Search Results (546)

Sliding mode control (SMC) is a robust nonlinear control framework that enforces system trajectories onto predefined manifolds, providing strong robustness guarantees against uncertainties. However, SMC inherently introduces unwanted transients or chattering in system state trajectories, which may cause issues especially for sensitive applications such as regulation of drug administration. This paper proposes a multi-input smooth sliding mode control (MISSMC) strategy that combines radiotherapy and chemotherapy for a nonlinear tumor–immune dynamical system described by ordinary differential equations. The closed-loop system is first analyzed to establish key qualitative properties: all state variables remain positive and bounded, the sliding surfaces exhibit asymptotic convergence, and explicit analytical upper bounds on the cumulative therapy doses are derived under clinically motivated constraints. On this basis, a smooth hyperbolic-tangent sliding manifold and associated control law are designed to regulate the radiation and drug infusion rates. While the use of a hyperbolic-tangent smoothing function effectively suppresses chattering, it introduces a small steady-state error due to the presence of a boundary layer. To address this limitation, integral action is incorporated into the sliding surfaces, ensuring asymptotic convergence of tumor state and reducing residual steady-state error, while enhancing robustness against model uncertainties and parameter variations. Numerical simulations, based on a brain-tumor case study, show that the proposed smooth SMC markedly suppresses transient overshoots in both states and control inputs, while preserving effective tumor reduction. Compared with a conventional (non-smooth) SMC scheme, the MISSMC controller reduces baseline radiation and chemotherapy intensities on average by roughly 70%. Similarly, MISSMC lowers the overall cumulative doses on average by about 40%, without degrading the therapeutic outcome. The resulting integral smooth SMC framework therefore offers a rigorous nonlinear-systems approach to designing combined radio-chemotherapy protocols with guaranteed positivity, boundedness, and asymptotic stabilization of the closed-loop system, together with explicit bounds on the control inputs. 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 $O\left(h^{s-\Omega }\right)$. 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

We investigate the Lagrange formulation for the one-dimensional Saint Venant–Exner system. The system describes shallow-water equations with a bed evolution, for which the bedload sediment flux depends on the velocity, $Q\left(t,x\right)=A_g{u}^m,m\ge 1$. In terms of the Lagrange variables, the nonlinear hyperbolic system is reduced to one master third-order nonlinear partial differential equation. We employ Lie’s theory and find the Lie symmetry algebra of this equation. It was found that for an arbitrary parameter m, the master equation possesses four Lie symmetries. However, for $m=3$, there exists an additional symmetry vector. We calculate a one-dimensional optimal system for the Lie algebra of the equation. We apply the latter for the derivation of invariant functions. The invariants are used to reduce the number of the independent variables and write the master equation into an ordinary differential equation. The latter provides similarity solutions. Finally, we show that the traveling-wave reductions lead to nonlinear maximally symmetric equations which can be linearized. The analytic solution in this case is expressed in closed-form algebraic form. Full article

With the advancement in deep-sea resource development, the creep behavior of deep-sea remolded sediments under coupled temperature, confining pressure (σ₃), and stress effects has become a critical issue threatening engineering stability. The traditional Singh–Mitchell model, limited by its neglect of temperature effects and prediction of infinite strain, struggles to meet deep-sea environmental requirements. Based on low-temperature, high-pressure triaxial tests (with temperatures ranging from 4 to 40 °C and confining pressures ranging from 100 to 300 kPa), this study proposes a modified model incorporating temperature–stress–time coupling. The model introduces a hyperbolic creep strain rate decay function to achieve strain convergence, establishes a saturated strain–stress exponential relationship, and quantifies the effect of temperature on characteristic time via coupling through the Arrhenius equation. The modified model demonstrates R² values > 0.96 for full-condition creep curves. The results show several key findings: a 10 °C increase in temperature leads to a 30–50% growth in the steady-state creep rate; a 100 kPa increase in confining pressure enhances long-term strength by 20–30%. 20 °C serves as a critical temperature point. At this point, strain amplification reaches 2.1 times that of low-temperature ranges. These experimental findings provide crucial theoretical foundations and technical support for incorporating soil creep effects in deep-sea engineering design. Full article

