On a boundary value problem for a nonlinear differential equation with a Riemann-Liouville fractional derivative of variable order and nonlocal boundary conditions

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Abstract

The paper studies the existence of at least two positive solutions for a nonlinear differential equation with a fractional Riemann-Liouville derivative of variable order and nonlocal boundary conditions. To transform the original equation into an integral equation, the Green function is constructed and boundedness conditions for the Green function are obtained. A theorem on the existence of at least two positive solutions to the problem is proven. An example is given that shows the existence of two positive solutions, and approximation graphs are constructed for clarity.

Keywords

fractional derivative / boundary conditions / Green function / non-stationary processes

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References

[1] V. V. Uchaikin, R. T. Sibatov, Fractional differential kinetics of dispersive transport as the consequence of its self-similarity, JETP Lett., 86 (2007), 512-516. https://doi.org/10.1134/S0021364007200040

[2] V. D. Beybalaev, A. A. Aliverdiev, J. Hristov, Transient heat conduction in a semi-infinite domain with a memory effect: Analytical solutions with a robin boundary condition, Fractal Fract., 7 (2023), 770. https://doi.org/10.3390/fractalfract7100770

[3] J. Hristov, Constitutive fractional modeling, in Mathematical Modelling: Principle and Theory (eds. Hemen Dutta), American Mathematical Society, 786 (2023), 37-140. https://doi.org/10.1090/conm/786/15795

[4] V. D. Beybalaev, A. A. Aliverdiev, A. Z. Yakubov, S. A. Ninalalov, A. A. Amirova, Mathematical model of heat conduction for a semi-infinite body, taking into account memory effects and spatial correlations, Fractal Fract., 7 (2023), 265. https://doi.org/10.3390/fractalfract7030265

[5] K. S. Miller, Fractional differential equations, J. Fract. Calc., 3 (1993), 49-57.

[6] A. M. Nakhushev, The Sturm-Liouville problem for a second order ordinary differential equation with fractional derivatives in the lower terms, Dokl. Akad. Nauk SSSR, 234 (1977), 308-311.

[7] I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering , Academic Press, 1999.

[8] I. Podlubny, The Laplace transform method for linear differential equations of the fractional order, Inst. Expe. Phys., Slov. Acad. Sci., 1994 (1994).

[9] Z. B. Bai, H. S. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl., 311 (2005), 495-505. https://doi.org/10.1016/j.jmaa.2005.02.052

[10] E. R. Kaufmann, E. Mboumi, Positive solutions of a boundary value problem for a nonlinear fractional differential equation, Electron. J. Qual. Theory Differ. Equ., 3 (2008), 1-11. https://doi.org/10.14232/ejqtde.2008.1.3

[11] C. Z. Bai, Triple positive solutions for a boundary value problem of nonlinear fractional differential equation, Electron. J. Qual. Theory Differ. Equ., 24 (2008), 1-10.

[12] Y. Sun, Y. Sun, Positive solutions and monotone iterative sequences for a fractional differential equation with integral boundary conditions, Adv. Differ. Equ., 2014 (2014), 29. https://doi.org/10.1186/1687-1847-2014-29

[13] C. B. Zhai, M. R. Hao, Mixed monotone operator methods for the existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value problems, Bound. Value Probl., 2013 (2013), 85. https://doi.org/10.1186/1687-2770-2013-85

[14] S. Q. Zhang, The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl., 252 (2000), 804-812. https://doi.org/10.1006/jmaa.2000.7123

[15] X. G. Zhang, Y. F. Han, Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations, Appl. Math. Lett., 25 (2012), 555-560. https://doi.org/10.1016/j.aml.2011.09.058

[16] X. G. Zhang, L. S. Liu, Y. H. Wu, Multiple positive solutions of a singular fractional differential equation with negatively perturbed term, Math. Comput. Modell., 55 (2012), 1263-1274. https://doi.org/10.1016/j.mcm.2011.10.006

[17] X. G. Zhang, L. S. Liu, Y. H. Wu, Y. N. Lu, The iterative solutions of nonlinear fractional differential equations, Appl. Math. Comput., 219 (2013), 4680-4691. https://doi.org/10.1016/j.amc.2012.10.082

