Khaled Helmi Khashan

King Saud University

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King Saud University

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Khaled Helmi Khashan has a total of 5 co-authors. Here are the top co-authors by number of shared publications:

Diaa Eldin Elgezouli
Department of Basic Sciences, Common First Year Deanship, King Saud University, P.O. Box 145111, Riyadh 11362, Saudi Arabia
3 shared publications
Mohamed A. Abdoon
Department of Basic Sciences, Common First Year Deanship, King Saud University, P.O. Box 145111, Riyadh 11362, Saudi Arabia
3 shared publications
Amr M. Y. Abdelaty
Department of Basic Sciences, Common First Year Deanship, King Saud University, Riyadh 12373, Saudi Arabia
1 shared publication
Ghadah Alhawael
Department of Basic Sciences, Common First Year Deanship, King Saud University, P.O. Box 145111, Riyadh 11362, Saudi Arabia
1 shared publication
Ibrahim S. Elshazly
Department of Basic Sciences, Common First Year Deanship, King Saud University, Riyadh 12373, Saudi Arabia
1 shared publication

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Open Access
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- 26 October 2026
- Published by MDPI AG in Mathematics
Physical-Admissibility Projection and Split-Conformal Calibration for Burgers Neural Operators
by Ibrahim S. Elshazly, Khaled H. Khashan, Amr M. Y. Abdelaty
Accurate PDE prediction does not ensure physical admissibility or calibrated uncertainty. For periodic viscous Burgers dynamics, we combine an input-dependent Euclidean projection, determined by the initial condition, onto discrete mean and centered-energy constraints with split-conformal bands based on one maximum-coordinate residual per function. For fixed input, the projection is nonexpansive and cannot increase Euclidean error to an admissible target. Under exchangeability, the bands provide finite-sample marginal coverage simultaneously over the 64-point terminal grid. The nonzero-mean Cole–Hopf reference agrees with an independent dealiased Fourier–RK4 solver, with maximum relative $L^{2}$ and scaled $L^{\infty}$ discrepancies of $3.62 \times 10^{- 8}$ and $5.83 \times 10^{- 8}$ . Projected FNO gives the lowest ID mean relative $L^{2}$ error, $0.0190$ , among the four implementations at $N = 64$ and $T = 0.5$ over five initializations. Full projection eliminates measured constraint violations and lowers mean error across models and regimes from $0.1984$ to $0.1904$ . At nominal 90%, constant-width ID simultaneous coverage ranges from $0.904$ to $0.930$ ; normalized FNO bands reduce mean width from $0.108$ to $0.085$ with coverage $0.901$ . Under distribution shift, undercoverage and occasional conservative width inflation restrict conclusions to empirical diagnostics. The framework separates admissibility with respect to the two selected discrete constraints, predictive accuracy, and statistical coverage without a new architecture.
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- 26 October 2026
- Published by MDPI AG in Mathematics
Open Access
Physical-Admissibility Projection and Split-Conformal Calibration for Burgers Neural Operators
by Ibrahim S. Elshazly, Khaled H. Khashan, Amr M. Y. Abdelaty
Accurate PDE prediction does not ensure physical admissibility or calibrated uncertainty. For periodic viscous Burgers dynamics, we combine an input-dependent Euclidean projection, determined by the initial condition, onto discrete mean and centered-energy constraints with split-conformal bands based on one maximum-coordinate residual per function. For fixed input, the projection is nonexpansive and cannot increase Euclidean error to an admissible target. Under exchangeability, the bands provide finite-sample marginal coverage simultaneously over the 64-point terminal grid. The nonzero-mean Cole–Hopf reference agrees with an independent dealiased Fourier–RK4 solver, with maximum relative $L^{2}$ and scaled $L^{\infty}$ discrepancies of $3.62 \times 10^{- 8}$ and $5.83 \times 10^{- 8}$ . Projected FNO gives the lowest ID mean relative $L^{2}$ error, $0.0190$ , among the four implementations at $N = 64$ and $T = 0.5$ over five initializations. Full projection eliminates measured constraint violations and lowers mean error across models and regimes from $0.1984$ to $0.1904$ . At nominal 90%, constant-width ID simultaneous coverage ranges from $0.904$ to $0.930$ ; normalized FNO bands reduce mean width from $0.108$ to $0.085$ with coverage $0.901$ . Under distribution shift, undercoverage and occasional conservative width inflation restrict conclusions to empirical diagnostics. The framework separates admissibility with respect to the two selected discrete constraints, predictive accuracy, and statistical coverage without a new architecture.