In this paper, we consider the solution of the initial-value problem in thermodiffusion in a solid body in three-dimensional space. Not only do we prove the behavior of the solution over time, but we identify some of its technical aspects as well. Stabilizing a thermodiffusion system in solids is essential for understanding the long-time behaviour of some materials, which are used in mechanical engineering; this is especially important for materials used in aviation, not only in civil aviation but in the air force as well. Full article

In this work, we investigate the solutions of fractional singular m-dimensional pseudo-hyperbolic equations. To address these equations effectively, we generalise the Natural transform to the multi-dimensional case and examine several of its essential properties. By integrating this transform with the Adomian Decomposition Method, we formulate the Multi-Dimensional Natural Adomian (MNA) Method, which provides a systematic framework for solving the considered equations under given initial conditions. The resulting solutions are expressed as rapidly convergent series that approach either exact or highly accurate approximations. To illustrate the practicality and robustness of the proposed method, two representative examples are included, demonstrating the construction of the solution series and the derivation of approximate or exact solutions. Full article

This paper presents the construction of exact wave solutions for the generalized nonlinear Schrödinger equation (NLSE) with second-order spatiotemporal dispersion using the modified exponential rational function method (mERFM). The NLSE plays a vital role in various fields such as quantum mechanics, oceanography, transmission lines, and optical fiber communications, particularly in modeling pulse dynamics extending beyond the traditional slowly varying envelope estimation. By incorporating higher-order dispersion and nonlinear effects, including cubic–quintic nonlinearities, this generalized model provides a more accurate representation of ultrashort pulse propagation in optical fibers and oceanic environments. A wide range of soliton solutions is obtained, including bright and dark solitons, as well as trigonometric, hyperbolic, rational, exponential, and singular forms. These solutions offer valuable insights into nonlinear wave dynamics and multi-soliton interactions relevant to shallow- and deep-water wave propagation. Conservation laws associated with the model are also derived, reinforcing the physical consistency of the system. The stability of the obtained solutions is investigated through the analysis of modulation instability (MI), confirming their robustness and physical relevance. Graphical representations based on specific parameter selections further illustrate the complex dynamics governed by the model. Overall, the study demonstrates the effectiveness of mERFM in solving higher-order nonlinear evolution equations and highlights its applicability across various domains of physics and engineering. Full article

The paper concerns the solution of the ordinary differential equation $y^{″}\pm x^{m}y=0$, which may be designated the generalized Airy equation, since the original Airy equation corresponds to the particular case $m=1$ with the + sign. The solutions may be designated generalized circular (hyperbolic) sines and cosines for the + (−) sign, since the particular case $m=0$ corresponds to the elementary circular (hyperbolic) sines and cosines. There are 3 cases of solution of the generalized Airy equation, depending on the parameter m: (I) for m a non-negative integer, the coefficient $x^m$ is an analytic function, and the solutions are also analytic series; (II) for m complex other than an integer, the coefficient $x^m$ has a branch point at the origin, and the solutions also have a branch point multiplied by an analytic series; (III) for m a negative integer, the coefficient $x^m$ has a pole of order m, and the generalized Airy equation is singular. Case III has four subcases: (III-A) for $m=-1$, the coefficient $x^{-1}$ is a simple pole, and the solutions are Frobenius–Fuchs series of two kinds; (III-B) for $m=-2$, the coefficient is a double pole, and the solutions are a combination of elementary functions, namely exponential, logarithmic, and circular (hyperbolic) sine and cosine for the + (−) sign; (III-C,D) for $m=-3,-4,\dots$, the coefficient is a pole of multiplicity $\left|m\right|$, and the generalized Airy differential equation has an irregular singularity of degree $\left|m\right|-2$ at the origin. In the sub-cases (III-C,D), the solutions can be obtained by inversion as asymptotic series of descending powers specified by (III-C) Frobenius–Fuchs series of two kinds for a triple pole $m=-3$; (III-D) for higher-order poles $m=-4,-5,\dots$ by generalized circular (hyperbolic) sines and cosines of $1/x$. It is shown that in all cases the ascending and descending series are absolutely and uniformly convergent with the n-th term decaying like $O\left(n^2\right)$. This enables the use of a few terms of the series to obtain tables and plot graphs of the solutions of the generalized Airy differential equation as generalized circular and hyperbolic sines and cosines for several values of the parameter m. As a physical application, it is shown that the generalized circular (hyperbolic) cosines and sines specify the motion of a linear oscillator with natural frequency a power of time in the oscillatory (monotonic) case when the origin is an attractor (repeller). Full article

