Search Results (96)

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

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

We consider a system of parabolic partial differential equations with a cross-diffusion phenomenon. Previous results showed that a weak solution exists to the semiconductor model with electron-hole scattering. In this work, we show that this weak solution exists uniquely. For weak solutions of cross-diffusion systems, few uniqueness results have been derived. Among these uniqueness results, we require that weak solutions are bounded. The weak solution of the semiconductor model may not be bounded, so its uniqueness is very difficult to prove. We rely on the structural character of this model to derive a sequence of weak solutions. By considering the limit of this sequence of solutions, we show that the weak solution of the semiconductor model is unique. Full article

Based on existing models, this paper incorporates some key ecological factors, thereby obtaining a class of eco-epidemiological models that can more objectively reflect natural phenomena. This model simultaneously integrates disease dynamics within the prey population and the Beddington–DeAngelis functional response, thus achieving an organic combination of ecological dynamics, epidemic transmission, and spatial movement under time-varying environmental conditions. The proposed framework significantly enhances ecological realism by simultaneously accounting for spatial dispersal, predator–prey interactions, disease transmission within prey species, and seasonal or temporal variations, providing a comprehensive mathematical tool for analyzing complex eco-epidemiological systems. The theoretical results obtained from this study can be summarized as follows: Firstly, the existence and uniqueness of globally positive solutions for any positive initial data are rigorously established, ensuring the well-posedness and biological feasibility of the model over extended temporal scales. Secondly, analytically tractable sufficient conditions for uniform population persistence are derived, which elucidate the mechanisms of species coexistence and biodiversity preservation even under sustained epidemiological pressure. Thirdly, by employing innovative applications of differential inequalities and fixed point theory, the existence and uniqueness of a positive spatially homogeneous periodic solution in the presence of time-periodic coefficients are conclusively demonstrated, capturing essential rhythmicities inherent in natural systems. Fourthly, through a sophisticated combination of the upper and lower solution method for parabolic partial differential equations and Lyapunov stability theory, the global asymptotic stability of this periodic solution is rigorously established, offering a powerful analytical guarantee for long-term predictive modeling. Beyond theoretical contributions, these research findings provide actionable insights and quantitative analytical tools to tackle pressing ecological and public health challenges. They facilitate the prediction of thresholds for maintaining ecosystem stability using real-world data, enable the analysis and assessment of disease persistence in spatially structured environments, and offer robust theoretical support for the planning and design of wildlife management and conservation strategies. The derived criteria support evidence-based decision-making in areas such as controlling zoonotic disease outbreaks, maintaining ecosystem stability, and mitigating anthropogenic impacts on ecological communities. A representative numerical case study has been integrated into the analysis to verify all of the theoretical findings. In doing so, it effectively highlights the model’s substantial theoretical value in informing policy-making and advancing sustainable ecosystem management practices. 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

A new class of poroelastic dynamic contact problems stemming from hydraulic fracture theory is introduced and studied. The two-phase medium consists of a solid phase and pores which are saturated with a Newtonian fluid. The porous body contains a fluid-driven crack endowed with non-penetration conditions for the opposite crack surfaces. The poroelastic model is described by a coupled system of hyperbolic–parabolic partial differential equations under the unilateral constraint imposed on displacement. After full discretization using finite-element and Hilber–Hughes–Taylor methods, the well-posedness of the resulting variational inequality is established. Formulation of the complementarity conditions with the help of a minimum-based merit function is used for the semi-smooth Newton method of solution presented in the form of a primal–dual active set algorithm which is tested numerically. Full article

The main purpose of the present paper is to investigate the problem of estimating the time-varying coefficient in a stochastic parabolic equation driven by a sub-fractional Brownian motion. More precisely, we introduce a kernel-type estimator for the time-varying coefficient $\theta \left(t\right)$ in the following evolution equation:$$\left\{\begin{array}{c}{\displaystyle du(t,x)=(A_0+\theta \left(t\right)A_1)u(t,x)dt+d{\xi }^H(t,x),x\in [0,1],t\in (0,T],}\hfill \\ {\displaystyle u(0,x)=u_0\left(x\right),}\hfill \end{array}\right.$$ where ${\xi }^H(t,x)$ is a cylindrical sub-fractional Brownian motion in ${\mathbb{L}}^2\left([0,T]\times [0,1]\right)$, and $A_0+\theta \left(t\right)A_1$ is a strongly elliptic differential operator. We obtain the asymptotic mean square error and the limiting distribution of the proposed estimator. These results are proved under some standard conditions on the kernel and some mild conditions on the model. Finally, we give an application for the confidence interval construction. Full article

