Results in Applied Mathematics最新文献

In this paper, we present fixed point theorem for mappings defined on $b$-metric spaces, which satisfies extended quasi-contractive inequality with nonlinear comparison function. Our result generalizes and improves several recent results from fixed point theory.

This article presents the implementation of a randomized neural network (RNN) approach in approximating the solution of fractional order boundary value problems using a Petrov–Galerkin framework with Lagrange basis test functions. Traditional methods, like Physics Informed Neural Networks (PINNs), use standard deep learning techniques, which suffer from a computational bottleneck. In contrast, RNNs offer an alternative by employing a random structure with random coefficients, only solving for the output layer. We allow for the application of numerical analysis principles by using RNNs as trial functions and piecewise Lagrange polynomials as test functions. The article covers the construction and properties of the RNN basis, the definition and solution of fractional boundary value problems, and the implementation of the RNN Petrov–Galerkin method. We derive the stiffness matrix and solve it using least squares. Error analysis shows that the method meets the requirements of the Lax–Milgram lemma along with a Ceá inequality, ensuring optimal error estimates, depending on the regularity of the exact solution. Computational experiments demonstrate the method’s efficacy, including multiples cases with both regular and irregular solutions. The results highlight the utility of RNN-based Petrov–Galerkin methods in solving fractional differential equations with experimental convergence.

The blow-up features of a shallow water wave equation on the line $R$ are investigated. The $L^2$ conservation law is utilized to derive several estimates of solutions for the equation. Sufficient conditions for wave breaking and lifespan of the solutions are established. Our main results contain parts of the wave breaking conditions for the Fornberg–Whitham and Degasperis–Procesi equations in the previous literatures.

The paper proves convergence of unsymmetric radial basis functions (RBFs) collocation for second order elliptic boundary value problems on the bounded domains. By using Schaback’s linear discretization theory, $L^2$ error is obtained based on the kernel-based trial spaces generated by the compactly supported radial basis functions. The present theory covers a wide range of kernel-based trial spaces including stationary and non-stationary approximation. The convergence rates depend on the regularity of the solution, the smoothness of the computing domain, and the approximation of scaled kernel-based spaces. Some numerical examples are added for illustration.

In Boffi et al. (2000), it was shown that the linear Lagrange element space on criss-cross meshes and its divergence exhibit spurious eigenvalues when applied in the mixed formulation of the Laplace eigenvalue problem, despite satisfying both the inf–sup condition and ellipticity on the discrete kernel. The lack of a Fortin interpolation is responsible for the spurious eigenvalues produced by the linear Lagrange space. In contrast, results in Boffi et al. (2022) confirm that quartic and higher-order Lagrange elements do not yield spurious eigenvalues on general meshes without nearly singular vertices, including criss-cross meshes as a special case. In this paper, we investigate quadratic and cubic Lagrange elements on criss-cross meshes. We prove the convergence of discrete eigenvalues by fitting the Lagrange elements on criss-cross meshes into a complex and constructing a Fortin interpolation. As a by-product, we construct bounded commuting projections for the finite element Stokes complex, which induces isomorphisms between cohomologies of the continuous and discrete complexes. We provide numerical examples to validate the theoretical results.

We analyse a reaction–diffusion system describing the growth of microbial species in a model of flocculation type that arises in biology. A generalized model is formulated on a one dimensional bounded domain with feed terms at one end of the interval. Existence of global classical positive solutions is proved under general growth assumptions, with polynomial flocculation and deflocculation rates that guarantee uniform sup norm bounds for all time t obtained by an $L^p-$energy functional estimate. We also show finite time blow up can occur when the yield coefficients are large enough. Also, using arguments relying on the spectral and fixed theory, we show persistence and existence of nonhomogeneous population steady-states. Finally, we present some numerical simulations to show the combined effects of motility coefficients and the flocculation–deflocculation rates on the coexistence of species.

In this paper, we study the application of Itô-vector projection [1] to the optimal filtering problem. The algorithm projects one SDE to another, possibly lower dimensional, SDE by minimizing an Itô–Taylor expansion of the local projection error’s $L^2$ norm. We explicitly derive the projection filter equation for a general class of parametric densities, and then specifically apply it to exponential families. We demonstrate that for the case where the measurement drift function is in the span of the natural statistics, the Itô-vector projection filter (IVPF) coincides with the Stratonovich-projection filter (SPF) [2]. We then compare the performance of the IVPF against the SPF (with both being implemented using the Gaussian bijection proposed in [3] and the sparse Gauss–Patterson numerical integration) for two-dimensional optimal filtering problem to show the effectiveness of the proposed algorithm. We vary the measurement drift function to four different functions that are not in the span of natural statistics. Based on one hundred Monte Carlo simulations for each measurement drift, we found that their performances are comparable, with the IVPF potentially offering a slightly more robust performance. However, in our current numerical implementation, the SPF consistently outperforms the IVPF in terms of speed.

In this paper we prove Poincaré inequalities for the Discrete de Rham (DDR) sequence on a general connected polyhedral domain $\Omega$ of $R^3$. We unify the ideas behind the inequalities for all three operators in the sequence, deriving new proofs for the Poincaré inequalities for the gradient and the divergence, and extending the available Poincaré inequality for the curl to domains with arbitrary second Betti numbers. A key preliminary step consists in deriving “mimetic” Poincaré inequalities giving the existence and continuity of the solutions to topological balance problems useful in general discrete geometric settings. As an example of application, we study the stability of a novel DDR scheme for the magnetostatics problem on domains with general topology.

In this study, the $\psi$-Caputo fractional derivative (as a generalization of the classical Caputo derivative where the fractional derivative is defined with respect to the function $\psi$) is considered to introduce a class of multi-term time fractional 2D diffusion equations. A numerical method based on the Chebyshev cardinal polynomials (CCPs) is proposed to solve this problem. In this way, a new operational matrix for the $\psi$-Caputo fractional derivative of the CCPs is provided. By approximating the solution of the problem by a finite series of the CCPs (with some unknown coefficients) and employing the derived fractional matrix, an algebraic system of equations is generated, which by solving it the expressed coefficients, and consequently, the problem’s solution are identified. The validity of the established method is investigated by solving some numerical examples.

In the traditional heterogeneous agent model, investors are assumed to be risk averse, and the wealth expected utility function maximization principle is used to form the optimal asset quantity demand. In such models, the risk aversion coefficient of investors is often assumed to be constant. This paper considers that the risk aversion coefficient of investors is time-varying and changes with the change of wealth, and establishes an endogenous evolutionary mechanism model formed by fundamental analysts, technical analysts, and market makers. Compared with the fixed risk aversion coefficient model, this paper analyzes the investor’s behavior, the interaction between investor behaviors, and the influence of different types of investors on the stability of the market. At the same time, we test asset price and asset behavior and conclude that investor behavior affects the stability of the system model. The numerical simulation of the corresponding stochastic model shows that the model can simulate the basic characteristics of financial time series, such as the partial peak and thick tail of asset return series, and the long memory of fluctuations.