Computers & Mathematics with Applications最新文献

We present a simple and robust numerical technique for a novel phase-field model of three-dimensional (3D) shape transformation. Shape transformation has been achieved using phase-field models. However, previous phase-field models have intrinsic drawbacks, such as shrinkage due to motion by mean curvature and unwanted growth. To overcome these drawbacks associated with previous models, we propose a novel phase-field model that eliminates these shortcomings. The proposed phase-field model is based on the Allen–Cahn (AC) equation with nonstandard mobility and a nonlinear source term. To numerically and efficiently solve the proposed phase-field equation, we adopt an operator splitting method, which consists of the AC equation with a nonstandard mobility and a fidelity equation. The modified AC equation is solved using a fully explicit finite difference method with a time step that ensures stability. For solving the fidelity equation, we use a semi-implicit scheme with a frozen coefficient. We have performed several numerical experiments with various 3D sources and target shapes to verify the robustness and efficacy of our proposed mathematical model and its numerical method.

Conventional freeze-drying takes a long drying time and makes the process expensive. High-quality biological materials, medicine, and vaccines may not find easy acceptance with this technology. To overcome the operative time, several engineering innovations are carried out. A long drying time during freeze-drying can be minimized by accelerating the sublimation rate. Obtaining a fast drying rate without harming the material properties is the prime focus of the accelerated freeze-drying (AFD) like-techniques. In connection with this, the study of temperature-dependent thermal-physical properties of the medium during sublimation is considered in this study. For example, a temperature-dependent volumetric heat source is assumed within the vapor region. An increase in the temperature field results in higher pressure. Therefore, a temperature-dependent specific heat of vapor pressure is also accounted for. Furthermore, the permeability of the medium and the specific heat of the water vapor are also assumed to be temperature-dependent. Exploring realistic theoretical models with variable-dependent characteristics and convection is essential since the experimental investigation of sublimation in a porous media may be challenging. Despite the previous available studies on sublimation heat and mass transfer, there is still a lack of mathematical modeling of this particular problem. To solve this non-linear sublimation problem, the Genocchi operational matrix of differentiation method ($GOMOD$) method is employed to obtain the numerical results. In case of full non-linear model, results obtained via current numerical technique are verified with finite-difference method (FDM). Furthermore, in a particular case, the accuracy test of the $GOMOD$ method against FDM is presented, and it is found that the current numerical technique is more accurate than FDM. In the current study, it is found that a temperature-dependent heat source offers a faster sublimation rate than a constant one. Similarly, temperature-dependent specific heat of vapor pressure accelerates the pressure distribution within the sublimated region. With temperature-dependent permeability, the concentration distribution within the medium decreases. Moreover, the temperature-dependent specific heat of water vapor delayed the sublimation rate. Results found from this study are expected to aid in AFD techniques, food industry and pharmaceuticals.

In this paper, a novel hybrid method based on the finite element method (FEM) and physics-informed kernel function neural network (PIKFNN) is proposed. The method is applied to predict underwater acoustic propagation induced by structural vibrations in diverse ocean environments, including the unbounded ocean, deep ocean, and shallow ocean. In the hybrid method, PIKFNN is regarded as an improved shallow physics-informed neural network (PINN) in which the activation function in the PINN is replaced with a physics-informed kernel function (PIKF). This ensures the integration of prior physical information into the neural network model. Moreover, PIKFNN circumvents embedding the governing equations into the loss function in the PINN and requires only training on boundary data. By using Green's function as PIKF and the structural-acoustic coupling response information obtained from the FEM as training data, PIKFNN can inherently capture the Sommerfeld radiation condition at infinity, which are naturally suitable for predicting ocean acoustic propagation. Numerical experiments demonstrate the accuracy and feasibility of FEM-PIKFNN in comparison with analytical solutions and finite element results.

Using elementary means, we derive the three most popular splittings of $e^{t(A+B)}$ and their error bounds in the case when A and B are (possibly unbounded) operators in a Hilbert space, generating strongly continuous semigroups, $e^{tA}$, $e^{tB}$ and $e^{t(A+B)}$. The error of these splittings is bounded in terms of the norm of the commutators $[A,B]$, $[A,[A,B\left]\right]$ and $[B,[A,B\left]\right]$.

The purpose of this article is to study how the integrals of the Gaussian radial basis function can be employed to produce the coefficients of approximations under the radial basis function - finite difference solver. Here these coefficients are reported for a five-point stencil. Error equations are derived to demonstrate that the convergence rate is four for approximating the 1st and 2nd differentiations of a function. Then the coefficients are used in solving multi-dimensional option pricing problems, which are modeled as time-dependent variable-coefficients parabolic partial differential equations with non-smooth initial conditions. The numerical simulations support the applicability and usefulness of the presented method.

In this study, we explore the theoretical and numerical aspects of the generalized Burgers-Huxley equation (a non-linear advection-diffusion-reaction problem) incorporating weakly singular kernels in a d-dimensional domain, where $d\in \{2,3\}$. For the continuous problem, we provide an in-depth discussion on the existence and the uniqueness of weak solution using the Faedo-Galerkin approximation technique. Further, regularity results for the weak solution are derived based on assumptions of smoothness for both the initial data and the external forcing. Using the regularity of the solution, the uniqueness of weak solutions has been established. In terms of numerical approximation, we introduce a semi-discrete scheme using the conforming finite element method (CFEM) for space discretization and derive optimal error estimates. Subsequently, we present a fully discrete approximation scheme that employs backward Euler discretization in time and CFEM in space. A priori error estimates for both the semi-discrete and fully discrete schemes are discussed under minimal regularity assumptions. To validate our theoretical findings, we provide computational results that lend support to the derived conclusions.

We introduce an efficient computational framework for data assimilation of fractional dynamical systems, extending traditional data assimilation techniques to fractional models. This framework offers effective computational methods that eliminate the need for complex adjoint model derivations and algorithm redesign. We establish the fundamental problem formulation, develop both the AtD and DtA approaches, and derive adjoint forms and numerical schemes for each method. Additionally, we create a unified fractional-order variational data assimilation framework applicable to both linear and nonlinear models, incorporating both explicit and implicit discrete methods. Specific discretization schemes and gradient formulas are derived for three distinct types of fractional-order models. The method's reliability and convergence are verified, and the effect of observation sparsity and quality is examined through numerical examples.