Research Article
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Year 2019, , 100 - 106, 28.06.2019
https://doi.org/10.32323/ujma.549942

Abstract

References

  • [1] KS. Miller, B. Ross, (Eds.), An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley, NY 1993.
  • [2] I. Podlubny, Fractional Differential Equations, Academic Press, New York 1999.
  • [3] A.A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Amsterdam, 2006.
  • [4] J. Sabatier, P. Lanusse, P. Melchior, A. Oustaloup, Fractional Order Differentiation and Robust Control Design, Springer, 2015.
  • [5] F. Mainardi, Fractional Calculus and Waves in linear Viscoelasticity: an Introduction to Mathematical Models, World Scientific, 2010.
  • [6] V.V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Springer, 2013.
  • [7] R. J. Greechie, S. P. Gudder, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, 111 Elsevier, 1974.
  • [8] A. N. Kochubei, Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential equations, to Methods of Their Solution and some of their Applications, 198 Academic Press 1998.
  • [9] F. Usta, Fractional Type Poisson Equations by Radial Basis Functions Kansa Approach, J. Inequal. Spec. Funct., (7)4, (2016),143-149.
  • [10] M. Z. Sarikaya and F. Usta, On Comparison Theorems for Conformable Fractional Differential Equations, Int. J. Anal. Appl., (12)2, (2016), 207-214.
  • [11] F. Usta, A mesh-free technique of numerical solution of newly defined conformable differential equations, Konuralp J. Math., (4)2,(2016) 149-157.
  • [12] F. Usta and M. Z. Sarıkaya, The analytical solution of Van der Pol and Lienard differential equations within conformable fractional operator by retarded integral inequalities, Demonstr. Math., 52(1), (2019), 204–212.
  • [13] Fuat Usta, Computational solution of Katugampola conformable fractional differential equations via RBF collocation method, AIP Conference Proceedings, 1833(1) (2017), 200461-200464., Doi: http://dx.doi.org/10.1063/1.4981694.
  • [14] M. Caputo, M. Fabrizio, A New Definition of Fractional Derivative without Singular Kerne, Progr. Fract. Differ. Appl., 1:1 (2015), 1-13.
  • [15] J. Losada, J.J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2016), 87-92.
  • [16] M. Caputo, M. Fabrizio, Applications of new time and spatial frac-tional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2 (2016), 1-11.
  • [17] Xiao-Jun Yang, H.M.Srivastava, J.A.Machado Tenreiro, A new fractional derivative without singular kernel, Thermal Science, (2015), doi:10.2298/TSCI151224222Y.
  • [18] Xiao-Jun Yang, H.M.Srivastava, J.A.Machado Tenreiro, Modeling diffusive transport with a fractional derivative without singular kernel, Physic A, 447 (2016), 467–481.
  • [19] J.F.G. Aguilar, H. Y. Martinez, C.C. Ramon, I.C. Ordunia, R.F. E. Jimenez, V.H.O. Peregrino, Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel, Entropy, 17 (2015), 6289-6303.
  • [20] M.Yavuz, N. Özdemir, European Vanilla Option Pricing Model of Fractional Order without Singular Kernel , Fractal and Fractional, (2)1, 3 (2018).
  • [21] F. Evirgen, M.Yavuz, An alternative approach for nonlinear optimization problem with Caputo-Fabrizio derivative , In ITM Web of Conferences , 22, EDP Sciences, (2018),p. 01009.
  • [22] M.Yavuz, N. Özdemir, Comparing the new fractional derivative operators involving exponential and Mittag Leffler kernel, Discrete Contin. Dyn. Syst., 13(3), (2019),1098-1107.
  • [23] D. Zhao, M. Luo, Representations of acting processes and memory effects: General fractional derivative and its application to theory of heat conduction with finite wave speeds, Appl. Math. Comput., 346 (2019), 531-544.

The Existence and Uniqueness of Initial-Boundary Value Problems of the Fractional Caputo-Fabrizio Differential Equations

Year 2019, , 100 - 106, 28.06.2019
https://doi.org/10.32323/ujma.549942

Abstract

In this paper, the existence and uniqueness problem of the initial and boundary value problems of the linear fractional Caputo-Fabrizio differential equation of order $\sigma \in (1,2]$ have been investigated. By using the Laplace transform of the fractional derivative, the fractional differential equations turn into the classical differential equation of integer order. Also, the existence and uniqueness of nonlinear boundary value problem of the fractional Caputo-Fabrizio differential equation has been proved. An application to mass spring damper system for this new fractional derivative has also been presented in details.

