AppliedMath

This work proposes an integral-enhanced nonlinear $P{I}^2$ state observer for the robust estimation of unmeasured states in nonlinear dynamic systems, with experimental validation on a flat-panel photobioreactor. The observer is designed as a virtual sensor to reconstruct key biological variables using a reduced set of online measurements and known operating conditions. Compared with a conventional extended Luenberger observer, the proposed structure improves estimation accuracy and robustness against constant disturbances and model mismatch, which are common in bioprocess applications. The experimental results show a clear performance advantage during transient growth phases while highlighting that the method relies on a locally valid model structure and appropriate gain tuning. Overall, the proposed observer provides a practical and scalable monitoring tool for nonlinear systems where the direct measurement of critical state is not feasible.

Nonlinear systems with multiple roots arise frequently in biomedical and engineering models, yet their reliable numerical solution remains a challenging task. Many classical methods suffer from sensitivity to initial guesses, reduced convergence rates, and loss of accuracy in the presence of multiple or clustered solutions. In addition, the exploitation of parallelism to improve robustness and computational efficiency has received limited attention. In this work, we propose a high-accuracy parallel numerical framework of fourth-order convergence for the simultaneous approximation of all solutions of nonlinear systems with multiple roots. The proposed scheme is derivative-free and structurally decoupled, enabling efficient parallel implementation and robust convergence even when reliable initial approximations are unavailable. The effectiveness of the method is demonstrated on representative biomedical engineering models, including a glucose–insulin–glucagon regulatory network and a multi-compartment pharmacokinetic system, both exhibiting strong nonlinearity and multistability. Numerical experiments confirm stable convergence toward distinct solution clusters, machine-level accuracy, reduced residual norms, and improved computational performance when compared with existing approaches. These results indicate that the proposed framework provides a reliable and efficient alternative for solving nonlinear systems with multiple roots in complex applied settings.

An advanced frequency study in thick-walled functionally graded material (FGM) spherical shells is investigated with advanced shear correction. The values of advanced shear correction can be greater than one, be a negative value, and be affected by a nonlinear term of third-order shear deformation theory (TSDT) of displacements, FGM power law index, and temperature. It is novel and interesting to consider using TSDT and advanced shear correction to derive a simple homogeneous equation with reasonable simplifications into a symmetrical sparse matrix subjected to free vibration. The zero determinant of the symmetrical sparse matrix can be expressed to calculate the natural frequency by Newton’s method. The parameter effects of advanced shear correction, a nonlinear TSDT term, temperature, and the FGM power-law index on the natural frequencies of thick-walled FGM spherical shells are presented. The natural-frequency data for the axial and circumferential mode shapes are obtained. This is a new finding, as the assumed simplification in a sparse matrix causes a numerical truncation error; the natural-frequency values of the presented sparse matrix are much greater than those in a full matrix for thick-walled FGM spherical shells.