Nonlinear higher-order time-fractional equations with purely integral boundary conditions: analytical results and numerical simulations

Ahcene Merad; Abdellah Menasri; Ahcene Merad; Abdellah Menasri

doi:10.3934/math.2026658

AIMS Mathematics

2026, Volume 11, Issue 6: 15990-16007. doi: 10.3934/math.2026658

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Nonlinear higher-order time-fractional equations with purely integral boundary conditions: analytical results and numerical simulations

Ahcene Merad ^{1
,
,},
Abdellah Menasri ^1,2

1.
System Dynamics and Control Laboratory, Department of Mathematics and Informatics, Oum El Bouaghi University, Algeria
2.
Higher National School of Forests, Khenchela, Algeria

Received: 05 April 2026 Revised: 08 May 2026 Accepted: 26 May 2026 Published: 05 June 2026
MSC : 35R11, 35K55, 65M06, 65M12

This paper studies a nonlinear higher-order time-fractional partial differential equation with purely integral boundary conditions in the strip $ Q_T = (0, 1)\times(0, T) $. The model involves a Caputo derivative of order $ 0 < \alpha < 1 $, a variable-coefficient principal part, and a nonlinear term depending on the solution and on its first spatial derivative. The analysis is formulated for any positive integer $ m $ under an explicit boundedness and coercivity hypothesis for the weak realization of the spatial operator on a moment-constrained space; this point is stated as a structural assumption, not as a consequence of positivity of the coefficient alone. We clarify the exact order of the variable-coefficient operator, construct a bounded lifting for the two imposed moments, give the weak duality formulation, and derive an a priori estimate with constants that do not depend on the unknown solution. Existence is obtained from a linear fractional solvability result and a fixed-point argument, while uniqueness and continuous dependence follow from a fractional energy inequality and a Mittag-Leffler version of the fractional Gronwall lemma. The numerical section is deliberately presented as a reproducible $ m = 1 $ validation: it includes the classical L1 finite-difference benchmark, a zero-moment forcing test, and an additional variable-coefficient nonlinear manufactured example solved by Picard iteration. The remaining extension to fully nonlinear higher-order discretizations for $ m > 1 $ is identified explicitly as future work.
- time-fractional partial differential equation,
- purely integral boundary conditions,
- Caputo derivative,
- weak solution,
- existence and uniqueness,
- continuous dependence,
- nonlocal constraints,
- L1 method
Citation: Ahcene Merad, Abdellah Menasri. Nonlinear higher-order time-fractional equations with purely integral boundary conditions: analytical results and numerical simulations[J]. AIMS Mathematics, 2026, 11(6): 15990-16007. doi: 10.3934/math.2026658

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Abstract

This paper studies a nonlinear higher-order time-fractional partial differential equation with purely integral boundary conditions in the strip $ Q_T = (0, 1)\times(0, T) $. The model involves a Caputo derivative of order $ 0 < \alpha < 1 $, a variable-coefficient principal part, and a nonlinear term depending on the solution and on its first spatial derivative. The analysis is formulated for any positive integer $ m $ under an explicit boundedness and coercivity hypothesis for the weak realization of the spatial operator on a moment-constrained space; this point is stated as a structural assumption, not as a consequence of positivity of the coefficient alone. We clarify the exact order of the variable-coefficient operator, construct a bounded lifting for the two imposed moments, give the weak duality formulation, and derive an a priori estimate with constants that do not depend on the unknown solution. Existence is obtained from a linear fractional solvability result and a fixed-point argument, while uniqueness and continuous dependence follow from a fractional energy inequality and a Mittag-Leffler version of the fractional Gronwall lemma. The numerical section is deliberately presented as a reproducible $ m = 1 $ validation: it includes the classical L1 finite-difference benchmark, a zero-moment forcing test, and an additional variable-coefficient nonlinear manufactured example solved by Picard iteration. The remaining extension to fully nonlinear higher-order discretizations for $ m > 1 $ is identified explicitly as future work.

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