appl. Math. 2(4), 163-197 (2002).
[25] D. Baleanu, A.K. Golmankhaneh, A.K. Golmankhaneh and R.R. Nigmatullin,Newtonian law with memory, Nonlinear Dyn. 60, 81-86 (2010).
[26] M. Caputo and M. Fabrizio M., Damage and fatigue described by a fractionalderivative model, in press, J. Comput. Phys., (2014)
[27] M. Caputo, C. Cametti and V. Ruggiero, Time and spatial concentration profileinside a membrane by means of a memory formalism, Physica A 387, 2010–2018 (2007).
[28] F. Cesarone, M. Caputo and C. Cametti, Memory formalism in the passive diffusion across a biological membrane, J. Membrane Sci. 250, 79-84 (2004).
[29] M. Caputo and C. Cametti, Memory diffusion in two cases of biological interest,J. Theor. Biol. 254,697-703 (2008).
[30] M. Caputo, Modeling social and economic cycles, in Alternative Public Economics, Forte F., Navarra P., Mudambi R. (Eds.), Elgar, Cheltenham, 2014.
[31] G. Jumarie, New stochastic fractional models of the Malthusian growth, thePoissonian birth process and optimal management of populations, Math.Comput. Modell. 44, 231-254 (2006).
[32] G. Iaffaldano, M. Caputo and S. Martino, Experimental and theoretical memorydiffusion of water in sand, Hydrology and Earth System Sciences, 10, 93–100(2006).
[33] M. El Shaed, A Fractional Calculus Model of Semilunar Heart Valve Vibrations, 2003 International Mathematica Symposium, London, 2003.
[34] R. L. Magin, Fractional Calculus in Bioengineering, Begell House Inc. Publishers, 2006.
[35] I.M. Gel’fand and G.E. Shilov, Generalized Functions 1–5, Academic Press,1968.
[36] E. Hewitt and K. Stromberg, Real and Abstract Analysis, Springer-Verlag,1963.
[37] G.Z. Voyiadjis and R. K. Abu Al-Rub, Nonlocal gradient-dependent thermodynamics for modeling scale-dependent plasticity, Int. J. Multiscale Comput. 5, 295-323 (2007).
[38] G. Z. Voyiadjis and A. H. Almasri, Variable material length scale associatedwith Nano indentation experiments, J. Eng. Mech. ASCE 135(3), 139-148(2009).
[39] M. Fabrizio, Fractional rheological models for thermomechanical systems.Dissipation and free energies, Fract. Calc. Appl. Anal. 17, 206-223 (2014).
[40] M. Fabrizio and A. Morro, Dissipativity and irreversibility of electromagnetic systems, Math. Mod. Meth. Appl. Sci. 10, 217-246 (2000).
[41] J.H.Williamson, Lebesgue Integration, Holt, Rinehart and Winston Inc. New York,1962.
[42] M. Caputo, Mean fractional-order-derivatives differential equations and filters,Annali Universit`a di Ferrara, Sez. VII, ScienzeMatematiche, 41, 73–83(1995).
[43] M. Caputo and M. Fabrizio, A New Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl. 1, No. 2, 73-85 (2015).
[44] L. Mourya, M. Sharma and R. Mishra: New Results by Using Advanced Definition of Fractional Derivative without Singular Kernel, Progr. Fract. Differ. Appl. 9, No. 4, 589-595 (2023).
[45] M. Al-Refai and D. Baleanu, On an extension of the operator with Mittag – Lefflerkernel, Fractals, Vol. 30, No. 5, 2240129-(1-7) (2022).
[46] A. Atangana, and D. Baleanu, New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model, the Journal Thermal Science (2016)
Published
2025-12-19
Section
Advanced Computational Methods for Fractional Calculus
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