Search Results (170)

In this paper, a singular stationary Gierer–Meinhardt-type activator–inhibitor system with homogeneous Neumann boundary conditions is investigated. The model incorporates singular interaction terms together with sublinear source nonlinearities, which describe production mechanisms that grow slowly at high concentrations. The main objective is to establish the existence of positive solutions for the associated elliptic system defined on a bounded domain with a smooth boundary. To overcome the lack of a variational structure caused by the singular nonlinearities, an approximation scheme combined with the Galerkin method and a variant of Brouwer’s fixed point theorem is employed. A suitable perturbed problem was introduced in order to regularize the singular terms and obtain uniform a priori estimates. It was proved that, for sufficiently small parameters $\lambda$ and $\mu$, the system admits a positive weak solution in $H^1(\Omega )$. Furthermore, lower bounds for the components of the solution were derived, and regularity results were obtained through bootstrap arguments. These results extend existing studies on singular Gierer–Meinhardt systems by incorporating sublinear nonlinearities and Neumann boundary conditions, which are relevant in models of biological pattern formation and diffusion–reaction processes. Full article

The inverse problem under consideration concerns a one-dimensional parabolic equation whose thermal diffusivity takes the quadratic-in-space form $a_s\left(\tau \right){\kappa }^2+b_s\left(\tau \right)\kappa +c_s\left(\tau \right)$. The unknowns are three time-dependent coefficients $a_s\left(\tau \right),b_s\left(\tau \right),c_s\left(\tau \right)$ together with the temperature field $T(\kappa ,\tau )$. The direct problem supplies initial data, Neumann boundary conditions, and three over-determination conditions: two boundary temperatures and the spatial integral of T. We prove two theorems. The first theorem establishes the local-in-time existence of a solution under explicit regularity and sign conditions on the given data $\xi ,{\nu }_k,{\delta }_{\ell },\theta$ and compatibility at $\tau =0$. The second theorem guarantees the uniqueness of this solution. Despite uniqueness, the inverse reconstruction remains ill-posed: small perturbations in the over-specified data can cause large deviations in the recovered coefficients. For the forward model, we implement two numerical schemes: (i) a Crank–Nicolson finite difference methodology (CN-FDM) on a uniform grid and (ii) a semi-discretized Crank–Nicolson approach combined with Bell spectral collocation in space (CN–Bell). The inverse step minimizes a Tikhonov-regularized least-squares functional using MATLAB’s (R2026a) lsqnonlin. Two numerical examples (smooth and non-smooth), tested with both exact synthetic data and artificially added noise, demonstrate stable and accurate coefficient reconstructions. The framework applies directly to heat conduction and porous media flow where diffusivity varies quadratically in space. Full article

The analytical investigations of 3-miktoarm star (3-arm $\mu$-star) copolymers of type $A_{2}B$ and $A{B}_2$ are performed in the framework of mean-field approximation and Flory–Huggins theory. The total entropy of mixing and the Helmholtz free energy of interaction are calculated for the number $N_A$ monomers of type A and number $N_B$ monomers of type B, respectively. The results confirm that the Helmholtz free energy of miktoarm star copolymers differs from that of polymer blends. The temperature dependence of the Helmholtz free energy allowed us to construct a phase diagram of the solution of miktoarm star copolymers, showing regions of stability, instability, and metastability. The analytical results confirm that a miktoarm star copolymer is not merely a mixture of different homo-arm star polymers and are consistent with a previous investigation performed by liquid chromatography under the critical conditions. Moreover, we performed molecular dynamics simulations of a dilute solution of 3-miktoarm star copolymer of type $A_{2}B$ with a certain number of beads (300 + 300 + 200 + 1) and star copolymer of type $A{B}_2$ with number of beads (300 + 200 + 200 + 1), accordingly. The calculations of the radius of gyration and monomer density profiles of the 3-miktoarm star copolymers of type $A_{2}B$ and $A{B}_2$ in confined geometry of two repulsive surfaces (Dirichlet–Dirichlet boundary conditions) and one repulsive and other one attractive surface (Dirichlet–Neumann boundary conditions) by molecular dynamics simulations are performed. The obtained analytical and numerical results indicate that a dilute solution of miktoarm star copolymers can be used in biotechnology and medicine for drug and gene transmission as well as for the production of new functional materials. Full article