In this article, we propose a novel nonlinear cross-diffusion framework to model the distribution of susceptible and infected individuals within their habitat using a reduced SIR model that incorporates saturated incidence and treatment rates. The study investigates solution boundedness through the theory of parabolic partial differential equations, thereby validating the proposed spatio-temporal model. Through the implementation of the suggested cross-diffusion mechanism, the model reveals at least one non-constant positive equilibrium state within the susceptible–infected (SI) system. This work demonstrates the potential coexistence of susceptible and infected populations through cross-diffusion and unveils Turing instability within the system. By analyzing codimension-2 Turing–Hopf bifurcation, the study identifies the Turing space within the spatial context. In addition, we explore the results for Turing–Bogdanov–Takens bifurcation. To account for seasonal disease variations, novel perturbations are introduced. Comprehensive numerical simulations illustrate diverse emerging patterns in the Turing space, including holes, strips, and their mixtures. Additionally, the study identifies non-Turing and Turing–Bogdanov–Takens patterns for specific parameter selections. Spatial series and surfaces are graphed to enhance the clarity of the pattern results. This research provides theoretical insights into the implications of cross-diffusion in epidemic modeling, particularly in contexts characterized by localized mobility, clinically evident infections, and community-driven isolation behaviors. Full article

This paper studies an averaged Linear Quadratic Regulator (LQR) problem for a parabolic partial differential equation (PDE), where the system dynamics are affected by uncertain parameters. Instead of assuming a deterministic operator, we model the uncertainty using a probability distribution over a set of possible system dynamics. This approach extends classical optimal control theory by incorporating an averaging framework to account for parameter uncertainty. We establish the existence and uniqueness of the optimal control solution and analyze its convergence as the probability distribution governing the system parameters changes. These results provide a rigorous foundation for solving optimal control problems in the presence of parameter uncertainty. Our findings lay the groundwork for further studies on optimal control in dynamic systems with uncertainty. Full article

Modeling of impurity diffusion processes in a multiphase randomly inhomogeneous body is performed using the Feynman diagram technique. The impurity diffusion equations are formulated for each of the phases separately. Their random boundaries are subject to non-ideal contact conditions for concentration. The contact mass transfer problem is reduced to a partial differential equation describing diffusion in the body as a whole, which accounts for jump discontinuities in the searched function as well as in its derivative at the stochastic interfaces. The obtained problem is transformed into an integro-differential equation involving a random kernel, whose solution is constructed as a Neumann series. Averaging over the ensemble of phase configurations is performed. The Feynman diagram technique is developed to investigate the processes described by parabolic partial differential equations. The mass operator kernel is constructed as a sum of strongly connected diagrams. An integro-differential Dyson equation is obtained for the concentration field. In the Bourret approximation, the Dyson equation is specified for a multiphase randomly inhomogeneous medium with uniform phase distribution. The problem solution, obtained using Feynman diagrams, is compared with the solutions of diffusion problems for a homogeneous layer, one having the coefficients of the base phase and the other having the characteristics averaged over the body volume. Full article

In this paper, a structure-preserving local discontinuous Galerkin (LDG) method is proposed for parabolic stochastic partial differential equations with periodic boundary conditions and multiplicative noise. It is proven that under certain conditions, this numerical method is stable in the $L^2$ sense and can preserve energy conservation. The optimal spatial error estimate in the mean square sense can reach $n+1$ if the degree of the polynomial is n. The correctness of the theoretical results is verified through numerical examples. Full article

This article introduces a new fuzzy double integral transformation called fuzzy double Yang transformation. We review some of the main properties of the transformation and find the conditions for its existence. We prove the theorems for partial derivatives and fuzzy unitary convolution. All of the new results are applied to find an analytical solution to the fuzzy parabolic Volterra integro-differential equation (FPVIDE) with a suitably selected memory kernel. In addition, a numerical example is provided to illustrate how the proposed method might be helpful for solving FPVIDE utilizing symmetric triangular fuzzy numbers. Compared with other symmetric transforms, we conclude that our new approach is simpler and needs less calculations. Full article

This paper focuses on the investigation of the Yu–Toda–Sasa–Fukuyama (YTSF) equation in its three-dimensional form. Based on the well-known Euler operator, a set of seven non-singular local multipliers is explored. Using these seven non-singular multipliers, the corresponding local conservation laws are derived. Additionally, an auxiliary potential-related system of partial differential equations (PDEs) is constructed. This study delves into nonlocal systems, which reveal numerous intriguing exact solutions of the YTSF equation. The nonlinear systems exhibit stable structures such as kink solitons, representing transitions, and breather or multi-solitons, modeling localized energy packets and complex interactions. These are employed in materials science, optics, communications, and plasma. Additionally, patterns such as parabolic backgrounds with ripples inform designs involving structured or varying media such as waveguides. Full article

This study presents a novel fifth-order iterative method for solving nonlinear systems derived from a modified combination of Jarratt and Newton schemes, incorporating a frozen derivative of the Jacobian. The method is applied to approximate solutions of the nonlinear convection–diffusion equation. A MATLAB script function was developed to implement the approach in two stages: first, discretizing the equation using the Crank–Nicolson Method, and second, solving the resulting nonlinear systems using Newton’s iterative method enhanced by a three-step Jarratt variant. A comprehensive analysis of the results highlights the method’s convergence and accuracy, comparing the numerical solution with the exact solution derived from linear parabolic partial differential transformations. This innovative fifth-order method provides an efficient numerical solution to the nonlinear convection–diffusion equation, addressing the problem through a systematic methodology that combines discretization and nonlinear equation solving. The study underscores the importance of advanced numerical techniques in tackling complex problems in physics and mathematics. Full article