A doubly nonlocal fractional integro-boundary problem with weakly singular weights is investigated for the order range $ 2 < \alpha < 3 $. The governing equation contains a weakly singular Volterra memory operator, while the boundary condition involves a weighted integral functional with endpoint singularities. This configuration is relevant to anomalous transport and hereditary diffusion models in which interior memory and boundary feedback act simultaneously. An explicit Green representation is derived for the associated linear problem and is normalized by an endpoint-aware singular profile, which leads to two-sided kernel estimates and to a positive cone adapted to the doubly singular structure. Sufficient conditions for at least three positive solutions are then obtained by means of a Leggett–Williams framework. In parallel, a product-integration Nyström discretization is constructed for the weakly singular state operator and the weakly singular boundary operator. A uniform-grid convergence estimate is proved under a local contractivity condition, and a graded-mesh extension is formulated for endpoint-dominated regimes. Numerical experiments are reported for mesh refinement, boundary-parameter sensitivity, singularity-intensity variation, and graded-mesh comparison. The results confirm that the proposed approach provides a coherent analytic-computational framework for fractional models with simultaneous nonlocal singularities on the equation and boundary sides.
Citation: Zhao-sheng Wang, Guan-fa Li, Shiyu Li. Triple positive solutions and product-integration Nyström approximation for a doubly nonlocal fractional integro-boundary problem with weakly singular weights[J]. AIMS Mathematics, 2026, 11(7): 20875-20898. doi: 10.3934/math.2026847
A doubly nonlocal fractional integro-boundary problem with weakly singular weights is investigated for the order range $ 2 < \alpha < 3 $. The governing equation contains a weakly singular Volterra memory operator, while the boundary condition involves a weighted integral functional with endpoint singularities. This configuration is relevant to anomalous transport and hereditary diffusion models in which interior memory and boundary feedback act simultaneously. An explicit Green representation is derived for the associated linear problem and is normalized by an endpoint-aware singular profile, which leads to two-sided kernel estimates and to a positive cone adapted to the doubly singular structure. Sufficient conditions for at least three positive solutions are then obtained by means of a Leggett–Williams framework. In parallel, a product-integration Nyström discretization is constructed for the weakly singular state operator and the weakly singular boundary operator. A uniform-grid convergence estimate is proved under a local contractivity condition, and a graded-mesh extension is formulated for endpoint-dominated regimes. Numerical experiments are reported for mesh refinement, boundary-parameter sensitivity, singularity-intensity variation, and graded-mesh comparison. The results confirm that the proposed approach provides a coherent analytic-computational framework for fractional models with simultaneous nonlocal singularities on the equation and boundary sides.
| [1] | I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999. |
| [2] | A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006. |
| [3] |
H. Sun, W. Chen, Y. 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 doi: 10.1016/j.physa.2009.07.024
|
| [4] |
Z. Li, H. Wang, R. Xiao, S. Yang, A variable-order fractional differential equation model of shape memory polymers, Chaos Soliton. Fract., 102 (2017), 473–485. https://doi.org/10.1016/j.chaos.2017.04.042 doi: 10.1016/j.chaos.2017.04.042
|
| [5] |
M. Cukic, S. Galovic, Mathematical modeling of anomalous diffusive behavior in transdermal drug-delivery including time-delayed flux concept, Chaos Soliton. Fract., 172 (2023), 113584. https://doi.org/10.1016/j.chaos.2023.113584 doi: 10.1016/j.chaos.2023.113584
|
| [6] |
Z. Bai, H. Lu, 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 doi: 10.1016/j.jmaa.2005.02.052
|
| [7] | A. Cabada, G. Wang, Positive solutions of nonlinear fractional differential equations with integral boundary value conditions, J. Math. Anal. Appl. 389 (2012), 403–411. https://doi.org/10.1016/j.jmaa.2011.11.065 |
| [8] |
W. Yang, Positive solutions for nonlinear Caputo fractional differential equations with integral boundary conditions, J. Appl. Math. Comput., 44 (2014), 39–59. https://doi.org/10.1007/s12190-013-0679-8 doi: 10.1007/s12190-013-0679-8
|
| [9] | S. W. Vong, Positive solutions of singular fractional differential equations with integral boundary conditions, Math. Comput. Model. 57 (2013), 1053–1059. https://doi.org/10.1016/j.mcm.2012.06.024 |
| [10] |
L. Zhang, W. Zhang, X. Liu, M. Jia, Existence of positive solutions for integral boundary value problems of fractional differential equations with $p$-Laplacian, Adv. Differ. Equ., 2017 (2017), Art. 36. 10.1186/s13662-017-1086-5 doi: 10.1186/s13662-017-1086-5