In this work, a partitioned coupling algorithm is developed by integrating the improved discrete velocity method (IDVM) with the lattice Boltzmann flux solver (LBFS) to address conjugate heat transfer (CHT) in microscale systems across all flow regimes. Specifically, the flow field is solved by the IDVM, generating a heat flux that acts as a Neumann boundary condition at the interface for the solid domain. Subsequently, the LBFS calculates the thermal distribution inside the solid, and the updated temperature at the interface is then applied to the fluid computations as a Dirichlet condition. The proposed framework effectively combines the strengths of the IDVM in modeling rarefied gas flows with the advantages of the LBFS in handling heat conduction in complex geometries. Crucially, the current approach implicitly captures temperature jump discontinuities at the conjugate boundary, bypassing the requirement for supplementary jump conditions. To evaluate its performance, several CHT test cases involving rarefied gas in microchannels were conducted. Computational evidence suggests that the scheme is robust across diverse flow regimes. Full article

In this paper, we consider a reaction–diffusion system governed by incommensurate fractional time derivatives based on the Degn–Harrison model. Its formulation incorporates various memory effects on axial position through Caputo derivatives of variable orders, producing a more realistic modeling of the temporal dynamics. This paper starts with a study of the spatially homogeneous system and establishes conditions for local stability by using the Matignon criterion. The spectral decomposition method under Neumann boundary condition is then applied to study the complete reaction–diffusion system and describe diffusion-induced instabilities. Our results indicate that the noninteger fractional orders lead to significant changes in stability regions, as well as the initiation of pattern formation. Specifically, the orders of fractions induced as a control variable are regarded to be effective in controlling the stability of the system, thus they are global (or positive) control variables when their values achieved at some levels apply to the entire saturation, etc. Our numerical simulations are in excellent agreement with the theoretical predictions and show that memory asymmetry induces complex spatiotemporal dynamics not seen for classical integer-order systems. Full article

The simulation of free-surface flows in hydraulic engineering presents several challenges due to the intrinsic complexity of modeling a fluid that continuously deforms and evolves over time. In this context, the Smoothed Particle Hydrodynamics (SPH) method, a Lagrangian approach that represents the fluid as a set of moving particles, is better suited than traditional grid-based methods. However, compared to the latter, the SPH method also exhibits certain drawbacks, including increased difficulty in handling wall boundary conditions and a higher computational cost. This work proposes an original wall boundary treatment technique that, to the best of our knowledge, is applied in the Incompressible SPH (ISPH) approach for the first time. The proposed treatment relies on boundary particles external to the fluid and internal extrapolation points, where pressure is computed to enforce Neumann boundary conditions in a consistent manner. During the development of this technique, several intrinsic advantages over existing methods in the literature are identified. A series of numerical benchmarks are conducted to verify the validity of the proposed ISPH model. Numerical results show good agreement with experimental data reported in the literature, confirming the effectiveness of the proposed numerical model in reproducing free-surface flow hydraulic phenomena. Full article

For heat transfer outside borehole heat exchanger (BHE) arrays in aquifers, existing analytical models mostly adopt Neumann or Robin boundary conditions, whereas constant-temperature (Dirichlet) boundaries are more practical and convenient for monitoring in engineering applications. Considering the coupled effects of heat advection and conduction induced by groundwater seepage, and based on the engineering reality that vertical heat flow is much smaller than horizontal heat flow, this study idealized the BHE array as a constant-temperature boundary and established a one-dimensional simplified model. The advection term of the governing equation was removed through the exponential transformation of the dependent variable, and an analytical solution was derived using Fourier transformation. A three-dimensional coupled hydro-thermal numerical model was established in FEFLOW for validation. The results indicate that relative errors between analytical and numerical solutions remain below 3% outside the BHE array; however, the analytical method is inapplicable inside the array due to significant thermal interference, and independent field validation is precluded by prior thermal disturbances. The proposed solution features fast computation and clear physical interpretation, providing a simple and efficient tool for rapid estimation of temperature variations during preliminary feasibility studies of ground-source heat-pump projects. Full article

