Search Results (124)

The generalized Burgers–Huxley (gBH) equation arises as a canonical model in nerve-pulse propagation (generalizing the Hodgkin–Huxley/FitzHugh–Nagumo excitable-media framework), in population dynamics with Allee-threshold reaction kinetics, and in nonlinear wave propagation in dispersive media; accurate benchmark solutions are essential for quantitative predictions in these domains. We couple the fourth-order exponential time differencing scheme ETDRK4 with a Chebyshev collocation spatial discretization and a linear boundary-lifting procedure to solve the gBH equation on a bounded interval with non-homogeneous Dirichlet data. On the canonical Ismail–Raslan–Rabboh travelling-wave benchmark the scheme attains $L^{\infty }$ errors at the level of floating-point round-off (∼10⁻¹⁹ absolute, ∼10⁻¹⁵ relative) with as few as $N=2$ collocation points and a single time step of size $\Delta t=1.0$—that is, three total nodes and one ETDRK4 advance. In strongly nonlinear regimes ($\gamma =0.1, 0.3, 0.5, 0.9$) the scheme exhibits approximately $\mathcal{O}(\Delta t^{2.45})$ temporal convergence across all four parameter values, consistent with the classical Hochbruck–Ostermann order reduction for exponential integrators on parabolic PDEs with non-homogeneous Dirichlet data. Used as a high-accuracy probe, the scheme provides a diagnostic of independent interest: the wave-speed formula of Wang, Zhu and Lu, still appearing as the exact-solution benchmark in numerical studies as recently as 2020, does not satisfy the partial differential equation. The corrected formula stated by Deng and verified symbolically by Appadu and Tijani is the unique value that makes the travelling-wave ansatz a genuine solution. We derive the residual associated with Wang’s formula in closed form, $R=\frac{\gamma A_1}{2}(A_2-A_2^{\mathrm{W}})(1-v^2)$, and show both analytically and numerically that reported errors for schemes benchmarked against Wang’s formula coincide with the analytical wave-profile gap $\frac{\gamma A_1}{2}|A_2-A_2^{\mathrm{W}}|$ rather than with true scheme accuracy. At the Ismail benchmark this gap equals $3.748\times {10}^{-7}$, which matches the N- and $\Delta t$-independent plateau observed when the scheme is measured against Wang’s profile. In the nerve-pulse and excitable-media interpretation, the two formulas correspond to action-potential propagation speeds of opposite sign at the Ismail benchmark, underscoring that the correction is not a mere algebraic curiosity but changes the qualitative physical prediction of the model. Full article

This paper presents a comprehensive numerical modeling of nonlinear groundwater flow in a synthetic heterogeneous four-layer porous medium. Multilayered aquifer systems present significant modeling challenges due to nonlinear filtration and interlayer exchange processes. The mathematical model consists of four coupled nonlinear parabolic partial differential equations, where the nonlinearity arises from the dependence of hydraulic conductivity on hydraulic head. Vertical exchange between layers is described by Darcy’s law through separating aquicludes. The system is solved using a fully implicit finite-difference scheme by employing an alternating-direction implicit approach, resulting in a block-tridiagonal system of equations. The model is verified using analytical solutions and mass conservation tests. Application to a synthetic aquifer system demonstrates the model’s ability to reproduce complex transient behavior, including delayed response of upper layers to pumping and asymmetry of water-level drawdown cones due to nonlinear conductivity. The model’s greatest sensitivity is observed to the conductivity of the pumped layer and the vertical conductivity of the separating layers. The proposed approach represents a robust tool for groundwater management in structurally complex geological settings. Full article

This paper addresses the input-to-state stability (ISS) in different $L^q$-norms for a class of coupled nonlinear partial differential equations of parabolic type subject to both in-domain disturbances and Dirichlet boundary disturbances, where $q\in [1,+\infty )$. Specifically, we first prove the continuous dependence of solutions to the system on initial data and disturbances in different $L^q$-norms by using the generalized Lyapunov method, and subsequently derive ISS estimates via a density argument. The main challenge arises in handling the nonlinear coupling terms and deriving ISS small-gain conditions within the generalized Lyapunov framework, as each coupling term depends on all other state variables of the system. Full article

