Search Results (1,633)

This paper focuses on three finite difference schemes for the interval advection equation based on a novel interval calculus framework. Different from classical interval arithmetic, this innovative framework equips the interval number space with a rigorous Hilbert space structure and enables a critical decoupling of the derivative for interval-valued functions, where the center component obeys classical differentiation rules and the radius component complies with multiplicative differentiation principles. Using this unique decoupling property, we construct interval counterparts of three representative classical finite difference schemes, namely the Upwind, Lax–Friedrichs, and Lax–Wendroff schemes, and conduct a comprehensive and rigorous theoretical assessment of their numerical properties. Utilizing the inherent isometric isomorphism between the interval space and ${\mathbb{R}}^2$, we rigorously establish the consistency of the proposed schemes and adopt the von Neumann method for systematic stability analysis. A sharp, explicit Courant–Friedrichs–Lewy (CFL) condition is derived, which jointly accounts for the center velocity and logarithmic radius velocity of the interval advection field, and the scheme convergence is strictly guaranteed via the Lax equivalence theorem. Extensive numerical experiments are carried out to validate the theoretical conclusions and verify the practical merits of the developed interval schemes. The numerical results demonstrate that the proposed methods retain nearly constant interval width in long-duration simulations, completely bypass the switching-point complexity that intrinsically exists in traditional generalized Hukuhara (gH)-based interval approaches, and deliver competitive computational efficiency. This work corroborates that the newly proposed interval calculus framework serves as an elegant, solid, and versatile foundation for the numerical computation and analysis of interval partial differential equations. Full article

We develop a reaction–diffusion–chemotaxis model for spatial tumour–immune–chemokine dynamics that couples logistic tumour growth, immune-mediated killing, chemokine-dependent immune recruitment, chemotactic migration, and signal production. For the non-dimensional system, we establish local classical solvability, nonnegativity, a uniform tumour-density bound, and global mass estimates for the immune and chemokine components. The tumour-free equilibrium is stable precisely when the baseline immune-control index satisfies ${\sigma }_0/\delta >1$, whereas positive homogeneous coexistence is characterized by a scalar nonlinear equation. Linearization in the Neumann Laplacian eigenbasis yields a mode-dependent cubic dispersion relation, showing that chemotaxis does not alter the tumour-invasion threshold but can destabilize homogeneous coexistence through a finite-wavelength oscillatory instability above a critical sensitivity ${\xi }_c$. A conservative finite-volume discretization with upwind chemotactic fluxes and implicit backward differentiation formula time integration is used to test these predictions. Numerical experiments recover the analytical equilibria and growth rates, identify the dominant unstable mode, reproduce the transition to spatial heterogeneity, and quantify the effects of immune recruitment, decay, and diffusion on the stability boundary. Grid-refinement, mass-balance, residual, and nonnegativity diagnostics support the computational reliability of the results. Full article

The paper investigates the asymptotic stability for the control of systems described by multi-order fractional differential equations. By utilizing generalized Lyapunov functions and the Kalman–Yakubovich–Popov lemma, frequency-domain criteria are derived to evaluate asymptotic stability. The formulated criteria are similar to the ‘Popov Circle Criterion,’ but the circle parameters are determined by the system’s fractional order and the control parameters. Additionally, the asymptotic stability condition requires that all polar plots associated with the multi-fractional-order system lie inside the circle defining the criterion. Human–Robot System applications highlight the investigation techniques and the particularities of the presented criteria. Full article

Nonlinear electromagnetic wave propagation in complex plasma environments has attracted considerable attention due to its important applications in nonlinear optics, plasma physics, space science, and communication technologies. In the present study, a time-fractional Drinfel’d–Sokolov–Wilson equation (DSWE) is investigated under the influence of electromagnetic wave perturbations. The fractional-order formulation incorporates memory and hereditary effects, providing a more realistic description of wave propagation in nonlinear dispersive media. By employing an appropriate fractional traveling-wave transformation, the governing nonlinear fractional partial differential equation is reduced to a nonlinear ordinary differential equation. Exact solitary wave solutions are subsequently constructed using the ${\displaystyle \frac{G^{\prime }}{G^2}}$-expansion technique. Furthermore, the nonlinear dynamical behavior of the reduced system is examined through phase portraits, bifurcation diagrams, Lyapunov exponents, sensitivity analysis, and multistability investigations. Particular attention is devoted to understanding the emergence of chaotic dynamics induced by electromagnetic wave effects and fractional-order interactions. The obtained results reveal that the fractional-order parameter significantly influences the stability, propagation characteristics, and dynamical evolution of nonlinear wave structures. The coexistence of multiple attractors, transitions between stable states, and chaotic regimes is identified for various parameter configurations. These findings provide deeper insight into the complex dynamics governed by the time-fractional DSWE and contribute to the understanding of nonlinear electromagnetic wave propagation in plasma and other nonlinear dispersive media. Full article

