International Journal of Nonlinear Analysis and Applications

Existence and uniqueness outcomes for a nonlinear fractional differential equation of high order featuring nonlocal boundary conditions

Articles in Press

Document Type : Research Paper

Authors

1 Department of Mathematics, Imam Khomeini International University, Qazvin, 34149-16818, Iran

2 Department of Mathematics, Buein Zahra Technical University, Buein Zahra, Qazvin, Iran

Abstract

This study centers on establishing the existence of a unique solution for a class of fractional differential equations that incorporate the Riemann-Liouville fractional derivative. The boundary conditions encompass a nonlocal condition involving integration in a sub-domain near the boundary. Initially, the precise solution is derived for the linear fractional differential equation. Subsequently, the Banach contraction mapping theorem is employed to establish the primary result for the general nonlinearity of the source term. Additionally, the validity and applicability of our primary result are illustrated through a specific example.

Keywords

[1] L. AhmadSoltani, Two-point nonlocal nonlinear fractional boundary value problem with Caputo derivative: Analysis and numerical solution, Nonlinear Engin. 11 (2022), no. 1, 71–79.
[2] N. Asaithambi and J. Garner, Pointwise solution bounds for a class of singular diffusion problems in physiology, Appl. Math. Comput. 30 (1989), no. 3, 215–222.
[3] A.C. Burton, Rate of growth of solid tumours as a problem of diffusion, Growth 30 (1966), no. 2, 157–176.
[4] K. Diethelm, The Analysis of Fractional Differential Equations: An Application-Oriented Exposition Using Differential Operators of Caputo Type, Springer Science & Business Media, 2010.
[5] A. Dinmohammadi, A. Razani, and E. Shivanian, Analytical solution to the nonlinear singular boundary value problem arising in biology, Bound. Value Prob. 2017 (2017), no. 1, 1–9.
[6] A. Dinmohammadi, E. Shivanian, and A. Razani, Existence and uniqueness of solutions for a class of singular nonlinear two-point boundary value problems with sign-changing nonlinear terms, Numer. Funct. Anal. Optim. 38 (2017), no. 3, 344–359.
[7] R. Duggan and A. Goodman, Pointwise bounds for a nonlinear heat conduction model of the human head, Bull. Math. Bio. 48 (1986), no. 2, 229–236.
[8] U. Flesch, The distribution of heat sources in the human head: A theoretical consideration, J. Theor. Bio. 54 (1975), no. 2, 285–287.
[9] B. Gray, The distribution of heat sources in the human head—theoretical considerations, J. Theor. Bio. 82 (1980), no. 3, 473–476.
[10] H. Greenspan, Models for the growth of a solid tumor by diffusion, Stud. Appl. Math. 51 (1972), no. 4, 317–340.
[11] D. Haghighi, S. Abbasbandy, and E. Shivanian, Study of the fragile points method for solving two-dimensional linear and nonlinear wave equations on complex and cracked domains, Engin. Anal. Bound. Elements 146 (2023), 44–55.
[12] S. Heidari and A. Razani, Existence and multiplicity of weak solutions for singular fourth-order elliptic systems, Sao Paulo J. Math. Sci. 16 (2022), no. 2, 1309–1326.
[13] A. Kilbas, Theory and Applications of Fractional Differential Equations, Vol. 204, Elsevier, 2006.
[14] R. Klages, G. Radons, and I.M. Sokolov, Anomalous Transport, Wiley Online Library, 2008.
[15] R. Kubo, The fluctuation-dissipation theorem, Rep. Prog. Phys. 29 (1966), no. 1, 255.
[16] R. Kubo, M. Toda, and N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics, vol. 31, Springer Science & Business Media, 2012.
[17] R.L. Magin, Fractional Calculus in Bioengineering, vol. 2, Begell House Redding, 2006.
[18] D. McElwain, A re-examination of oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics, J. Theor. Bio. 71 (1978), no. 2, 255–263.
[19] S. Rezabeyk, S. Abbasbandy, E. Shivanian, and H. Derili, A new approach to solve weakly singular fractional-order delay integro-differential equations using operational matrices, J. Math. Model. 11 (2023), no. 2, 257–275.
[20] E. Shivanian, Error estimate and stability analysis on the study of a high-order nonlinear fractional differential equation with Caputo-derivative and integral boundary condition, Comput. Appl. Math. 41 (2022), no. 8, 395.
[21] E. Shivanian, On the solution of Caputo fractional high-order three-point boundary value problem with applications to optimal control, J. Nonlinear Math. Phys. 31 (2024), no. 1, 2.
[22] E. Shivanian and A. Dinmohammadi, On the solution of a nonlinear fractional integro-differential equation with non-local boundary condition, Int. J. Nonlinear Anal. Appl. 15 (2024), no. 1, 125–136.
[23] E. Shivanian and H. Fatahi, To study existence of unique solution and numerically solving for a kind of three-point boundary fractional high-order problem subject to Robin condition, Comput. Meth. Differ. Equ. 11 (2023), no. 2, 332–344.
[24] E. Shivanian, S.J.H. Ghoncheh, and H. Goudarzi, A unique weak solution for the fractional integro-differential Schrodinger equations, Math. Sci. 17 (2023), no. 1, 15-–19.
[25] E. Shivanian, Z. Hajimohammadi, F. Baharifard, K. Parand, and R. Kazemi, A novel learning approach for different profile shapes of convecting–radiating fins based on shifted Gegenbauer LSSVM, New Math. Natural Comput. 19 (2023), no. 1, 195–215.
International Journal of Nonlinear Analysis and Applications

Articles in Press, Corrected Proof
Available Online from 12 July 2025
  • Receive Date: 06 February 2024
  • Revise Date: 06 March 2024
  • Accept Date: 19 November 2024