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This study examines the nonlinear behaviors of a clamped-clamped porous pipe made of a functionally graded material (FGM) that conveys fluids and is equipped with a retaining clip, focusing on primary resonance and subcritical dynamics. The nonlinear governing equations for the FGM pipe are derived by the extended Hamilton's principle, and subsequently discretized through the application of the Galerkin method. The direct method of multi-scales is then used to solve the derived equations. A thorough analysis of various parameters, including the clip stiffness, the power-law index, the porosity, and the clip location, is conducted to gain a comprehensive understanding of the system's nonlinear dynamics. Through the analysis of the first natural frequency, the study highlights the influence of the flow velocity and the clip stiffness, while the comparisons with metallic pipes emphasize the role of FGM composition. The examination of the forced response curves reveals saddle-node bifurcations and their dependence on parameters such as the detuning parameter and the power-law index, offering valuable insights into the system's nonlinear resonant behavior. Furthermore, the frequency-response curves illustrate the hardening nonlinearities influenced by factors such as the porosity and the clip stiffness, revealing nuanced effects on the system response and resonance characteristics. This comprehensive analysis enhances the understanding of nonlinear behaviors in FGM porous pipes with a retaining clip, providing key insights for practical engineering applications in system design and optimization. [ABSTRACT FROM AUTHOR]

In the paper, we study a Balakrishnan–Taylor quasilinear wave equation | z t | α ⁢ z t ⁢ t - Δ ⁢ z t ⁢ t - (ξ 1 + ξ 2 ⁢ ∥ ∇ ⁡ z ∥ 2 + σ ⁢ (∇ ⁡ z , ∇ ⁡ z t) ) ⁢ Δ ⁢ z - Δ ⁢ z t + β ⁢ (x) ⁢ f ⁢ (z t) + g ⁢ (z) = 0 in a bounded domain of ℝ n with Dirichlet boundary conditions. By using Faedo–Galerkin method, we prove the existence of global weak solutions. By the help of the perturbed energy method, the exponential stability of solutions is also established. [ABSTRACT FROM AUTHOR]

Integrating evolutionary partial differential equations (PDEs) is an essential ingredient for studying the dynamics of the solutions. Indeed, simulations are at the core of scientific computing, but their mathematical reliability is often difficult to quantify, especially when one is interested in the output of a given simulation, rather than in the asymptotic regime where the discretization parameter tends to zero. In this paper we present a computer-assisted proof methodology to perform rigorous time integration for scalar semilinear parabolic PDEs with periodic boundary conditions. We formulate an equivalent zero-finding problem based on a variation of constants formula in Fourier space. Using Chebyshev interpolation and domain decomposition, we then finish the proof with a Newton–Kantorovich type argument. The final output of this procedure is a proof of existence of an orbit, together with guaranteed error bounds between this orbit and a numerically computed approximation. We illustrate the versatility of the approach with results for the Fisher equation, the Swift–Hohenberg equation, the Ohta–Kawasaki equation and the Kuramoto–Sivashinsky equation. We expect that this rigorous integrator can form the basis for studying boundary value problems for connecting orbits in partial differential equations. [ABSTRACT FROM AUTHOR]

In the present work, we study the global existence and uniform decay rates of solutions to the initial-boundary value problem related to the dynamic behavior of evolution equations accounting for rotational inertial forces along with a linear time-nonlocal Kelvin-Voigt damping arising in viscoelastic materials. By constructing appropriate Lyapunov functional, we show that the solution converges to the equilibrium state exponentially in the energy space. [ABSTRACT FROM AUTHOR]

We consider a nonlinear parabolic partial differential equation in the case where the unknown function depends on two spatial variables and time, which is a generalization of the well-known Kuramoto–Sivashinsky equation. We consider homogeneous Dirichlet boundary-value problems for this equation. We examine local bifurcations when spatially homogeneous equilibrium states change stability. We show that post-critical bifurcations are realized in the boundary-value problems considered. We obtain asymptotic formulas for solutions and examine the stability of spatially inhomogeneous solutions. [ABSTRACT FROM AUTHOR]