This paper investigates the controller design problem for switched systems with input delays. A predictor-based switched controller is proposed to compensate for the effect of input delays. By introducing an appropriate transformation, the switched system with input delays is converted into an equivalent switched partial differential equation system. The exponential stability of the resulting closed-loop system is established by constructing multiple Lyapunov–Krasovskii functionals, which allows for arbitrarily large input delays. Furthermore, the robustness of the proposed controller against perturbations in switching signals is rigorously analyzed based on the Lyapunov–Krasovskii framework. A dynamic predictor-based switched controller is also developed, and the exponential stability of the corresponding closed-loop system is guaranteed. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed control schemes. Full article

This paper presents a numerical method for solving transient one-dimensional heat conduction problems in multilayer slabs with temperature-dependent thermal conductivities. The governing nonlinear partial differential equations are formulated separately in each layer, allowing for distinct material properties. Perfect thermal contact at internal interfaces is enforced through continuity of temperature and heat flux, while general boundary conditions are imposed at the external boundaries, including prescribed temperature, specified heat flux, and convective exchange. A key feature of the proposed approach is to discretize the partial differential equations first in time using the implicit Euler method, thereby reducing the original problem to a sequence of nonlinear two-point boundary value problems with interface (transmission) conditions. A second-order finite difference scheme is employed for spatial discretization, and the resulting system is expressed in global form using a unified indexing strategy. The system is solved at each time step by Newton linearization, yielding a sparse Jacobian matrix that is tridiagonal in the interior and locally extended at the interfaces. Efficient banded solvers lead to O(N) cost per time step, where N is the number of spatial nodes. Numerical experiments confirm the expected accuracy, unconditional stability, and computational complexity of the method. Full article

To overcome the limitations of existing three-degree-of-freedom milling stability models in representing axial cutting conditions, this study develops a stability prediction framework that accounts for both axial segmentation and the axial contact angle. A three-degree-of-freedom dynamic model of the milling system is first formulated by introducing the axial contact angle. The tool axis is then discretized, so that the cutting force coefficients can be evaluated in different axial sections and the non-uniform distribution of cutting forces along the tool can be captured more accurately. After incorporating the regenerative mechanism, the milling dynamics are expressed in the form of a linear time-delay differential equation. To enhance the numerical accuracy of the time-delay system solution, a full-discretization scheme using third-order Lagrange–Hermite interpolation is developed for constructing the state transition matrix. The stability boundary is subsequently determined based on Floquet theory, from which the stability lobe diagram is generated. The proposed model and solution procedure are validated by comparison with existing methods and by time-domain simulation. The results show that, when the spindle speed ranges from 5000 to 10,000 rpm and the axial depth of cut ranges from 0 to 8 mm, the overall variation rate of the predicted stability region is 11.19% after incorporating axial discretization and 59.88% after considering the axial contact angle. The stable and unstable cutting responses obtained from time-domain simulations are consistent with the regions predicted by the stability lobe diagram, which supports the validity of the proposed approach. Further investigation shows that, for the established three-degree-of-freedom milling model and the specified cutting parameters, the axial contact angle exerts a pronounced nonlinear effect on the stability boundary. Specifically, as the axial contact angle $\epsilon$ increases within the range $0^{°}<\epsilon \le {45}^{°}$, the stable region gradually shrinks; when $\epsilon$ increases from 45° to 90°, the stable region expands instead. These observations can provide useful guidance for selecting milling parameters and identifying stable machining conditions. Full article

In this paper, the new fractal modified Zakharov–Kuznetsov equation (fmZKe) defined on Cantor sets is investigated. The fmZKe is a non-differentiable model that arises naturally in mathematical physics, nonlinear wave theory, and plasma physics. The extended rational sine–cosine method is utilized to construct new optical soliton solutions of the model. The fmZKe is reduced to a non-differentiable ordinary differential equation by applying a non-differentiable wave transformation defined on Cantor sets. This reduction leads to a system of linear algebraic equations, which upon solving yields several exact solutions of the model. Furthermore, to establish a clear understanding of the model’s behavior, non-smooth graphical representations of the solutions are presented for various parameter values. The stability analysis of the newly obtained solutions in a classical sense is examined using stability theory, and the real-life applications of the results are highlighted. Full article

