Results in Applied Mathematics最新文献

We consider single-phase flow with solute transport where ions in the fluid can precipitate and form a mineral, and where the mineral can dissolve and release solute into the fluid. Such a setting includes an evolving interface between fluid and mineral. We approximate the evolving interface with a diffuse interface, which is modeled with an Allen–Cahn equation. We also include effects from temperature such that the reaction rate can depend on temperature, and allow heat conduction through fluid and mineral. As Allen–Cahn is generally not conservative due to curvature-driven motion, we include a reformulation that is conservative. This reformulation includes a non-local term that makes the use of standard Newton iterations for solving the resulting non-linear system of equations very slow. We instead apply L-scheme iterations, which can be proven to converge for any starting guess, although giving only linear convergence. The three coupled equations for diffuse interface, solute transport and heat transport are solved via an iterative coupling scheme. This allows the three equations to be solved more efficiently compared to a monolithic scheme, and only few iterations are needed for high accuracy. Through numerical experiments we highlight the usefulness and efficiency of the suggested numerical scheme and the applicability of the resulting model.

In this work, nonstandard finite difference method is presented for the numerical solution of time-fractional singularly perturbed convection–diffusion problems with a delay in time. The time-fractional derivative is considered in the Caputo sense and discretized using Crank–Nicholson technique. Then, a nonstandard finite difference scheme is constructed on a uniform mesh discretization along the spatial direction. The parameter-uniform convergence of the proposed method is proved rigorously and shown to be $\varepsilon$-uniform convergent with order of convergence $O\left({\left(\Delta t\right)}^2\right)$ along the temporal domain and $M^{-1}$ along the spatial domain. Finally, the proposed scheme is validated using model examples and the computational results are in agreement with the theoretical expectation.

This study focuses on the efficient and precise computation of Bessel transforms, defined as ${\int }_a^{b}f\left(x\right)J_{\nu }\left(\omega x\right)dx$. Exploiting the integral representation of $J_{\nu }\left(\omega x\right)$, these Bessel transformations are reformulated into the oscillatory integrals of Fourier-type. When $a>0$, these Fourier-type integrals are transformed through distinct complex integration paths for cases with $b<+\infty$ and $b=+\infty$. Subsequently, we approximate these integrals using the generalized Gauss–Laguerre rule and provide error estimates. This approach is further extended to situations where $a=0$ by partitioning the integral’s interval into two separate subintervals. Several numerical experiments are provided to demonstrate the efficiency and accuracy of the proposed algorithms.

This paper addresses the numerical analysis of a class of a degenerate second-order evolution equations. We employ a finite element method for spatial discretization and a family of implicit finite difference schemes for time discretization. Introducing a stabilization parameter, denoted by $\theta$, we propose a well-posed fully-discrete scheme. Sufficient conditions for its well-posedness and for quasi-optimal error estimates are established. The abstract theory is illustrated through the application to the degenerate wave equation, and numerical results validate our theoretical findings.

This paper explores an iterative approach to solve linear thermo-poroelasticity problems, with its application as a high-fidelity discretization utilizing finite elements during the training of projection-based reduced order models. One of the main challenges in addressing coupled multi-physics problems is the complexity and computational expenses involved. In this study, we introduce a decoupled iterative solution approach, integrated with reduced order modeling, aimed at augmenting the efficiency of the computational algorithm. The iterative technique we employ builds upon the established fixed-stress splitting scheme that has been extensively investigated for Biot’s poroelasticity. By leveraging solutions derived from this coupled iterative scheme, the reduced order model employs an additional Galerkin projection onto a reduced basis space formed by a small number of modes obtained through proper orthogonal decomposition. The effectiveness of the proposed algorithm is demonstrated through numerical experiments, showcasing its computational prowess.

In this paper, we present an efficient numerical algorithm for approximating integrals involving highly oscillatory Bessel functions with Cauchy-type singularities. By employing the technique of complex line integration, the highly oscillatory Bessel integrals are transformed into oscillatory integrals with a Fourier kernel. When the integration interval does not contain zeros, we use Cauchy’s theorem to transform the integration path to the complex plane and then use the Gaussian–Laguerre formula to compute the integral. For cases in which the integration interval contains zeros, we decompose the integral into two parts: the ordinary and the singular integral. We give a stable recursive formula based on Chebyshev polynomials and Bessel functions for ordinary integrals. For singular integrals, we utilize the MeijerG function for efficient computation. Numerical examples verify the effectiveness of the new algorithm and the fast convergence.

A nonlinear shallow water wave equation containing the Fornberg–Whitham model is considered. The phase portrait analytical technique is employed to establish the existence of the smooth, peaked and cusped solitary wave solutions of the equation under inhomogeneous boundary conditions. Asymptotic and numerical analysis illustrates the dynamical features for the smooth, peaked and cusped solitary wave solutions. Our results are helpful to further understand the dynamical tendency of the solutions when the space variable tends to positive or negative infinite.

We consider a perturbed linear boundary-value problem for a system of fractional differential equations with Caputo derivative. The boundary-value problem is specified by a linear vector functional, the number of components of which does not coincide with the dimension of the system of differential equations. This formulation of the problem is being considered for the first time and includes both underdetermined and overdetermined boundary-value problems. Under the condition that the solution of the homogeneous generating boundary-value problem is not unique and that the inhomogeneous generating boundary-value problem is unsolvable, the conditions for the bifurcation of solutions of this problem are determined. An iterative procedure for constructing a family of solutions of the perturbed linear boundary-value problem in the form of Laurent series in powers of a small parameter $\varepsilon$ with singularity at the point $\varepsilon =0$ is proposed. The results obtained by us generalize the known results of perturbation theory for boundary-value problems for ordinary differential equations.

The primary objective of this research is to examine a nonlinear elastic membrane equation incorporating delay and source terms within a bounded domain. We obtain sufficient conditions on the initial data and the involved functionals for which the energy of solutions with non positive initial energy as well as positive initial energy blow up in a finite-time. In addition, this research work provides estimates for the lifespan of these solutions.

In this paper, we present a wavelet collocation method for efficiently solving singularly perturbed differential–difference equations (SPDDEs) and one-parameter singularly perturbed differential equations (SPDEs) taking into account the singular perturbations inherent in control systems. These equations represent a class of mathematical models that exhibit a combination of differential and difference equations, making their analysis and solution challenging. The terms that include negative and positive shifts were approximated using Taylor series expansion. The main aim of this technique is to convert the problems by using operational matrices of integration of Haar wavelets into a system of algebraic equations that can be solved using Newton’s method. The adaptability and multi-resolution properties of wavelet functions offer the ability to capture system behavior across various scales, effectively handling singular perturbations present in the equations. Numerical experiments were conducted to showcase the effectiveness and accuracy of the wavelet collocation method, demonstrating its potential as a reliable tool for analyzing and solving SPDDEs in control system.