This paper focused on an $ hp $-version mixed discontinuous Galerkin method without penalty terms for the biharmonic equation with Navier boundary conditions. By introducing the auxiliary variable $ v = u_{xx} $, we reduced the fourth-order problem to a second-order system and derived its penalty-free variational formulation. The analysis replaces the standard coercivity condition with a polynomial-degree-dependent inf-sup condition for the bilinear form $ B(\cdot, \cdot) $. While $ h $-convergence rates for both $ u_h $ and $ v_h $ were optimal, the $ p $-convergence exhibited contrasting behavior: suboptimal in $ L^2 $-norm but optimal in the energy norm, regardless of the $ p^2 $ scaling in the inf-sup condition. Numerical results revealed that $ p $-convergence order-doubling for boundary-aligned singularities significantly enhanced the efficacy for singular solutions. Furthermore, the method was shown to extend to nonlinear biharmonic equations, while the treatment of Dirichlet boundary conditions necessitated the introduction of penalty terms.

An adaptive moving grid method is developed to solve the time-fractional integrodifferential governing equations of viscoelastic nanofluid. The momentum equation is derived based on a dual-parameter fractional Maxwell constitutive relation, and the energy equation employs a generalized Cattaneo heat conduction relation. To improve solution accuracy, a monitor function based on the equidistribution principle is constructed, and an adaptive mesh redistribution strategy is developed in the spatial domain. The temporal fractional-order operators are approximated by the L1 algorithm and the weighted-shifted Grünwald difference method. Numerical experiments demonstrate that the adaptive grid achieves 77.6–88.4% higher accuracy compared to uniform grids at the same grid scale, along with enhanced stability in convergence. Parametric analysis indicates that increasing the fractional-order derivative in the energy equation results in a thickening of both the velocity and thermal boundary layers. Furthermore, the dual-fractional Maxwell model exhibits a thicker velocity boundary layer than its classical single-parameter counterpart. The proposed method offers an efficient and robust approach for simulating complex viscoelastic nanofluid systems with memory effects and multi-field coupling.

A novel numerical simulation was conducted for boundary layer flow and heat transfer of viscoelastic fluid in a non-Darcy porous medium. The double fractional Maxwell model and generalized Fourier's law were employed in constitutive relation and heat transport, respectively. The strongly coupled and nonlinear fractional governing equations were formulated, which were solved and validated by the finite volume method combined with the fractional L1 scheme. The influences of fractional derivative parameters, Darcy number, porosity, and inertial parameter on the transport fields are discussed. Our results demonstrated that the different fractional derivative parameters exhibit opposite influences on velocity/temperature distributions. Moreover, the inertial effects induce velocity profile crossover that marks the transition between viscous and inertial dominance. These findings redefine our understanding of energy transport in complex porous systems.

Is it possible to interpret the modeling decisions made by a neural network trained to simulate the constitutive behavior of simple or complex materials? The problem of the interpretability of a neural network is a crucial aspect that has been studied since the first appearance of this type of modeling tool and it is certainly not specific to applications related to constitutive modeling of heterogeneous materials. All areas of application, such as computer vision, biomedicine, and speech, suffer from this fuzziness, and for this reason, neural networks are often referred to as 'black-box models'. The present work highlighted the efforts dedicated to this aspect in the constitutive modeling of the behavior of path independent materials, reviewing both more standard neural networks and those adopting, more or less strongly, the specific point of view of interpretability.

<p>Using the lattice dynamics approach, we discussed the propagation of interfacial waves localized near the interface in an infinite square lattice. The interface has been modeled as a single-particle layer of material particles with masses and elastic bonds different from those in the bulk. In this lattice structure there were anti-plane interface waves, i.e., waves that decayed exponentially with distance from the interface. Such waves could be useful for determining material properties in the vicinity of the interface. We obtained equations of motion and analyzed the corresponding dispersion relations for steady-state solutions. Here, the dispersion equation related the circular frequency to the wave number. In addition, we provided a comparison of the dispersion relations with those derived within the Gurtin-Murdoch surface elasticity. To do this, we have used the scaling law that links the continuum and discrete models. Unlike the continuum model, in the discrete model the wave number was limited by the first Brillouin zone, whereas in the continuum model it took a range from zero to infinity. The detailed parametric analysis was given for the discrete model. Finally, other models of interfaces in the case of a square lattice were discussed.</p>

<p>We formulate and investigate a nonlinear parabolic reaction–diffusion equation describing the oxygen concentration in encapsulated pancreatic cells with a general core-shell geometry. This geometry introduces a discontinuous diffusion coefficient as the material properties of the core and shell differ. We apply monotone operator theory to show the well-posedness of the problem in the strong form. Furthermore, the stationary solutions are unique and asymptotically stable. These results rely on the gradient structure of the underlying PDE. Our results provide necessary theoretical steps for validation of the model.</p>

In this study, we showed the existence of global, finite energy, weak solutions to a class of compressible Euler equations, describing the dynamics of a quantum fluid interacting with an external magnetic field. To do so, we exploited suitable a priori estimates for weak solutions of the underlying wave-function dynamics, governed by a magnetic semilinear Schrödinger equation.

<p>Several Krylov subspace methods are based on the Arnoldi process, such as the full orthogonalization method (FOM), GMRES, and in general all the Arnoldi-type methods. In fact, the Arnoldi process is an algorithm for building an orthogonal basis of the Krylov subspace. Once the inner products are performed inexactly, which cannot be avoided due to round-off errors, the orthogonality of Arnoldi vectors is lost. In this paper, we presented a new analysis framework to show how the inexact inner products influence the Krylov subspace methods that are based on the Arnoldi process. A new metric was developed to quantify the inexactness of the Arnoldi process with inexact inner products. In addition, the proposed metric can be used to approximately estimate the loss of orthogonality in the practical use of the Arnoldi process. The discrepancy in residual gaps between Krylov subspace methods employing inexact inner products and their corresponding exact counterparts was discussed. Numerical experiments on several examples were reported to illustrate our theoretical findings and final observations were presented.</p>

Recently, the variable coefficient homogeneous differential equations (VCHDE) have been widely applied to real-world problems, such as wave propagation and material science. However the exploration and research on higher-order VCHDE is relatively lagging. Given this, this work focuses on the solutions of fourth-order and nth-order VCHDE with polynomial coefficients. By means of the sufficient conditions for the existence of solutions to differential equations, a connection is established between the rank of the variable coefficient matrix and the existence of polynomial particular solutions. The main results show that: (1) the necessary and sufficient conditions for the existence of polynomial particular solutions of fourth-order VCHDE are derived; (2) the necessary and sufficient conditions for the existence of only one polynomial particular solution, or the existence of two, three, or four linearly independent polynomial particular solutions of fourth-order VCHDE are proved; (3) the necessary and sufficient conditions for the existence of only one polynomial particular solution, or the existence of two, three, or four linearly independent polynomial particular solutions of nth-order VCHDE are proved. These results not only extend the class of solvable differential equations, but also provide a new way of thinking about the existence of solutions to VCHDE.