In this study, we utilize the ${\varphi }^6$-model expansion method to derive a diverse set of Jacobi elliptic function solutions for the conformable resonant Nonlinear Schrödinger Equation (NLSE) with parabolic law nonlinearity. As the modulus of the Jacobi elliptic functions approaches 1 and 0, the solutions transform into hyperbolic and trigonometric functions, respectively. This methodology yields various exact traveling wave solutions, including kink solitons, singular solitons, periodic solutions, and singular periodic solutions. Notably, this work represents the first investigation into identifying Jacobi elliptic function solutions for the conformable resonant NLSE. These results enhance the understanding of the nonlinear dynamical properties intrinsic to the NLSE. We use graphical illustrations to highlight the dynamical features of the solutions. Moreover, our approach showcases versatility in addressing other nonlinear partial differential equations, offering insights applicable to nonlinear optics, fluid dynamics, and quantum physics. Full article

This paper investigates altitude control of the Unmanned Aerial Vehicle (UAV) through the elevator. Elevators are flight control surfaces, which control lateral altitude by changing the pitch balance. The angle deflection along with the thrust from propulsion system is matched and guided by the system for the gain or loss of altitude over desired range of distance. A linear time-invariant elevator–altitude channel model is obtained by linearizing the six-degree-of-freedom equations of motion about a steady, level-flight trim condition. The resulting transfer function is analyzed using state-space representation and root-locus techniques, revealing that the uncompensated unity-feedback system is unstable. A proportional-integral (PI) controller is then designed and implemented in a unity-feedback configuration. The closed-loop dynamics are evaluated through time-domain simulations under step, ramp, and parabolic altitude commands, and key performance indices such as rise time, settling time, overshoot, and steady-state error are extracted. The Routh–Hurwitz criterion is used to derive an admissible gain range and to select a gain that balances response speed and robustness. The steady-state error is quantified analytically for step, ramp, and parabolic inputs, confirming a finite error for step inputs and infinite error for ramp and parabolic inputs, consistent with a type-0 system. The results demonstrate that a simple PI-based elevator controller can stabilize the linearized altitude channel and significantly improve transient performance, providing a useful baseline for more advanced nonlinear or adaptive designs in UAV flight-control applications. Full article

In this paper, we study the singular limit $q\to \infty$ for non-negative weak solutions of a doubly nonlinear parabolic Cauchy problem with absorption. Under the assumptions $m>1$, $m(p-1)>1$ and $q>p(m+2)$, we first prove that the family $\left\{u^{\left(q\right)}\right\}$ is precompact, uniformly on compact subsets of ${\mathbb{R}}^N\times (0,\infty )$, and that the whole family converges to the unique bounded weak solution of the homogeneous problem with initial datum $v_0\left(x\right)=min\{u_0\left(x\right),1\}$. We then show that the limit operator ${\mathcal{T}}_t$ is given by the composition of the homogeneous semigroup with the truncation $u_0\mapsto min\{u_0,1\}$; it forms an order-preserving semigroup, satisfies the $L^1$-contraction property, and exhibits an initial layer precisely characterized by the excess mass ${(u_0-1)}_{+}$. Finally, we compare the singular limit with the homogeneous solution and give a sharp description of the corresponding convergence defect. Full article

This paper investigates the spatial asymptotic behavior of solutions to a class of nonlinear parabolic equations defined on an exterior region in ${\mathbb{R}}^3$. By constructing a suitable weighted energy functional and employing a fractional-order differential inequality technique, we establish a sharp Phragmén–Lindelöf type alternative: the solution either ceases to exist at a finite radial distance or decays to zero as the radial variable $r\to \infty$ when the power $p>2$. In the decay case, we derive explicit polynomial type decay estimates. The analysis is conducted in unbounded exterior domains where traditional compactness arguments are not applicable, extending previous studies on semi-infinite cylinders to more complex geometric settings. Our results reveal distinct spatial behaviors compared to those observed in linear or differently nonlinear parabolic problems and can be seen as a version of Saint-Venant principle in exterior regions. Full article

The well-known shooting algorithm has produced important results in solving various linear as well as nonlinear BVPs, defined on unbounded intervals, but has become obsolete. The main difficulty lies in the numerical handling of the domain’s infiniteness. This paper presents a three-step strategy that significantly improves the traditional truncation algorithm. It consists of Chebyshev collocation, implemented as Chebfun, in conjunction with rational AAA interpolation and analytic continuation. Furthermore, and more importantly, this approach enables us to provide a thorough analysis of both possible errors in dealing with and the hidden singularities of some BVPs of real interest. A singular second-order eigenvalue problem and a fourth-order nonlinear degenerate parabolic equation, all defined on the real axis, are considered. For the latter, Chebfun provides properties-preserving solutions. Travelling wave solutions are also studied. They are highly nonlinear BVPs. The problem arises from the analysis of thin viscous film flows down an inclined plane under the competing stress due to the surface tension gradients and gravity, a long-standing concern of ours. By extending the solutions to these problems in the complex plane, we observe that the complex poles do not influence their behaviour. On the other hand, the real ones involve singularities and indicate how long solutions can be extended through continuity. Full article