References

  • [1] KS. Miller, B. Ross, (Eds.), An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley, NY 1993.
  • [2] I. Podlubny, Fractional Differential Equations, Academic Press, New York 1999.
  • [3] A.A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies, Amsterdam, 2006.
  • [4] J. Sabatier, P. Lanusse, P. Melchior, A. Oustaloup, Fractional Order Differentiation and Robust Control Design, Springer, 2015.
  • [5] F. Mainardi, Fractional Calculus and Waves in linear Viscoelasticity: an Introduction to Mathematical Models, World Scientific, 2010.
  • [6] V.V. Uchaikin, Fractional Derivatives for Physicists and Engineers, Springer, 2013.
  • [7] R. J. Greechie, S. P. Gudder, The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, 111 Elsevier, 1974.
  • [8] A. N. Kochubei, Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential equations, to Methods of Their Solution and some of their Applications, 198 Academic Press 1998.
  • [9] F. Usta, Fractional Type Poisson Equations by Radial Basis Functions Kansa Approach, J. Inequal. Spec. Funct., (7)4, (2016),143-149.
  • [10] M. Z. Sarikaya and F. Usta, On Comparison Theorems for Conformable Fractional Differential Equations, Int. J. Anal. Appl., (12)2, (2016), 207-214.
  • [11] F. Usta, A mesh-free technique of numerical solution of newly defined conformable differential equations, Konuralp J. Math., (4)2,(2016) 149-157.
  • [12] F. Usta and M. Z. Sarıkaya, The analytical solution of Van der Pol and Lienard differential equations within conformable fractional operator by retarded integral inequalities, Demonstr. Math., 52(1), (2019), 204–212.
  • [13] Fuat Usta, Computational solution of Katugampola conformable fractional differential equations via RBF collocation method, AIP Conference Proceedings, 1833(1) (2017), 200461-200464., Doi: http://dx.doi.org/10.1063/1.4981694.
  • [14] M. Caputo, M. Fabrizio, A New Definition of Fractional Derivative without Singular Kerne, Progr. Fract. Differ. Appl., 1:1 (2015), 1-13.
  • [15] J. Losada, J.J. Nieto, Properties of a new fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2016), 87-92.
  • [16] M. Caputo, M. Fabrizio, Applications of new time and spatial frac-tional derivatives with exponential kernels, Progr. Fract. Differ. Appl., 2 (2016), 1-11.
  • [17] Xiao-Jun Yang, H.M.Srivastava, J.A.Machado Tenreiro, A new fractional derivative without singular kernel, Thermal Science, (2015), doi:10.2298/TSCI151224222Y.
  • [18] Xiao-Jun Yang, H.M.Srivastava, J.A.Machado Tenreiro, Modeling diffusive transport with a fractional derivative without singular kernel, Physic A, 447 (2016), 467–481.
  • [19] J.F.G. Aguilar, H. Y. Martinez, C.C. Ramon, I.C. Ordunia, R.F. E. Jimenez, V.H.O. Peregrino, Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel, Entropy, 17 (2015), 6289-6303.
  • [20] M.Yavuz, N. Özdemir, European Vanilla Option Pricing Model of Fractional Order without Singular Kernel , Fractal and Fractional, (2)1, 3 (2018).
  • [21] F. Evirgen, M.Yavuz, An alternative approach for nonlinear optimization problem with Caputo-Fabrizio derivative , In ITM Web of Conferences , 22, EDP Sciences, (2018),p. 01009.
  • [22] M.Yavuz, N. Özdemir, Comparing the new fractional derivative operators involving exponential and Mittag Leffler kernel, Discrete Contin. Dyn. Syst., 13(3), (2019),1098-1107.
  • [23] D. Zhao, M. Luo, Representations of acting processes and memory effects: General fractional derivative and its application to theory of heat conduction with finite wave speeds, Appl. Math. Comput., 346 (2019), 531-544.
There are 23 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Şuayip Toprakseven 0000-0003-3901-9641

Publication Date June 28, 2019
Submission Date April 5, 2019
Acceptance Date April 24, 2019
Published in Issue Year 2019

Cite

APA Toprakseven, Ş. (2019). The Existence and Uniqueness of Initial-Boundary Value Problems of the Fractional Caputo-Fabrizio Differential Equations. Universal Journal of Mathematics and Applications, 2(2), 100-106. https://doi.org/10.32323/ujma.549942
AMA Toprakseven Ş. The Existence and Uniqueness of Initial-Boundary Value Problems of the Fractional Caputo-Fabrizio Differential Equations. Univ. J. Math. Appl. June 2019;2(2):100-106. doi:10.32323/ujma.549942
Chicago Toprakseven, Şuayip. “The Existence and Uniqueness of Initial-Boundary Value Problems of the Fractional Caputo-Fabrizio Differential Equations”. Universal Journal of Mathematics and Applications 2, no. 2 (June 2019): 100-106. https://doi.org/10.32323/ujma.549942.
EndNote Toprakseven Ş (June 1, 2019) The Existence and Uniqueness of Initial-Boundary Value Problems of the Fractional Caputo-Fabrizio Differential Equations. Universal Journal of Mathematics and Applications 2 2 100–106.
IEEE Ş. Toprakseven, “The Existence and Uniqueness of Initial-Boundary Value Problems of the Fractional Caputo-Fabrizio Differential Equations”, Univ. J. Math. Appl., vol. 2, no. 2, pp. 100–106, 2019, doi: 10.32323/ujma.549942.
ISNAD Toprakseven, Şuayip. “The Existence and Uniqueness of Initial-Boundary Value Problems of the Fractional Caputo-Fabrizio Differential Equations”. Universal Journal of Mathematics and Applications 2/2 (June 2019), 100-106. https://doi.org/10.32323/ujma.549942.
JAMA Toprakseven Ş. The Existence and Uniqueness of Initial-Boundary Value Problems of the Fractional Caputo-Fabrizio Differential Equations. Univ. J. Math. Appl. 2019;2:100–106.
MLA Toprakseven, Şuayip. “The Existence and Uniqueness of Initial-Boundary Value Problems of the Fractional Caputo-Fabrizio Differential Equations”. Universal Journal of Mathematics and Applications, vol. 2, no. 2, 2019, pp. 100-6, doi:10.32323/ujma.549942.
Vancouver Toprakseven Ş. The Existence and Uniqueness of Initial-Boundary Value Problems of the Fractional Caputo-Fabrizio Differential Equations. Univ. J. Math. Appl. 2019;2(2):100-6.

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