This paper presents a probabilistic–statistical approach to the analysis of diffusion processes in randomly nonhomogeneous multilayered bodies under conditions of incomplete experimental information on the boundary. The boundary condition is reconstructed from experimental data using linear regression, while the solution of the corresponding contact initial-boundary value problem is obtained in the form of a Neumann series and averaged over an ensemble of phase configurations. A system of statistical estimates for the solution is developed, including confidence intervals and two-sided critical regions, which provide complementary characteristics of uncertainty. Numerical experiments are performed for six representative samples differing in sample size, variance, and observation interval. It is shown that, despite significant differences in the statistical properties of the input data, the averaged concentration field preserves a qualitatively stable spatio-temporal structure. The results of the article address gaps in existing research by applying a probabilistic-statistical approach that consistently integrates two key elements for the analysis of diffusion processes in multilayer media. The first of these is the reconstruction of boundary conditions using linear regression to recover the conditions at the body boundary based on incomplete experimental data. The second key point is the analysis of uncertainty propagation by combining the regression model with a probabilistic analysis of the corresponding contact initial-boundary value problem, which allows us to quantitatively assess how the errors in the experimental data affect the final solution. From the point of view of mathematical modeling methods, the novelty of the approach lies in the creation of a structural-hierarchical scheme that synthesizes the approaches of mathematical statistics and the theory of random fields. The developed method is a theoretical and computational innovative basis for the analysis of specific physical and technological processes. Full article

Despite its importance in modeling subdiffusion in fractal and heterogeneous media, a rigorous and computational scheme for solving the fractional diffusion equation on generalized comb structures over unbounded domains has remained elusive, mainly due to the nonlocal memory effect and slow spatial decay of solutions. To the best of our knowledge, we address this long-standing gap by presenting a fully integrated framework that simultaneously resolves both challenges. We derive the governing equation from constitutive relations and establish exact absorbing boundary conditions (ABCs) for the multi-skeleton comb model, a result absent in prior work. A transparent Dirichlet-to-Neumann (DtN) map, constructed via Laplace analysis, rigorously handles skeletal Dirac delta singularities and eliminates spurious reflections without empirical parameters. Furthermore, we propose a novel structure-preserving finite difference scheme that applies the sum-of-exponentials (SOE) approximation not only to the interior Caputo derivative but also to the convolution kernels arising from the ABCs. This yields a dramatic reduction in computational complexity, from quadratic $O\left(N_t^2\right)$ to quasi-linear $O(N_{t}log{N}_t)$, while preserving the physics of anomalous transport. We prove the well-posedness, unconditional stability, and convergence of the method. Numerical results confirm theoretical error estimates and show excellent agreement between simulated particle distributions, mean square displacement profiles, and exact asymptotics, validating both accuracy and robustness. The speedup (CPU time ratio Direct/Fast) is about $1.00\times$–$1.23\times$ for $N_t=5000$ in our tests. Our approach sets a new benchmark for simulating anomalous dynamics in fractal-inspired media. Full article

We consider two steady-state heat conduction systems called, S and $S_{\alpha }$, in a multidimensional bounded domain D for the Poisson equation with source energy g. In one system, we impose mixed boundary conditions (temperature b on the boundary ${\mathsf{\Gamma }}_1$, heat flux q on ${\mathsf{\Gamma }}_2$ and an adiabatic condition on ${\mathsf{\Gamma }}_3$). In the other system, the condition on ${\mathsf{\Gamma }}_1$ is replaced by a convective heat flux condition with coefficient $\alpha$. For each of these systems, we consider three associated optimization problems $\left(P_i\right)$ and $\left(P_{i\alpha }\right)$, $i=1,2,3$, where the variable is the source energy g, the heat flux q and the environmental temperature b, respectively. In the particular case where D is a rectangle, the explicit continuous optimization variables and the corresponding state of the systems are known. In the present work, by using a finite difference scheme, we obtain the discrete systems $\left(S^h\right)$ and $\left(S_{\alpha }^h\right)$ and discrete optimization problems $\left(P_i^h\right)$ and $\left(P_{i\alpha }^h\right)$, $i=1,2,3$, where h is the space step in the discretization. Explicit discrete solutions are found, and convergence and estimation errors results are proved when h goes to zero and when $\alpha$ goes to infinity. Moreover, some numerical simulations are provided in order to test theoretical results. Finally, we note that the use of a three-point finite-difference approximation for the Neumann or Robin boundary condition at the boundary improves the global order of convergence from $O\left(h\right)$ to $O\left(h^2\right)$. Full article

Physics-informed neural networks (PINNs) solve partial differential equations (PDEs) by embedding physical conditions as soft penalties into the loss function. However, the coexistence of multiple loss components often leads to gradient conflicts, degrading convergence and solution accuracy. To address this issue, we propose a dynamic domain–gradient loss reweighting PINN (DDR-PINN). The proposed method introduces a dual-residual reweighting mechanism based on gradient variations, where adaptive weights are derived from the L2 norm of the dot product between loss gradients and residuals. These weights are further normalized through a nonlinear hyperbolic tangent transformation, enabling dynamic and balanced reweighting of interior, initial, and boundary domain losses throughout training. Extensive numerical experiments on PDEs with both Dirichlet and Neumann boundary conditions demonstrate that the DDR-PINN consistently outperforms the standard PINN, APINN, and VI-PINN with the fewest trainable parameters. Full article