This paper develops validated computational methods for studying infinite dimensional stable manifolds at equilibrium solutions of parabolic PDEs, synthesizing disparate errors resulting from numerical approximation. To construct our approximation, we decompose the stable manifold into three components: a finite dimensional slow component, a fast-but-finite dimensional component, and a strongly contracting infinite dimensional "tail". We employ the parameterization method in a finite dimensional projection to approximate the slow-stable manifold, as well as the attached finite dimensional invariant vector bundles. This approximation provides a change of coordinates which largely removes the nonlinear terms in the slow stable directions. In this adapted coordinate system we apply the Lyapunov-Perron method, resulting in mathematically rigorous bounds on the approximation errors. As a result, we obtain significantly sharper bounds than would be obtained using only the linear approximation given by the eigendirections. As a concrete example we illustrate the technique for a 1D Swift-Hohenberg equation. [ABSTRACT FROM AUTHOR]

This paper aims to introduce a measure of the non-univalency of a harmonic mapping. By using it, we find the best approximation of an analytic function by a univalent harmonic mapping. [ABSTRACT FROM AUTHOR]

We consider fully discrete numerical schemes for a downscaling dataassimilation algorithm aimed at approximating the velocity field of the 2DNavier-Stokes equations corresponding to given coarse mesh observationalmeasurements. The time discretization is done by considering semi- andfully-implicit Euler schemes, and the spatial discretization is based on aspectral Galerkin method. The two fully discrete algorithms are shown to beunconditionally stable, with respect to the size of the time step, number oftime steps and the number of Galerkin modes. Moreover, explicit, uniform intime error estimates between the fully discrete solution and the referencesolution corresponding to the observational coarse mesh measurements areobtained, in both the $L^2$ and $H^1$ norms. Notably, the two-dimensionalNavier-Stokes equations, subject to the no-slip Dirichlet or periodic boundaryconditions, are used in this work as a paradigm. The complete analysis that ispresented here can be extended to other two- and three-dimensional dissipativesystems under the assumption of global existence and uniqueness.

We apply the Postprocessing Galerkin method to a recently introducedcontinuous data assimilation (downscaling) algorithm for obtaining a numericalapproximation of the solution of the two-dimensional Navier-Stokes equationscorresponding to given measurements from a coarse spatial mesh. Under suitableconditions on the relaxation (nudging) parameter, the resolution of the coarsespatial mesh and the resolution of the numerical scheme, we obtain uniform intime estimates for the error between the numerical approximation given by thePostprocessing Galerkin method and the reference solution corresponding to themeasurements. Our results are valid for a large class of interpolant operators,including low Fourier modes and local averages over finite volume elements.Notably, we use here the 2D Navier-Stokes equations as a paradigm, but ourresults apply equally to other evolution equations, such as the Boussinesqsystem of Benard convection and other oceanic and atmospheric circulationmodels.

In this paper, we consider the boundary-value problem for the Kuramoto–Sivashinsky equation with homogeneous Neumann conditions. The problem on the existence and stability of second-kind equilibrium states was studied in two ways: by the Galerkin method and by methods of the modern theory of infinite-dimensional dynamical systems. Some differences in results obtained are indicated. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of solutions for a class of elliptic equations on the unit ball with zero Dirichlet boundary condition. The nonlinearity is supercritical in the sense of Trudinger–Moser. Using a suitable approximating scheme we obtain the existence of at least one positive solution. [ABSTRACT FROM AUTHOR]