This study introduces a mathematical framework for analyzing unsteady gas filtration in porous media with pressure-dependent porosity variations. The physical process is formulated as a nonlinear parabolic boundary value problem that captures the coupled interaction between pressure evolution and porosity changes during gas production. To solve the equation, a numerical strategy is developed by integrating the Alternating Direction Implicit (ADI) scheme with quasi-linearization iterations, employing finite difference discretization on a two-dimensional spatial grid. Extensive computational experiments are performed to investigate the influence of key reservoir parameters—including porosity coefficient, permeability, gas viscosity, and well production rate—on the spatiotemporal behavior of pressure and porosity during long-term extraction. The results indicate significant porosity variations near the wellbore driven by local pressure depletion, reflecting strong sensitivity of the system to formation properties. The validated numerical model provides valuable quantitative insights for optimizing reservoir management and improving production forecasting in gas field development. Overall, the proposed methodology serves as a practical tool for oil and gas engineers to assess long-term reservoir performance under diverse operational conditions and to design efficient extraction strategies that incorporate pressure-dependent formation property changes. Full article

Kolmogorov–Arnold Networks employ learnable univariate activation functions on edges rather than fixed node nonlinearities. Standard B-spline implementations require $O\left(3KW\right)$ parameters per layer (K basis functions, W connections). We introduce shared Gaussian radial basis functions with learnable centers ${\mu }_k^{\left(l\right)}$ and widths ${\sigma }_k^{\left(l\right)}$ maintained globally per layer, reducing parameter complexity to $O(KW+2LK)$ for L layers—a threefold reduction, while preserving Sobolev convergence rates $O\left(h^{s-\mathrm{\Omega }}\right)$. Width clamping at ${\sigma }_{min}={10}^{-6}$ and tripartite regularization ensure numerical stability. On MNIST with architecture $[784,128,10]$ and $K=5$, RBF-KAN achieves $87.8\%$ test accuracy versus $89.1\%$ for B-spline KAN with $1.4\times$ speedup and 33% memory reduction, though generalization gap increases from $1.1\%$ to $2.7\%$ due to global Gaussian support. Physics-informed neural networks demonstrate substantial improvements on partial differential equations: elliptic problems exhibit a $45\times$ reduction in PDE residual and maximum pointwise error, decreasing from $1.32$ to $0.18$; parabolic problems achieve a $2.1\times$ accuracy gain; hyperbolic wave equations show a $19.3\times$ improvement in maximum error and a $6.25\times$ reduction in $L^2$ norm. Superior hyperbolic performance derives from infinite differentiability of Gaussian bases, enabling accurate high-order derivatives without polynomial dissipation. Ablation studies confirm that coefficient regularization reduces mean error by 40%, while center diversity prevents basis collapse. Optimal basis count $K\in [3,5]$ balances expressiveness and overfitting. The architecture establishes Gaussian RBFs as efficient alternatives to B-splines for learnable activation networks with advantages in scientific computing. Full article

In this work, we develop a disturbance suppression-oriented fuzzy sliding mode secured sampled-data controller for third-order parabolic partial differential equations that ought to cope with nonlinearities, hybrid cyber attacks, and modeled disturbances. This endeavor is mainly driven by formulating an observer model with a T–S fuzzy mode of execution that retrieves the latent state variables of the perceived system. Progressing onward, the disturbance observers are formulated to estimate the modeled disturbances emerging from the exogenous systems. In due course, the information received from the system and disturbance estimators, coupled with the sliding surface, is compiled to fabricate the developed controller. Furthermore, in the realm of security, hybrid cyber attacks are scrutinized through the use of stochastic variables that abide by the Bernoulli distributed white sequence, which combat their unpredictability. Proceeding further in this framework, a set of linear matrix inequality conditions is established that relies on the Lyapunov stability theory. Precisely, the refined looped Lyapunov–Krasovskii functional paradigm, which reflects in the sampling period that is intricately split into non-uniform intervals by leveraging a fractional-order parameter, is deployed. In line with this pursuit, a strictly $({\Phi }_1,{\Phi }_2,{\Phi }_3)-\varrho$ dissipative framework is crafted with the intent to curb norm-bounded disturbances. A simulation-backed numerical example is unveiled in the closing segment to underscore the potency and efficacy of the developed control design technique. Full article