Open Access
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Google Scholar
- 24 July 2026
- Published by MDPI AG in Mathematics
Variable-Order Fractional Calculus-Based Chaos Analysis of a Novel Eight-Dimensional Hyperchaotic System
by Khaled Helmi Khashan, Diaa Eldin Elgezouli, Mohamed A. Abdoon
In this study, we develop a novel variable-order fractional extension of an eight-dimensional (8D) hyperchaotic differential equation system modeled via the Liouville–Caputo operator. Moving beyond constant fractional-order models, our system implements time-variable orders, which enable its historical memory structure to evolve dynamically over time. To numerically approximate the trajectories of this complex 8D system, a second-order Lagrange numerical integration approach is formulated. An extensive dynamic analysis explores the behavior of this variable-order framework under two distinct configurations: a slowly periodic memory function and a smooth, monotonic hyperbolic tangent function. Topological complexity and multidimensional chaos are characterized using parameter-dependent bifurcation diagrams, phase portraits, Kaplan–Yorke fractal dimensions, and Kolmogorov–Sinai metric entropy. Numerical results show that both variable-order configurations display robust hyperchaotic dynamics characterized by four positive Lyapunov exponents. Crucially, the proposed variable-order extension enhances the phase space footprint of the baseline system, achieving a maximum Kaplan–Yorke dimension of 7.100, thereby offering excellent topological density for secure cryptographic applications.
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- 24 July 2026
- Published by MDPI AG in Mathematics
Open Access
Variable-Order Fractional Calculus-Based Chaos Analysis of a Novel Eight-Dimensional Hyperchaotic System
by Khaled Helmi Khashan, Diaa Eldin Elgezouli, Mohamed A. Abdoon
In this study, we develop a novel variable-order fractional extension of an eight-dimensional (8D) hyperchaotic differential equation system modeled via the Liouville–Caputo operator. Moving beyond constant fractional-order models, our system implements time-variable orders, which enable its historical memory structure to evolve dynamically over time. To numerically approximate the trajectories of this complex 8D system, a second-order Lagrange numerical integration approach is formulated. An extensive dynamic analysis explores the behavior of this variable-order framework under two distinct configurations: a slowly periodic memory function and a smooth, monotonic hyperbolic tangent function. Topological complexity and multidimensional chaos are characterized using parameter-dependent bifurcation diagrams, phase portraits, Kaplan–Yorke fractal dimensions, and Kolmogorov–Sinai metric entropy. Numerical results show that both variable-order configurations display robust hyperchaotic dynamics characterized by four positive Lyapunov exponents. Crucially, the proposed variable-order extension enhances the phase space footprint of the baseline system, achieving a maximum Kaplan–Yorke dimension of 7.100, thereby offering excellent topological density for secure cryptographic applications.
Open Access
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Cited by 1
- 14 July 2026
- Published by MDPI AG in Mathematics
Dynamics and Chaos Analysis of a Novel 4D Chaotic System Using Constant- and Variable-Order Fractional Calculus
by Khaled Helmi Khashan, Diaa Eldin Elgezouli, Mohamed A. Abdoon
In this work, a new 4D chaotic system is presented with Liouville–Caputo fractional derivatives, including constant-order (C) and variable-order (V). In this regard, the paper examines the equilibrium of the system, local stability, and dissipative property. The use of variable order through incorporating time-dependent fractional order in the system, which includes periodic and exponential functions, is an appropriate way of simulating the adaptive memory effect of the fractional-order systems. The analytical and numerical studies reveal that the proposed system is able to show the chaotic behaviour and sensitive dependence on parameters. Specifically, in terms of maximum Lyapunov exponent and Kaplan–Yorke dimension, the performance of the variable-order system is superior to the constant-order one, reaching the values of $λ_{max} \approx 0.275$ and $D_{K Y} \approx 2.223$ . The approach presented here offers a more realistic framework to model memory-dependent chaotic systems and finds applications in the design of nonlinear circuits, secure communications, neuromorphic computing, and advanced control systems.