[18] C. F. Li, X. N. Luo, Y. Zhou, Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations, Comput. Math. Appl., 59 (2010), 1363-1375. https://doi.org/10.1016/j.camwa.2009.06.029

[19] C. F. M. Coimbra, Mechanics with variable-order differential operators, Ann. Phys., 515 (2003), 692-703. https://doi.org/10.1002/andp.200310032

[20] D. Ingman, J. Suzdalnitsky, Control of damping oscillations by fractional differential operator with time-dependent order, Comput. Methods Appl. Mech. Eng., 193 (2004), 5585-5595. https://doi.org/10.1016/j.cma.2004.06.029

[21] C. F. Lorenzo, T. T. Hartley, Variable order and distributed order fractional operators, Nonlinear Dyn., 29 (2009), 57-98. https://doi.org/10.1023/A:1016586905654

[22] B. P. Moghaddam, J. A. T. Machado, Extended algorithms for approximating variable order fractional derivatives with applications, J. Sci. Comput., 71 (2017), 1351-1374. https://doi.org/10.1007/s10915-016-0343-1

[23] L. Ramirez, C. Coimbra, On the selection and meaning of variable order operators for dynamic modeling, Int. J. Differ. Equ., 1 (2010), 1-16. https://doi.org/10.1155/2010/846107

[24] H. G. Sun, W. Chen, Y. Q. Chen, Variable order fractional differential operators in anomalous diffusion modeling, Physica A, 388 (2009), 4586-4592. https://doi.org/10.1016/j.physa.2009.07.024

[25] S. Zhang, L. Hu, Unique existence result of approximate solution to initial value problem for fractional differential equation of variable order involving the derivative arguments on the half-axis, Mathematics, 7 (2019), 286. https://doi.org/10.3390/math7030286

[26] D. Valerio, J. Sa da Costa, Variable-order fractional derivative and their numerical approximations, Signal Process., 91 (2011), 470-483. https://doi.org/10.1016/j.sigpro.2010.04.006

[27] D. Tavares, R. Almeida, D. F. M. Torres, Caputo derivatives of fractional variable order: Numerical approximations, Commun. Nonlinear Sci. Numer. Simul., 35 (2016), 69-87. https://doi.org/10.1016/j.cnsns.2015.10.027

[28] A. Razminia, A. F. Dizaji, V. J. Majd, Solution existence for non-autonomous variable-order fractional differential equations, Math. Comput. Modell., 55 (2012), 1106-1117. https://doi.org/10.1016/j.mcm.2011.09.034

[29] A. A. Alikhanov, Boundary value problems for the equation of the variable order in differential and difference settings, Appl. Math. Comput., 219 (2012), 3938-3946. https://doi.org/10.1016/j.amc.2012.10.029

[30] A. Babaei, H. Jafari, S. Banihashemi, Numerical solution of variable order fractional nonlinear quadratic integro-differential equations based on the sixth-kind Chebyshev collocation method, J. Comput. Appl. Math., 377 (2020), 112908. https://doi.org/10.1016/j.cam.2020.112908

[31] C. J. Zuniga-Aguilar, H. M. Romero-Ugalde, J. F. Gomez-Aguilar, R. F. Escobar-Jimenez, M. Valtierra-Rodríguez, Solving fractional differential equations of variable-order involving operator with Mittag-Leffler kernel using artificial neural networks, Chaos, Solitons Fractals, 103 (2017), 382-403. https://doi.org/10.1016/j.chaos.2017.06.030

[32] C. M. Chen, F. Liu, V. Anh, I. Turner, Numerical schemes with high spatial accuracy for a variableorder anomalous subdiffusion equation, SIAM J. Sci. Comput., 32 (2010), 1740-1760. https://doi.org/10.1137/090771715

[33] M. O. Korpusov, A. A. Panin, Lectures on Linear and Nonlinear Functional Analysis, Part III: Nonlinear Analysis , Physics Faculty of Moscow State University, 2015.