We consider the ergodic (or additive eigenvalue) problem for the Neumann-type boundary-value problem for Hamilton–Jacobi equations and the corresponding discounted problems. Denoting by u^λ the solution of the discounted problem with discount factor λ > 0, we establish the convergence of the whole family to a solution of the ergodic problem as λ → 0, and give a representation formula for the limit function via the Mather measures and Peierls function. As an interesting by-product, we introduce Mather measures associated with Hamilton–Jacobi equations with the Neumann-type boundary conditions. These results are variants of the main results in a recent paper by Davini et al., who study the same convergence problem on smooth compact manifolds without boundary.

The present study introduced the unsteady squeezing flow of two-dimensional viscous fluid with nanoparticles between two disks by using the Levenberg–Marquardt backpropagated neural network (LMB-NN). Conversion of the partial differential equations (PDEs) into equivalent ordinary differential equations (ODEs) is performed by suitable similarity transformation. The data collection for suggested (LMB-NN) is made for various magnetohydrodynamic squeezing flow (MHDSF) scenarios in terms of the squeezing parameter, Prandtl number, Brownian motion parameter, and the thermophoresis parameter by employing the Runge–Kutta technique with the help of Mathematica software. The worth of the proposed methodology has been established for the proposed solver (LMB-NN) with different scenarios and cases, and the outcomes are compared through the effectiveness and reliability of mean square error (MSE) for the squeezing flow problem MHDSF. Moreover, the state transition, Fitness outline, histogram error, and regression presentation also endorse the strength and reliability of the solver LMB-NN. The high convergence between the reference solutions and the solutions obtained by incorporating the efficacy of a designed solver LMB-NN indicates the strength of the proposed methodology, where the accuracy level is achieved in the ranges from 10${}^{-6}$ to ${10}^{-12}$.

In this study, numerical treatment with the Lobatto IIIA technique is presented to analyze the dynamics of Darcy–Forchheimer flow for carbon nanotubes (CNTs) in a revolving frame. Stretching the surface causes fluid flow and the model is implemented for the transport of hybrid nanofluids. The governing partial differential system is simplified with the involvement of the boundary layer approximation. Similarity variables are used for the transformation of governing partial differential equations into a system of nonlinear ordinary differential equations. The results for single and multiwalled CNTs are attained by the Lobatto IIIA technique for the analysis of the resulting system. Adequate numerical and graphical illustrations are presented to investigate and understand how different flow parameters affect the velocities and temperature fields. Furthermore, the coefficients of the skin friction and the local Nusselt number are inspected numerically.

Bogoya, Böttcher, Grudsky, and Maximenko have recently obtained the precise asymptotic expansion for the eigenvalues of a sequence of Toeplitz matrices {T _n (f)}, under suitable assumptions on the associated generating function f. In this paper, we provide numerical evidence that some of these assumptions can be relaxed and extended to the case of a sequence of preconditioned Toeplitz matrices {T n−1(g)T _n (f)}, for f trigonometric polynomial, g nonnegative, not identically zero trigonometric polynomial, r = f/g, and where the ratio r plays the same role as f in the nonpreconditioned case. Moreover, based on the eigenvalue asymptotics, we devise an extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices with a high level of accuracy, with a relatively low computational cost, and with potential application to the computation of the spectrum of differential operators.

We study the dynamical boundary value problem for Hamilton‐Jacobi equations of the eikonal type with a small parameter. We establish two results concerning the asymptotic behavior of solutions of the Hamilton‐Jacobi equations: one concerns with the convergence of solutions as the parameter goes to zero and the other with the large‐time asymptotics of solutions of the limit equation.