This study introduces a high-order numerical scheme for solving 1D second-order hyperbolic telegraph equations (HTEs) with variable coefficients. We employ a generalized temporal discretization (TD) of order p via the Rothe approach, combined with a spatial spectral collocation (SCM) method using generalized shifted Jacobi polynomials (GSJPs). By utilizing a Galerkin-type basis that structurally satisfies homogeneous boundary conditions (HBCs)—including Dirichlet or Neumann types—we achieve a global error bound of $\mathcal{O}({(\Delta \tau )}^p+N^{-s})$, where $\Delta \tau$ denotes the temporal step size and s represents the spatial regularity of the exact solution (ExaS). The proposed algorithm, Rothe-GSJP, allows for an optimal balance between the temporal and spatial parameters, minimizing computational effort for high-precision engineering applications such as Phase-Locked Loop (PLL) modeling. Numerical experiments performed on an i9-10850 workstation show that the scheme always reaches the machine precision floor of ${10}^{-16}$. While the framework supports temporal orders up to $p=6$, the results indicate that $p\in \{2,3,4\}$ provides an optimal balance between high-order precision and absolute stability. The Rothe-GSJP method proves to be a robust, efficient, and highly accurate alternative to traditional solvers for hyperbolic systems. Full article

This paper presents a complete analysis of the Helfrich membrane energy functional in the product space ${\mathbb{H}}^2\times \mathbb{R}$. We address the analytical challenges posed by the ideal boundary of the space by developing a renormalization scheme, allowing us to formulate a well-posed variational problem. We derive the Euler-Lagrange equations for the renormalized functional, characterizing the equilibrium configurations through a coupled system of partial differential equations and a Neumann-type boundary condition. A central result of our work is a rigidity theorem, proven via a Killing field argument, which establishes that any admissible critical surface is necessarily axially symmetric. Finally, we connect this mathematical theory to biophysics by proposing a new variational principle for the Solvent Accessible Surface (SAS) under geometric confinement, demonstrating that our classified surfaces represent the optimal elastic energy shapes for such systems. Full article

Physics-informed neural networks (PINNs) often struggle to balance multiple loss terms in thermally coupled multiphysics problems. We propose Cooperative Soft-Hard PINNs (s-hPINN/s-HB-PINN), which apply soft constraints to fields with Neumann conditions while enforcing hard constraints on others to balance exact boundary enforcement with training stability. Validated on thermoelasticity and thermal convection, our method reduces training time by approximately 56%. In thermal convection experiments, incorporating partial data further reduces velocity errors by up to 78% compared to standard PINNs. We subsequently assessed the framework’s robustness against varying relative Gaussian white noise levels and different data sampling locations. The result demonstrate that s-HB-PINN maintains high-fidelity predictions even under noise interference, consistently outperforming baseline methods. This confirms that the proposed collaborative strategy offers a superior trade-off between accuracy, efficiency, and robustness in complex multiphysics environments. Full article

Fractional calculus approach is used to analyze the model of surfactant transport by anomalous diffusion and its adsorption on an interface in a liquid-liquid system. The anomalous diffusion is modeled by time-fractional partial differential equations in the bulk phases. The adsorption of surfactant is described by the corresponding time-fractional Neumann boundary conditions at the interface. The adsorption process is considered under mixed barrier-diffusion control, described by first-order ordinary differential equation, which relates the subsurface concentration with that on the interface. A second relation between these concentrations is derived in terms of a fractional equation by application of Laplace transform technique. By combining both relations the subsurface concentration is eliminated and a single multi-term fractional ordinary differential equation for the surfactant concentration on the interface is derived. Different adsorption kinetic models are considered. In the case of Henry adsorption isotherm the model is linear and possesses analytical solution in terms of multinomial Mittag-Leffler functions. In the cases of Volmer and van der Waals adsorption isotherms nonlinear differential equations of fractional order are obtained. They are reformulated in equivalent integral form, which is used for computer simulation of the process of adsorption. Numerical results are presented and compared with analytical asymptotic predictions. Full article