On the Solutions of the Higher Order Fractional Differential Equations of Riesz Space Derivative with Anti-Periodic Boundary Conditions

On the Solutions of the Higher Order Fractional Differential Equations of Riesz Space Derivative with Anti-Periodic Boundary Conditions

We present existence and uniqueness results for a class of higher order anti-periodic fractional boundary value problems with Riesz space derivative which is two-sided fractional operator. The obtained results are established by applying some fixed point theorems. Various numerical examples are given to illustrate the obtained results.

___

  • [1] R. L. Magin, Fractional calculus in bioengineering, Begell House Publisher, Inc., Connecticut, 2006.
  • [2] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999.
  • [3] G. M. Zaslavsky, Hamiltonian chaos and fractional dynamics, Oxford University Press, Oxford 2005.
  • [4] K. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley, NY, 1993.
  • [5] S. Shen, F. Liu, V. Anh, Numerical approximations and solution techniques for the Caputo-time Riesz–Caputo fractional advection–diffusion equation, Numer. Algorithms, 56 (2011), 383-403.
  • [6] D. R. Smart, Fixed point Theorems, Cambridge University Press, Cambridge 1980.
  • [7] C. Pinto, A. R. M. Carvalho, New findings on the dynamics of HIV and TB coinfection models, Appl. Math. Comput., 242(2014), 36-46.
  • [8] R. Gorenflo, F. Mainardi, D. Moretti, G. Pagnini, P. Paradisi, Discrete random walk models for space–time fractional diffusion, Chem. Phys., 284 (2012), 521-541.
  • [9] L. Guo, L. Liu, W. Ye, Uniqueness of iterative positive solutions for the singular fractional differential equations with integral boundary conditions, Comput. Math. Appl., 59(8) (2010), 2601–2609.
  • [10] J. W. Negele, E. Vogt (Eds.), Volume 23 of advances in the physics of particles and nuclei, Advances in nuclear physics, Springer Science and Business Media, 1996.
  • [11] R. Agarwal, D, O’Regan, S. Stanek, Positive solutions for Dirichlet problems of singular nonlinear fractional differential equations, J. Math. Anal. Appl., 371 (2010), 57-68.
  • [12] A. Babakhani, V. Gejji, Existence of positive solutions of nonlinear fractional differential equations, J. Math. Anal.Appl., 278 (2003), 434-442.
  • [13] C. Celik, M. Duman, Crank–Nicolson method for the fractional diffusion equation with the Riesz fractional derivative, J. Comput. Phys., 231 (2012), 1743–1750.
  • [14] F. Chen, A. Chen, X. Wu, Anti-periodic boundary value problems with Riesz-Caputo derivative, Adv. Dif. Eq., 2019 (2019), 119.
  • [15] M.Darwish, S. Ntouyas, On initial and boundary value problems for fractional order mixed type functional differential inclusion, Comput. Math. Appl., 59 (2010), 1253–1265.
  • [16] H. Sun, S. Hu, Y. Chen, W. Chen, Z. Yu, A dynamic–order fractional dynamic system Chinese Phys. Lett., 30 (2013), Article 046601 pp.4.
  • [17] S.Toprakseven, Existence and uniqueness of solutions to anti-periodic Riezs-Caputo impulsive fractional boundary value problems, Tbil. Math. J. 14(1) (2021), 71-82.
  • [18] S. Toprakseven, Existence and uniqueness of solutions to Riesz-Caputo impulsive fractional boundary value problems, Journal of Interdisciplinary Mathematics, (2021), DOI: 10.1080/09720502.2020.1826629.
  • [19] S. Toprakseven, Positive solutions for two-point conformable fractional differential equations by monotone iterative scheme, TWMS J. App. Eng. Math., 11(1) (2021), 289-301.
  • [20] S. Toprakseven, Solvability of fractional boundary value problems for a combined caputo derivative, Konuralp J. Math., 9(1) (2021), 119-126.
  • [21] F. Usta, M. Z. Sarıkaya, The analytical solution of Van der Pol and Lienard differential equations within conformable fractional operator by retarded integral inequalities, Demo. Math., 52(1) (2019), 204–212.
  • [22] B. Ahmad, Existence of solutions for fractional differential equations of order q 2 (2;3] with anti-periodic boundary conditions, J. Appl. Math. Comput., 34 (2010), 385-391.