The exact analytical solutions of a new combined Kairat-II-X differential equation are presented. The related model is investigated by combining the enhanced modified extended tanh function method and the modified $tan(\varphi /2)$-expansion method. Then, a wide range of solitary wave solutions with unknown coefficients are extracted in a variety of shapes, including dark, bright, bell-shaped, kink-type, combine, and complex solitons, exponential, hyperbolic, and trigonometric function solutions. To offer physical insight, some of the identified solutions are presented in figures. Also, 3D, 2D, and 2D density profiles of the obtained outcomes are illustrated in order to examine their dynamics with the choices of parameters involved. Based on the obtained findings, we can assert that the suggested computational approaches are efficient, dynamic, well-structured, and valuable for tackling complex nonlinear problems in several fields, including symbolic computations. The bifurcation analysis and sensitivity analysis are employed to comprehend the dynamical system. We assume that our findings will be very beneficial in improving our understanding of the waves that manifest in solids. Full article

We introduce a BiHom-type skew-symmetric bracket on general linear Lie algebra $GL\left(V\right)$ built from two commuting inner automorphisms $\alpha =A{d}_{\psi }$ and $\beta =A{d}_{\varphi }$, with $\psi ,\varphi \in GL\left(V\right)$ and integers $i,j$. We prove that $(GL\left(V\right),{[·,·]}_{(\psi ,\varphi )}^{(i,j)},\alpha ,\beta )$ is a BiHom–Lie algebra, and we study the Lax equation obtained by replacing the commutator in the finite nonperiodic Toda lattice by this bracket. For the symmetric choice $\varphi =\psi$ with $(i,j)=(0,0)$, the deformed flow is equivariant under conjugation and becomes gauge-equivalent, via $\tilde{L}={\psi }^{-1}L\psi$, to a Toda-type Lax equation with a conjugated triangular projection. In particular, scalar deformations amount to a constant rescaling of time. On embedded $2\times 2$ blocks, we derive explicit trigonometric and hyperbolic formulae that make symmetry constraints (e.g., tracelessness) transparent. In the asymmetric hyperbolic case, we exhibit a trace obstruction showing that the right-hand side is generically not a commutator, which amounts to symmetry breaking of the isospectral property. We further extend the construction to the weakly coupled Toda lattice with an indefinite metric and provide explicit $2\times 2$ solutions via an inverse-scattering calculation, clarifying and correcting certain formulas in the literature. The classical Toda dynamics are recovered at special parameter values. Full article

This study presents an efficient numerical scheme for solving the generalized nonlinear time-fractional Klein–Gordon equation. The Caputo time-fractional derivative is discretized using a conventional finite-difference approach, while the spatial domain is approximated with uniform hyperbolic polynomial B-splines. These discretizations are coupled through the $\theta$-weighted scheme. The uniform hyperbolic polynomial B-spline framework extends classical spline theory by incorporating hyperbolic functions, thereby enhancing flexibility and smoothness in curve and surface representations—features particularly useful for problems exhibiting hyperbolic characteristics. A rigorous stability and convergence analysis of the proposed method is provided. The effectiveness of the scheme is further validated through numerical experiments on benchmark problems. The results demonstrate up to two orders of magnitude improvement in $L_{\infty }$ error norms compared to prior spline methods. This substantial accuracy enhancement highlights the robustness and efficiency of the proposed approach for fractional partial differential equations. Full article