Transfer operators such as Perron–Frobenius and Koopman operator play a key role in modeling and analysis of complex dynamical systems, which allow linear representations of nonlinear dynamics by transforming the original state variables to feature spaces. However, it remains challenging to identify the optimal low-dimensional feature mappings from data. The variational approach for Markov processes (VAMP) provides a comprehensive framework for the evaluation and optimization of feature mappings based on the variational estimation of modeling errors, but it still suffers from a flawed assumption on the transfer operator and therefore sometimes fails to capture the essential structure of system dynamics. In this paper, we develop a powerful alternative to VAMP, called kernel embedding based variational approach for dynamical systems (KVAD). By using the distance measure of functions in the kernel embedding space, KVAD effectively overcomes theoretical and practical limitations of VAMP. In addition, we develop a data-driven KVAD algorithm for seeking the ideal feature mapping within a subspace spanned by given basis functions, and numerical experiments show that the proposed algorithm can significantly improve the modeling accuracy compared to VAMP. [ABSTRACT FROM AUTHOR]

In this work, the numerical solutions of advection-diffusion equation are investigated through the finite element method. The quartic-trigonometric tension (QTT) B-spline which presents advantages over the well-known existing B-splines is adapted as the base of the numerical algorithm. Space integration of the model partial differential equation is achieved through QTT B-spline Galerkin method. The resultant system of time-dependent differential equations is integrated using the Crank-Nicolson technique. The stability of the current scheme is accomplished and proved to be unconditionally stable. Simulation of several sample problems are carried out for verification of the proposed numerical scheme. Solutions obtained by numerically computed scheme are compared to the existing literature. [ABSTRACT FROM AUTHOR]

This paper combines image metamorphosis with deep features. To this end, images are considered as maps into a high-dimensional feature space and a structure-sensitive, anisotropic flow regularization is incorporated in the metamorphosis model proposed by Miller and Younes (Int J Comput Vis 41(1):61–84, 2001) and Trouvé and Younes (Found Comput Math 5(2):173–198, 2005). For this model, a variational time discretization of the Riemannian path energy is presented and the existence of discrete geodesic paths minimizing this energy is demonstrated. Furthermore, convergence of discrete geodesic paths to geodesic paths in the time continuous model is investigated. The spatial discretization is based on a finite difference approximation in image space and a stable spline approximation in deformation space; the fully discrete model is optimized using the iPALM algorithm. Numerical experiments indicate that the incorporation of semantic deep features is superior to intensity-based approaches. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to study a class of semilinear elliptic Dirichlet boundary value problems in the ball, where the nonlinearities involve the sum of a sublinear variable exponent and a superlinear (may be supercritical) variable exponents of the form 0 ≤ f (r , u) ≤ a 1 | u | p (r) - 1 , if u ≥ 0 , where r = | x | , p (r) = 2 ∗ + r α , with α > 0 , and 2 ∗ = 2 N / (N - 2) is the critical Sobolev embedding exponent. We do not impose the Ambrosetti–Rabinowitz condition on the nonlinearity (or some additional conditions) to obtain Palais–Smale or Cerami compactness condition. We employ techniques based on the Galerkin approximations scheme, combining with a Sobolev type embeddings for radial functions into variable exponent Lebesgue spaces (due to do Ó et al. in Calc Var Partial Differ Equ 55:83, 2016), to establish the existence result. [ABSTRACT FROM AUTHOR]

We consider spectral Galerkin approximations for the strong solutions of a system of incompressible flows through granular porous medium in a bounded domain of R n , n = 2 , 3 . We obtain uniform in time error bounds in the spatial L 2 and H 1 -norms for approximations of the velocity. Finally, we present some numerical simulations to verify the good behavior of spectral Galerkin approximations. [ABSTRACT FROM AUTHOR]

Transfer operators such as the Perron–Frobenius or Koopman operator play an important role in the global analysis of complex dynamical systems. The eigenfunctions of these operators can be used to detect metastable sets, to project the dynamics onto the dominant slow processes, or to separate superimposed signals. We propose kernel transfer operators, which extend transfer operator theory to reproducing kernel Hilbert spaces and show that these operators are related to Hilbert space representations of conditional distributions, known as conditional mean embeddings. The proposed numerical methods to compute empirical estimates of these kernel transfer operators subsume existing data-driven methods for the approximation of transfer operators such as extended dynamic mode decomposition and its variants. One main benefit of the presented kernel-based approaches is that they can be applied to any domain where a similarity measure given by a kernel is available. Furthermore, we provide elementary results on eigendecompositions of finite-rank RKHS operators. We illustrate the results with the aid of guiding examples and highlight potential applications in molecular dynamics as well as video and text data analysis. [ABSTRACT FROM AUTHOR]