This study examines the generalized nonlinear Hunter–Saxton (HS) model: ${\Phi }_{tx}=\Phi {\Phi }_{xx}+\gamma {\Phi }_x^2,\gamma \ne 0,$ that describes the evolution of spatial potential and angular velocity in the vector field of nematic liquid crystals. Closed-form nematicons are derived via the order reduction of the traveling wave ODE. The qualitative structures are analyzed for different values of the nonlinear parameter $\gamma$. The solutions are graphically depicted to discover rich nematicon geometries including parabolic, cuspon, kink, and singular wave structures. A comprehensive dynamic analysis of the reduced nonlinear ordinary system is performed using the phase plane method, which helps to reveal the non-isolated continuity of equilibrium and the role of singular manifolds in shaping the system’s sensitivity and stability. Bifurcation cases are investigated for distinct values of $\gamma$, and various transitions in trajectory geometry and semi-stability features are shown. The novelty appears in the comprehensive integrating of analytic and dynamic characterizations, through global phase and bifurcation analysis, of the generalized HS equation (HSE), which uncovers the control of nonlinear coefficient $\gamma$ in governing the geometry and stability of the nematicons. Also, the analysis confirms the non-chaotic nature of the associated two-dimensional system, compatible with the Poincaré–Bendixson theorem. Full article

By solving the three-dimensional Schrödinger equation with a second-order implicit Finite Difference Method (FDM), the combined effects of temperature, morphology, hydrostatic pressure, and transverse electric field on the nonlinear optical rectification (NOR) of GaAs/Al_εGa_1−εAs elliptical quantum rings are examined. The NOR amplitude is twelve times enhanced and a noticeable blue shift is induced in the THz region when the electric field is increased. Consequently, with the electric field of 1 × 10⁵ V/m, the NOR magnitude achieves its maximum value of 17.116 × 10⁵ m/V. Additionally, when the electric field is aligned along one side of the system’s in-plane cross-section, the strongest amplification takes place. However, with corresponding spectrum shifts, the NOR intensity rises with temperature and falls with hydrostatic pressure. Additionally, changing the transverse profile of the quantum ring from triangular to parabolic broadens the carrier wave functions and has a considerable impact on the NOR coefficient. These findings provide important information for the construction of high-performance, tunable THz optoelectronic devices by demonstrating effective external and structural tuning of NOR. Full article

Tumor angiogenesis models based on coupled nonlinear parabolic partial differential equations require solving stiff systems where explicit time-stepping methods impose severe stability constraints on the time step size. Implicit–Explicit (IMEX) schemes relax this constraint by treating diffusion terms implicitly and reaction–chemotaxis terms explicitly, reducing each time step to a single linear system solution. However, standard Gaussian elimination with partial pivoting exhibits cubic complexity in the number of spatial grid points, dominating computational cost for realistic discretizations in the range of 400–800 grid points. This work presents a CUDA-based parallel algorithm that accelerates the IMEX scheme through GPU implementation of three core computational kernels: pivot finding via atomic operations on double-precision floating-point values, row swapping with coalesced memory access patterns, and elimination updates using optimized two-dimensional thread grids. Performance measurements on an NVIDIA H100 GPU demonstrate speedup factors, achieving speedup factors from 3.5× to 113× across spatial discretizations spanning $M\in [25,800]$ grid points relative to sequential CPU execution, approaching 94.2% of the theoretical maximum speedup predicted by Amdahl’s law. Numerical validation confirms that GPU and CPU solutions agree to within twelve digits of precision over extended time integration, with conservation properties preserved to machine precision. Performance analysis reveals that the elimination kernel accounts for nearly 90% of total execution time, justifying the focus on GPU parallelization of this component. The method enables parameter studies requiring $\sim {10}^4$ PDE solves, previously computationally prohibitive, facilitating model-driven investigation of anti-angiogenic therapy design. Full article

This paper considers the question of local dynamics of the simplest non-periodic chains of nonlinear first-order equations with two-sided couplings. The main attention is paid to the study of chains with a large number N of elements. The critical cases in the problem of stability of the zero equilibrium state are identified. Questions about bifurcations of regular and irregular solutions are considered. Analogues of normal forms are constructed, the so-called quasinormal forms, which are special nonlinear equations of parabolic type. Their nonlocal dynamics determine the local structure of solutions to the original problem. Bifurcation problems for quasinormal forms are considered, and interestingly, the boundary conditions for them are not classical. The asymptotics of both regular and irregular solutions are constructed. The latter have the most complex structure. In particular, for negative values of the coupling parameter between elements, continual families of equilibrium states, cycles, and more complex structures can arise in the chain. Full article