This study employs the modified extended direct algebraic method (MEDAM) to investigate the generalized nonlinear fractional $(3+1)$-dimensional wave equation with gas bubbles. This advanced analytical framework is used to construct a comprehensive class of exact wave solutions and explore the associated dynamical characteristics of diverse wave structures. The analysis yields several categories of soliton solutions, including rational, hyperbolic (sech, tanh), and trigonometric (sec, tan) function forms. To the best of our knowledge, these soliton solutions have not been previously documented in the existing literature. By selecting appropriate standards for the permitted constraints, the qualitative behaviors of the derived solutions are illustrated using polar, contour, and two- and three-dimensional surface graphs. Furthermore, a stability analysis is performed on the obtained soliton solutions to ascertain their robustness and dynamical stability. The suggested analytical approach not only deepens the theoretical understanding of nonlinear wave phenomena but also demonstrates substantial applicability in various fields of applied sciences, particularly in engineering systems, mathematical physics, and fluid mechanics, including complex gas–liquid interactions. Full article

In this paper, we introduce the concept of s-convex-averaging domains, which are extensions of circular and irregular convex domains, by using s-convex functions and generalized Orlicz norms. Based on the quasi-hyperbolic metric and L^p-averaging domains, several fundamental properties of s-convex-averaging domains are characterized. These properties are applied to the domains of a class of two-dimensional diffusion-wave equations. Furthermore, we establish intrinsic relationships between the considered partial differential equations and the geometric structure of s-convex-averaging domains. Finally, the embedding inequality for the solutions of these kinds of partial differential equations is derived. Full article

This paper investigates the reverse space-time nonlocal nonlinear Schrödinger (NNLS) equation, which arises in nonlinear optics, Bose–Einstein condensation, integrable systems, and plasma physics. Several classes of exact solutions are constructed using multiple analytical techniques. First, traveling wave solutions of Jacobi elliptic, hyperbolic, and trigonometric function types are ultimately obtained by employing a traveling wave transformation combined with a Weierstrass-type Riccati equation expansion method. Second, Lie symmetry analysis is applied to the NNLS equation, and the corresponding infinitesimal generators are determined. Using these generators, the original equation is reduced to local and nonlocal ordinary differential equations (ODEs), whose invariant solutions are subsequently obtained through integration. Finally, the NNLS equation is generalized to a multi-component system, for which the general form of the infinitesimal symmetries is derived. Symmetry reductions of the extended system yield further classes of reduced ODEs. In particular, the general form of the multi-component solutions is derived. Full article

This article investigates the impact of Arrhenius energy and the radius of a nanoparticle subject to an irregular heat source on tangent-hyperbolic nanofluid flow over a stretching sheet with nonlinear radiation. The convective boundary effect, higher-order slip, and micropolarity are all included for a water-based $Cu$ nanofluid. The present study investigates the significance of a nanoparticle’s radius under inclined MHD conditions. The thermally convective flow of the nanofluid is optimized for the heat-transfer rate using the response surface technique. The modeled governing equations are converted into a system of first-order ODEs using the proper similarity transformations, and the BVP5C algorithm—a finite-difference-based solver—is then used to solve these ODEs numerically. Microrotation, thermal boundary-layer thickness, and the skin-friction coefficient all decrease as the nanoparticle radius increases. The thermal layer thickens as the Biot number increases. As the higher-order slip parameter coefficient increases, the results indicate that the skin friction and local Nusselt number fall. Using tables, figures, contour plots, and surface plots, the effects of several influencing factors on the rates of heat and mass transfer, as well as on the skin-friction factor, are demonstrated. The study uses “Response Surface Methodology” (RSM) in conjunction with “Analysis of Variance” (ANOVA) to optimize the most important factors, which are probably the magnetic parameter and the nanoparticle radius that control the flow and heat-transfer properties. Additionally, with a Nusselt number $R^2$ value of $99.96$, indicating an excellent fit, the suggested model exhibits amazing precision. The reliability and efficiency of the estimated model are assessed using the residual versus fitted plot. Full article