  • [23] Y. Chen, J.J. Nieto, D. O’Regan, Anti-periodic solutions for evolution equations associated with maximal monotone mappings, Appl. Math. Lett., 24 (3) (2011), 302-307.
  • [24] Y. Cui, Uniqueness of solution for boundary value problems for fractional differential equations, Appl. Math. Lett., 51 (2016), 48–54.
  • [25] C. Gu, G. Wu, Positive solutions of fractional differential equations with the Riesz space derivative, Appl. Math. Lett., 95 (2019), 59–64.
  • [26] A. Kilbas, H. H. Srivastava, J. Trujillo, Theory and applications of fractional differential equations, vol. 204, North–Holland mathematics studies, Elsevier, Amsterdam, 2006.
  • [27] I. Podlubny, Geometric and physical interpretation of fractional integration and fractional differentiation, Fract. Calculus Appl. Anal., 5 (2002), 367–386.
  • [28] M. Z. Sarıkaya, F. Usta, On comparison theorems for conformable fractional differential equations, Int. J. Anal. App., 12(2) (2016), 207-214.
  • [29] Sierociuk, T. Skovranek, M. Macias, I. Podlubny, I. Petras, A. Dzielinski, P. Ziubinski, Diffusion process modeling by using fractional–order models, Appl. Math. Comput., 15 (257) (2015), 2-11.
  • [30] S. Toprakseven, The existence and uniqueness of initial-boundary value problems of the fractional Caputo-Fabrizio differential equations, Uni. J. Math. App., 2 (2) (2019), 100-106.
  • [31] S. Toprakseven, The existence of positive solutions and a Lyapunov-type inequality for boundary value problems of the fractioanl Caputo-Fabrizio differential equations, Sigma J. Eng. Nat. Sci., 37 (4) (2019), 1125-1133.
  • [32] F. Usta, Numerical analysis of fractional Volterra integral equations via Bernstein approximation method, J. Comput. Appl. Math., 384(2021), 113198, DOI: 10.1016/j.cam.2020.113198.
  • [33] F. Usta, Fractional type Poisson equations by radial basis functions Kansa approach, J. Ineq. Special Func., 7(4) (2016), 143-149.
  • [34] F. Usta, Numerical solution of fractional elliptic PDE’s by the collocation method, Applications and Applied Mathematics: An International Journal, 12(1) (2017), 470- 478.
  • [35] F. Usta, H. Budak, M. Z. Sarıkaya, Yang-Laplace transform method Volterra and Abel’s integro-differential equations of fractional order, Int. J. Nonlinear Anal. App., 9(2) (2018), 203-214, DOI: 10.22075/ijnaa.2018.13630.1709.
  • [36] F. Usta, A mesh free technique of numerical solution of newly defined conformable differential equations, Konuralp J. Math., 4(2) (2016), 149-157.
  • [37] M. Yavuz, T. A. Sulaiman, F. Usta, H. Bulut, [Analysis and numerical computations of the fractional regularized long wave equation with damping term, Math. Meth. Appl. Sci., In Press, DOI: 10.1002/mma.6343.
  • [38] G. Wu, D. Baleanu et al., Lattice fractional diffusion equation in terms of a Riesz–Caputo difference, Physics A., 438 (2015), 335-339.
  • [39] X. Zhang, L. Liu, Y. Wu, The uniqueness of positive solution for a fractional order model of turbulent flow in a porous medium, Appl. Math. Lett., 37 (2014), 26–33.
  • [40] A. R. Aftabizadeh, Y. K. Huang, N. H. Pavel, Nonlinear third-order differential equations with anti-periodic boundary conditions and some optimal control problems, J. Math. Anal. Appl., 192 (1995), 266-293.
  • [41] M. Yavuz, N. O¨ zdemir, H.M. Baskonus, Solutions of partial differential equations using the fractional operator involving Mittag-Leffler kernel, Eur. Phys. J. Plus, 133(6) (2018), 1-11.
  • [42] M. Yavuz, Characterizations of two different fractional operators without singular kernel, Math. Model. Nat. Phenom, 14(3) (2019), 302.
  • [43] M. Yavuz, N. O¨ zdemir Comparing the new fractional derivative operators involving exponential and Mittag-Leffler kernel, Discrete Contin. Dyn. Syst. Ser. S, 13(3) (2020), 995-1006.
  • [44] A. Yokus¸, Construction of different types of traveling wave solutions of the relativistic wave equation associated with the Schr¨odinger equation, Math. Model. Numer. Simul. Appl., 1(1) (2021), 24-31.
  • [45] P. Kumar, V.S. Erturk, Dynamics of cholera disease by using two recent fractional numerical methods, Math. Model. Numer. Simul. Appl., 1(2) (2021), 102-111.