Inference, prediction, and control of complex dynamical systems from time series is important in many areas, including financial markets, power grid management, climate and weather modeling, or molecular dynamics. The analysis of such highly nonlinear dynamical systems is facilitated by the fact that we can often find a (generally nonlinear) transformation of the system coordinates to features in which the dynamics can be excellently approximated by a linear Markovian model. Moreover, the large number of system variables often change collectively on large time- and length-scales, facilitating a low-dimensional analysis in feature space. In this paper, we introduce a variational approach for Markov processes (VAMP) that allows us to find optimal feature mappings and optimal Markovian models of the dynamics from given time series data. The key insight is that the best linear model can be obtained from the top singular components of the Koopman operator. This leads to the definition of a family of score functions called VAMP-r which can be calculated from data, and can be employed to optimize a Markovian model. In addition, based on the relationship between the variational scores and approximation errors of Koopman operators, we propose a new VAMP-E score, which can be applied to cross-validation for hyper-parameter optimization and model selection in VAMP. VAMP is valid for both reversible and nonreversible processes and for stationary and nonstationary processes or realizations. [ABSTRACT FROM AUTHOR]

An algorithm is proposed to calculate traveling dissipative solitons for the FitzHugh–Nagumo equations. It is based on the application of the steepest descent method to a certain functional. This approach can be used to find solitons whenever the problem has a variational structure. Since the method seeks the lowest energy configuration, it has robust performance qualities. It is global in nature, so that initial guesses for both the pulse profile and the wave speed can be quite different from the correct solution. Also, bifurcations have a minimal effect on the performance. In the literature, there is a conjecture that no stable traveling pulse exists for a 2-component system in 2D unbounded domains. In many instances, such numerical studies investigate only solutions with a small speed, as they rely on good initial guesses based on stable standing pulse profiles. Studying a modified problem with a 2D strip domain R × [ - L , L ] with zero Dirichlet boundary conditions at y = ± L , by using our algorithm we establish the existence of fast-moving solitons. With an appropriate set of physical parameters in this unbounded rectangular strip domain, we observe the co-existence of single-soliton and 2-soliton solutions together with additional unstable traveling pulses. The algorithm automatically calculates these various pulses as the energy minimizers at different wave speeds. In addition to finding individual solutions, we anticipate that this approach could be used to augment or initiate continuation algorithms. We also note that the rectangular strip domain can serve as a first step to investigating waves in the whole of R 2 . [ABSTRACT FROM AUTHOR]

We present a framework for the systematic numerical approximation of nonlinear evolution problems with dissipation. The approach is based on rewriting the problem in a canonical form that complies with the underlying energy-dissipation structure. We show that the corresponding weak formulation then allows for a dissipation-preserving approximation by Galerkin methods in space and discontinuous Galerkin methods in time. The proposed methodology is rather general and can be applied to a wide range of applications. This is demonstrated by discussion of some typical examples ranging from diffusive partial differential equations to dissipative Hamiltonian systems. [ABSTRACT FROM AUTHOR]

In the present paper, a new orthonormal wavelet basis, called Chelyshkov wavelet, is constructed from a class of orthonormal polynomials. These wavelet basis and their properties are utilized to obtain their operational matrix of fractional integration in the Riemann-Liouville sense and delay operational matrix. Convergence and error bound of the expansion by this kind of wavelet functions are investigated. Then, these operational matrices along with the Galerkin approach have been implemented to solve systems of fractional delay differential equations (SFDDEs). The main superiority of the proposed technique is that it reduces SFDDEs to a system of algebraic equations. Moreover, accuracy and efficiency of the suggested Chelyshkov wavelet approach are verified through some linear and nonlinear SFDDEs. Finally, the obtained numerical results are compared with those previously reported in the literature. [ABSTRACT FROM AUTHOR]