|
| [11] |
Y. Zi, Y. Wang, Positive solutions for Caputo fractional differential system with coupled boundary conditions, Adv. Differ. Equ., 2019 (2019), Art. 80. https://doi.org/10.1186/s13662-019-2016-5 doi: 10.1186/s13662-019-2016-5
|
| [12] |
R. W. Leggett, L. R. Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J., 28 (1979), 673–688. https://doi.org/10.1512/iumj.1979.28.28046 doi: 10.1512/iumj.1979.28.28046
|
| [13] |
R. I. Avery, A. C. Peterson, Three positive fixed points of nonlinear operators on ordered Banach spaces, Comput. Math. Appl., 42 (2001), 313–322. https://doi.org/10.1016/S0898-1221(01)00156-0 doi: 10.1016/S0898-1221(01)00156-0
|
| [14] |
D. R. Anderson, R. I. Avery, Fixed point theorem of cone expansion and compression of functional type, J. Difference Equ. Appl., 8 (2002), 1073–1083. https://doi.org/10.1080/10236190290015344 doi: 10.1080/10236190290015344
|
| [15] |
X. Hu, L. Wang, Positive solutions to integral boundary value problems for singular delay fractional differential equations, AIMS Math., 8 (2023), 25550–25563. https://doi.org/10.3934/math.20231304 doi: 10.3934/math.20231304
|
| [16] |
H. Li, X. Liu, Positive solutions for a Caputo-type fractional differential equation with a Riemann–Stieltjes integral boundary condition, Electron. Res. Arch., 33 (2025), 4027–4044. https://doi.org/10.3934/era.2025179 doi: 10.3934/era.2025179
|
| [17] |
L. Guo, U. Riaz, A. Zada, M. Alam, On implicit coupled Hadamard fractional differential equations with generalized Hadamard fractional integro-differential boundary conditions, Fractal Fract., 7 (2023), 13. https://doi.org/10.3390/fractalfract7010013 doi: 10.3390/fractalfract7010013
|
| [18] |
D. Min, L. Guo, Existence and nonexistence of nontrivial solutions for fractional advection–dispersion equation with instantaneous and non-instantaneous impulses, Fractal Fract., 9 (2025), 571. https://doi.org/10.3390/fractalfract9090571 doi: 10.3390/fractalfract9090571
|
| [19] |
L. Guo, C. Li, N. Qiao, J. Zhao, Convergence analysis of positive solution for Caputo–Hadamard fractional differential equation, Nonlinear Anal. Model. Control, 30 (2025), 212–230. https://doi.org/10.15388/namc.2025.30.38509 doi: 10.15388/namc.2025.30.38509
|
| [20] |
A. P. Orsi, Product integration for Volterra integral equations of the second kind with weakly singular kernels, Math. Comp., 65 (1996), 1201–1212. https://doi.org/10.1090/S0025-5718-96-00736-3 doi: 10.1090/S0025-5718-96-00736-3
|
| [21] |
P. Baratella, A. P. Orsi, A new approach to the numerical solution of weakly singular Volterra integral equations, J. Comput. Appl. Math., 163 (2004), 401–418. https://doi.org/10.1016/j.cam.2003.08.047 doi: 10.1016/j.cam.2003.08.047
|
| [22] |
Y. Chen, T. Tang, Spectral methods for weakly singular Volterra integral equations with smooth solutions, J. Comput. Appl. Math., 233 (2009), 938–950. https://doi.org/10.1016/j.cam.2009.08.057 doi: 10.1016/j.cam.2009.08.057
|
| [23] |
Z. Ma, M. Stynes, A Nyström method based on product integration for weakly singular Volterra integral equations with variable exponent, J. Comput. Appl. Math., 454 (2025), 116164. https://doi.org/10.1016/j.cam.2024.116164 doi: 10.1016/j.cam.2024.116164
|
| [24] |
R. Katani, S. McKee, Numerical solution of two-dimensional weakly singular Volterra integral equations with nonsmooth solutions, J. Comput. Appl. Math., 402 (2022), 113779. https://doi.org/10.1016/j.cam.2021.113779 doi: 10.1016/j.cam.2021.113779
|
| [25] |
G. Wang, T. Wang, Singular asymptotic expansion and Legendre collocation method for two-term weakly singular Volterra integral equation of the second kind, Appl. Numer. Math., 197 (2024), 344–362. https://doi.org/10.1016/j.apnum.2023.12.002 doi: 10.1016/j.apnum.2023.12.002
|
| [26] |
F. Dell'Accio, D. Mezzanotte, F. Nudo, D. Occorsio, Numerical approximation of Fredholm integral equation by the constrained mock-Chebyshev least squares operator, J. Comput. Appl. Math., 447 (2024), 115886. https://doi.org/10.1016/j.cam.2024.115886 doi: 10.1016/j.cam.2024.115886
|
| [27] |
F. Dell'Accio, D. Mezzanotte, F. Nudo, D. Occorsio, Product integration rules by the constrained mock-Chebyshev least squares operator, BIT Numer. Math., 63 (2023), 24. https://doi.org/10.1007/s10543-023-00968-w doi: 10.1007/s10543-023-00968-w
|
| [28] |
F. Dell'Accio, F. Di Tommaso, F. Nudo, Generalizations of the constrained mock-Chebyshev least squares in two variables: Tensor product vs total degree polynomial interpolation, Appl. Math. Lett., 125 (2022), 107732. https://doi.org/10.1016/j.aml.2021.107732 doi: 10.1016/j.aml.2021.107732
|
Zhao-sheng Wang, Guan-fa Li, Shiyu Li. Triple positive solutions and product-integration Nyström approximation for a doubly nonlocal fractional integro-boundary problem with weakly singular weights[J]. AIMS Mathematics, 2026, 11(7): 20875-20898. doi: 10.3934/math.2026847
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