In this paper, a controlled Picard semi-analytical technique is improved for a branch of nonlinear fractional stochastic delay differential equations since the nonlinear terms always prevent the establishment of closed-form solutions. The proposed approach extends the known Picard iteration by embedding a convergence-control parameter that improves the flexibility and stability of the iterative scheme while keeping the original mathematical formulation. A careful theoretical analysis is presented to establish the existence of the iterative sequence, its mean-square boundedness, convergence, and an explicit error estimate under standard Lipschitz continuity and linear growth assumptions. Moreover, a Numerical Picard implementation is updated to rebuild stochastic sample trajectories and to give an independent illustration through comparison with a predictor–corrector scheme. The presented methodology is applied to fractional stochastic models of human postural sway and logistic population models. The numerical results illustrate that the semi-analytical framework evaluates the expectation of and variance in the stochastic response for various fractional orders accurately. Although the semi-analytical controlled Picard method is incapable of performing a lot of iterations, it generates statistical moments that agree with those obtained using both the Numerical Picard and predictor–corrector methods, while explicit semi-analytical representations of the solution. These results show that the presented technique gives an accurate and effective approach for examining nonlinear fractional stochastic delay systems from biological, ecological, and engineering applications. Full article

Three-dimensional Riemann–Liouville time-fractional integro-differential equations provide useful prototype models for diffusion and transport processes with temporal memory and weakly singular hereditary effects. Their numerical solution is challenging because of the nonlocal fractional derivative, history-dependent fractional integral term, and large-scale discrete systems arising from three-dimensional spatial discretization. In this work, an efficient high-order compact finite difference scheme is developed for solving such problems. The Riemann–Liouville fractional derivative is approximated by the weighted and shifted Grünwald difference formula, the fractional integral term is discretized by the product trapezoidal formula, and the Laplace operator is approximated by compact difference operators. The proposed scheme achieves second-order accuracy in time and fourth-order accuracy in space. Moreover, the solvability, stability, and convergence of the fully discrete three-dimensional scheme are analyzed under suitable regularity assumptions. Numerical experiments, including examples with smooth and non-smooth solutions, verify the theoretical convergence orders and demonstrate the effectiveness of the proposed method for different fractional parameters. Full article

This work examines a new methodology for the nonlinear electrohydrodynamic interfacial stability of dielectric viscoelastic liquids to enhance the predictive accuracy of microfluidic and biological applications. It tackles the intricacies of nonlinear coupled dynamics, encompassing interfacial deformation and viscoelastic stress influences. This study examines nonlinear stability, as linear stability has previously been thoroughly scrutinized. The interacting fluids are distinguished by differences in density, dielectric permittivity, permeability, viscoelastic parameters, surface tension, and their dynamic response at the perturbed interface. To simplify the mathematical organization, viscous potential flow theory is adopted. Further reduction is achieved by coupling linearized governing partial differential equations with the applicable nonlinear interfacial boundary conditions. This formulation leads to a nonlinear Mathieu oscillator, which governs the evolution of interface displacement. By adopting a non-perturbative approach, the achieved nonlinear ordinary differential equation is transformed into an equivalent linear one. Numerical solutions to the derived stability conditions reveal that the fundamental stability behavior remains qualitatively identical to both the real and complex coefficients associated with nonlinear characteristic equations describing the movement of interfacial displacement. The findings demonstrate that the Darcy number negatively influences the stability region, whereas kinematic viscosities, the Weber number, and Ohnesorge number facilitate the system’s stabilizing impact. Full article

Nonlinear biological systems exhibit complex interactions, uncertain parameters, and strong treatment-dependent dynamics, making mathematical modeling and control essential for designing reliable therapeutic intervention strategies. This study proposes a multi-input adaptive smooth sliding mode control (AMIS-SMC) framework for regulating combined radiotherapy and chemotherapy in a nonlinear tumor–immune dynamical system described by ordinary differential equations. The proposed controller integrates a hyperbolic tangent smoothing mechanism with adaptive parameter-estimation laws to compensate for uncertainty in tumor and healthy-cell growth dynamics. In this way, the method explicitly links biological-system modeling, feedback control, treatment-dose management, and parameter adaptation within a single mathematically analyzable framework. Fundamental closed-loop properties are established analytically, including positivity and boundedness of all biological state variables, asymptotic convergence of the sliding surfaces, and explicit upper bounds on the administered radiation and chemotherapeutic drug dosages. Lyapunov-based stability analysis is used to guarantee boundedness of the closed-loop signals and convergence of the sliding manifold. Numerical simulations based on a brain-tumor case study demonstrate that the proposed AMIS-SMC algorithm achieves rapid tumor suppression while administering substantially lower treatment intensities than conventional and integral SMC approaches. Under nominal conditions, the proposed controller reduces cumulative radiation and chemotherapy dosages while maintaining effective tumor mitigation. Under mismatched parameter conditions, AMIS-SMC consistently drives the tumor-cell population toward the desired equilibrium across all tested scenarios, whereas conventional and integral SMC show limited adaptability. Statistical analysis using Mann–Whitney U and Fisher’s tests further indicates that AMIS-SMC provides an effective mathematical-control and treatment-management strategy for tumor suppression in a simplified nonlinear tumor–immune biological system under parameter uncertainty. Full article