In this review paper, we will present different data-driven dimension reduction techniques for dynamical systems that are based on transfer operator theory as well as methods to approximate transfer operators and their eigenvalues, eigenfunctions, and eigenmodes. The goal is to point out similarities and differences between methods developed independently by the dynamical systems, fluid dynamics, and molecular dynamics communities such as time-lagged independent component analysis, dynamic mode decomposition, and their respective generalizations. As a result, extensions and best practices developed for one particular method can be carried over to other related methods. [ABSTRACT FROM AUTHOR]

We develop and implement a semi-numerical method for computing high-order Taylor approximations of unstable manifolds for hyperbolic fixed points of compact infinite-dimensional maps. The method can follow folds in the embedding and describes precisely the dynamics on the manifold. In order to ensure the accuracy of our computations in spite of the many truncation and round-off errors, we develop a posteriori error bounds for the approximations. Deliberate control of round-off errors (using interval arithmetic) in conjunction with explicit analytical estimates leads to mathematically rigorous computer-assisted theorems describing precisely the truncation errors for our approximation of the invariant manifold. The method is applied to the Kot-Schaffer model of population dynamics with spatial dispersion. [ABSTRACT FROM AUTHOR]

We demonstrate the application of an algorithmic trading strategy based upon the recently developed dynamic mode decomposition on portfolios of financial data. The method is capable of characterizing complex dynamical systems, in this case financial market dynamics, in an equation-free manner by decomposing the state of the system into low-rank terms whose temporal coefficients in time are known. By extracting key temporal coherent structures (portfolios) in its sampling window, it provides a regression to a best fit linear dynamical system, allowing for a predictive assessment of the market dynamics and informing an investment strategy. The data-driven analytics capitalizes on stock market patterns, either real or perceived, to inform buy/sell/hold investment decisions. Critical to the method is an associated learning algorithm that optimizes the sampling and prediction windows of the algorithm by discovering trading hot-spots. The underlying mathematical structure of the algorithms is rooted in methods from nonlinear dynamical systems and shows that the decomposition is an effective mathematical tool for data-driven discovery of market patterns. [ABSTRACT FROM AUTHOR]

In this paper two examples are studied; the Orr-Sommerfeld and nonlinear heat transfer equation. He's variational iteration method (VIM) and Adomian's decomposition method (ADM) are applied to the Orr-Sommerfeld equation. He's VIM and differential transformation method (DTM) are applied to the nonlinear heat transfer equation. Comparison of different methods represents while the effect of the nonlinear term in the heat transfer equation is negligible, VIM and DTM lead approximately to the same answers; and when the kinematic viscosity in the Orr-Sommerfeld equation is small, VIM and ADM lead nearly to the same results. [ABSTRACT FROM AUTHOR]

This paper is focused on the analytic and numerical dissipativity of nonlinear neutral delay integro-differential equations of the form . The dissipativity criteria of the theoretical solutions are obtained by applying the generalization of the Halanay inequality. Furthermore, the dissipativity of one-leg θ-methods and linear θ-methods for the underlying system is investigated. It is shown that, for ½<θ≤1, both the one-leg θ-methods and linear θ-methods are dissipative. Numerical example illustrates the theoretical results. [ABSTRACT FROM PUBLISHER]

In this article, we consider a classical ill-posed problem which is the backward problem for parabolic equation with time-dependent coefficient [image omitted] Applying the method of Fourier transform and the modified quasi-boundary value method, we shall give a regularization for the problem. A sharp error estimate between the regularized solution and the exact solution is given in this article. A numerical experiment is provided illustrating the main results. [ABSTRACT FROM AUTHOR]