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- 14 July 2026
- Published by MDPI AG in Mathematics
Open Access
Dynamics and Chaos Analysis of a Novel 4D Chaotic System Using Constant- and Variable-Order Fractional Calculus
by Khaled Helmi Khashan, Diaa Eldin Elgezouli, Mohamed A. Abdoon
In this work, a new 4D chaotic system is presented with Liouville–Caputo fractional derivatives, including constant-order (C) and variable-order (V). In this regard, the paper examines the equilibrium of the system, local stability, and dissipative property. The use of variable order through incorporating time-dependent fractional order in the system, which includes periodic and exponential functions, is an appropriate way of simulating the adaptive memory effect of the fractional-order systems. The analytical and numerical studies reveal that the proposed system is able to show the chaotic behaviour and sensitive dependence on parameters. Specifically, in terms of maximum Lyapunov exponent and Kaplan–Yorke dimension, the performance of the variable-order system is superior to the constant-order one, reaching the values of $λ_{max} \approx 0.275$ and $D_{K Y} \approx 2.223$ . The approach presented here offers a more realistic framework to model memory-dependent chaotic systems and finds applications in the design of nonlinear circuits, secure communications, neuromorphic computing, and advanced control systems.
Cited by 1
Open Access
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Cited by 3
- 15 December 2025
- Published by MDPI AG in Mathematics
Chaos Analysis of the Fractional Genesio-Tesi System with Constant and Variable-Order Dynamics
by Ghadah Alhawael, Mohamed A. Abdoon, Khaled Helmi Khashan, Diaa Eldin Elgezouli
Fractional calculus extends conventional differentiation and integration to non-integer orders, facilitating a more suitable modeling framework for complicated dynamical processes characterized by memory and long-range dependence. The fractional and variable-order fractional Genesio-Tesi systems have recently attracted significant interest owing to their rich nonlinear dynamics and the added flexibility introduced by variable fractional orders. This work comparatively studies the stability and chaotic behavior of constant-order versus variable-order formulations of the fractional Genesio–Tesi system. The study of the system dynamics is carried out by numerical simulations, including time series, bifurcation diagrams, Lyapunov exponents, and phase portraits. We identify further stability boundaries and chaotic regimes through analytical investigations based on the Jacobian eigenvalue spectrum. It is found that variable-order derivatives intensify sensitivity and transient responses, disclosing chaotic patterns that contribute to a more profound understanding of fractional nonlinear dynamics.
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- 15 December 2025
- Published by MDPI AG in Mathematics
Open Access
Chaos Analysis of the Fractional Genesio-Tesi System with Constant and Variable-Order Dynamics
by Ghadah Alhawael, Mohamed A. Abdoon, Khaled Helmi Khashan, Diaa Eldin Elgezouli
Fractional calculus extends conventional differentiation and integration to non-integer orders, facilitating a more suitable modeling framework for complicated dynamical processes characterized by memory and long-range dependence. The fractional and variable-order fractional Genesio-Tesi systems have recently attracted significant interest owing to their rich nonlinear dynamics and the added flexibility introduced by variable fractional orders. This work comparatively studies the stability and chaotic behavior of constant-order versus variable-order formulations of the fractional Genesio–Tesi system. The study of the system dynamics is carried out by numerical simulations, including time series, bifurcation diagrams, Lyapunov exponents, and phase portraits. We identify further stability boundaries and chaotic regimes through analytical investigations based on the Jacobian eigenvalue spectrum. It is found that variable-order derivatives intensify sensitivity and transient responses, disclosing chaotic patterns that contribute to a more profound understanding of fractional nonlinear dynamics.
Cited by 3