Research Article
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Year 2021, Volume: 4 Issue: 4, 171 - 179, 27.12.2021

Şuayip Toprakseven

https://doi.org/10.33434/cams.1016464
Cited By: 2

Abstract

References

  • [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.

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

Year 2021, Volume: 4 Issue: 4, 171 - 179, 27.12.2021

Şuayip Toprakseven

https://doi.org/10.33434/cams.1016464
Cited By: 2

Abstract

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.

Keywords

existence fractional boundary value problems anti-periodic boundary conditions

References

  • [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.
There are 45 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Şuayip Toprakseven

Submission Date October 30, 2021
Acceptance Date December 16, 2021
Publication Date December 27, 2021
Published in Issue Year 2021 Volume: 4 Issue: 4

Cite

APA Toprakseven, Ş. (2021). On the Solutions of the Higher Order Fractional Differential Equations of Riesz Space Derivative with Anti-Periodic Boundary Conditions. Communications in Advanced Mathematical Sciences, 4(4), 171-179. https://doi.org/10.33434/cams.1016464
AMA Toprakseven Ş. On the Solutions of the Higher Order Fractional Differential Equations of Riesz Space Derivative with Anti-Periodic Boundary Conditions. Communications in Advanced Mathematical Sciences. December 2021;4(4):171-179. doi:10.33434/cams.1016464
Chicago Toprakseven, Şuayip. “On the Solutions of the Higher Order Fractional Differential Equations of Riesz Space Derivative With Anti-Periodic Boundary Conditions”. Communications in Advanced Mathematical Sciences 4, no. 4 (December 2021): 171-79. https://doi.org/10.33434/cams.1016464.
EndNote Toprakseven Ş (December 1, 2021) On the Solutions of the Higher Order Fractional Differential Equations of Riesz Space Derivative with Anti-Periodic Boundary Conditions. Communications in Advanced Mathematical Sciences 4 4 171–179.
IEEE Ş. Toprakseven, “On the Solutions of the Higher Order Fractional Differential Equations of Riesz Space Derivative with Anti-Periodic Boundary Conditions”, Communications in Advanced Mathematical Sciences, vol. 4, no. 4, pp. 171–179, 2021, doi: 10.33434/cams.1016464.
ISNAD Toprakseven, Şuayip. “On the Solutions of the Higher Order Fractional Differential Equations of Riesz Space Derivative With Anti-Periodic Boundary Conditions”. Communications in Advanced Mathematical Sciences 4/4 (December2021), 171-179. https://doi.org/10.33434/cams.1016464.
JAMA Toprakseven Ş. On the Solutions of the Higher Order Fractional Differential Equations of Riesz Space Derivative with Anti-Periodic Boundary Conditions. Communications in Advanced Mathematical Sciences. 2021;4:171–179.
MLA Toprakseven, Şuayip. “On the Solutions of the Higher Order Fractional Differential Equations of Riesz Space Derivative With Anti-Periodic Boundary Conditions”. Communications in Advanced Mathematical Sciences, vol. 4, no. 4, 2021, pp. 171-9, doi:10.33434/cams.1016464.
Vancouver Toprakseven Ş. On the Solutions of the Higher Order Fractional Differential Equations of Riesz Space Derivative with Anti-Periodic Boundary Conditions. Communications in Advanced Mathematical Sciences. 2021;4(4):171-9.

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