This study addresses the inverse problem of parameter identification in mathematical models of tumor growth under limited and noisy experimental data. Three classical growth models—logistic, Richards, and Gompertz—are investigated in the context of structural and practical identifiability. It is demonstrated that, despite structural identifiability, parameter estimation remains highly unstable due to the ill-posed nature of the inverse problem. A comparative analysis of the Levenberg–Marquardt method and a genetic algorithm shows that improvements in optimization strategies alone do not resolve this instability and may lead to overfitting. To overcome this limitation, a Tikhonov regularization framework is introduced for the Gompertz model, ensuring stable and physically interpretable parameter estimates. The regularized formulation provides a balance between data fidelity and parameter stability, resulting in improved agreement with experimental observations. To further validate the identified parameters, a reaction–diffusion partial differential equation model is employed. Numerical simulations demonstrate that regularized parameters lead to significantly different spatial tumor morphologies, including more compact structures with sharper interfaces, highlighting the impact of inverse problem regularization on forward model predictions. The results confirm that the primary limitation in tumor growth modeling lies in the ill-posedness of the inverse problem rather than in the choice of optimization algorithm. The proposed framework provides a robust approach for parameter identification and improves the reliability of predictive tumor growth models. Full article

Non-rigid image registration is a fundamental problem in medical imaging and a representative example of continuous transformation modeling in image processing. Diffeomorphic registration methods, such as Large Deformation Diffeomorphic Metric Mapping (LDDMM) and its PDE-constrained variants (PDE-LDDMM), provide mathematically grounded formulations with strong geometric guarantees for transformation quality. However, existing approaches face persistent trade-offs between numerical stability, accuracy, and computational efficiency. Recent work has explored implicit neural representations (INRs) and neural ordinary differential equations (NODEs) as flexible neural representations for modeling continuous transformations. Despite their increasing adoption, their practical behavior and limitations in diffeomorphic registration remain insufficiently understood. In this paper, we present a unified formulation of INR- and NODE-based registration methods within LDDMM and PDE-LDDMM, enabling a systematic and controlled comparison across architectures, sampling strategies, and numerical solvers. Our analysis reveals fundamental trade-offs between these approaches. In particular, we show that MLP-based INR formulations introduce significant computational overhead and rely on sampling strategies that can degrade smoothness and lead to the increased occurrence of non-diffeomorphic transformations at higher resolutions. Moreover, these approximations do not fully alleviate the computational cost, with some variants exceeding the costs of expensive classical optimization-based methods. In contrast, NODE-based formulations and downsampling strategies consistently provide transformations with more controlled Jacobian extrema while maintaining competitive computational performance. Among the evaluated methods, the original NODE-LDDMM and NODE-PDE-LDDMM formulations achieve the most favorable trade-offs between registration accuracy, geometric consistency, and computational efficiency. These findings provide clear insights into the design of neural representations for continuous transformation modeling, with practical implications for diffeomorphic registration and computational anatomy applications. Full article

Introduction: The recognition of Complex Post-Traumatic Stress Disorder (CPTSD) as a distinct diagnosis in ICD-11 has intensified the need to clarify its boundaries with Borderline Personality Disorder (BPD), given their symptom overlap and shared traumatic origins. Therefore, this systematic review aimed to examine whether CPTSD is distinct from BPD, assess their comorbidity and symptom overlap, identify key criteria for differential diagnosis, and explore therapeutic implications. Method: A systematic search was conducted in PubMed, Scopus, and Web of Science following PRISMA 2020 guidelines. Seven empirical studies (2015–2025) were included, comprising 2574 adults (mean age 40.4 years; 73.3% women). Methods included latent class analysis, structural equation modeling, and network analysis. The methodological quality of the included studies was evaluated independently by two reviewers using the JBI Critical Appraisal Checklist for Analytical Cross-Sectional Studies. Results: Findings consistently supported that CPTSD and BPD are empirically distinguishable, though substantially correlated, particularly within the ICD-11 framework. High symptom co-occurrence was observed, with affective dysregulation identified as the symptom most centrally connecting the two symptom networks. Self-concept emerged as the most robust differentiator: stable and persistently negative in CPTSD versus unstable and fragmented in BPD. Shame was a key affective marker of more severe presentations, and trauma severity, rather than diagnostic category, was associated with symptom variation across studies. Conclusions: CPTSD and BPD are distinct yet frequently co-occurring conditions. Differential diagnosis should focus on self-concept stability, patterns of behavioral dysregulation, and shame. Treatment requires individualized, trauma-informed, and shame-sensitive approaches, with transdiagnostic emotion regulation strategies across presentations. These conclusions should be interpreted with caution, given the predominantly cross-sectional design and methodological heterogeneity of the available evidence. Full article