Search Results (16)

This paper presents a finite element methodology for generating continuous digital elevation models (DEMs) from discrete terrain data using the Poisson equation under steady-state conditions. Unlike conventional DEM interpolation techniques, the proposed methodology formulates terrain reconstruction as a constrained harmonic problem, solved directly on scattered point sets using standard finite element procedures, without requiring structured grids or intermediate interpolation stages. The approach interprets the elevation field as a harmonic scalar function whose smoothness is enforced by the variational formulation of the Poisson problem. The governing equation is solved using standard finite element procedures with Dirichlet boundary conditions applied at the measurement points, ensuring that the reconstructed surface passes exactly through the known elevations. The isotropic conductivity coefficient is set to unity and the source term to zero, which simplifies the formulation and yields a harmonic interpolation independent of any physical parameters. The resulting surfaces exhibit continuous slopes and reduced sensitivity to irregular data distributions. Numerical tests comprising two analytical examples and a real terrain case show that, compared with thin-plate FEM and RBF–NURBS reconstructions, the proposed Poisson-based approach yields smoother and more stable surfaces, with global errors of the same order of magnitude and reduced computational cost. Full article

In this paper, we develop a concise differential–potential framework for the functions of a generalized quaternionic variable in the two-parameter algebra ${\mathbb{H}}_{\alpha ,\beta }$, with $\alpha ,\beta \in \mathbb{R}\setminus \left\{0\right\}$. Starting from left/right difference quotients, we derive complete Cauchy–Riemann (CR) systems and prove that, away from the null cone where the reduced norm N vanishes, these first-order systems are necessary and, under ${\mathcal{C}}^1$ regularity, sufficient for left/right differentiability, thereby linking classical one-dimensional calculus to a genuinely four-dimensional setting. On the potential theoretic side, the Dirac factorization ${\Delta }_{\alpha ,\beta }=\overline{\mathcal{D}}\mathcal{D}=\mathcal{D}\overline{\mathcal{D}}$ shows that each real component of a differentiable mapping is ${\Delta }_{\alpha ,\beta }$-harmonic, yielding a clean second-order theory that separates the elliptic (Hamiltonian) and split (coquaternionic) regimes via the principal symbol. In the classical case $(\alpha ,\beta )=(-1,-1)$, we present a Poisson-type representation solving a model Dirichlet problem on the unit ball $B\subset {\mathbb{R}}^4$, recovering mean-value and maximum principles. For computation and symbolic verification, real $4\times 4$ matrix models for left/right multiplication linearize the CR systems. Examples (polynomials, affine CR families, and split-signature contrasts) illustrate the theory, and the outlook highlights boundary integral formulations, Green kernel constructions, and discretization strategies for quaternionic PDEs. Full article

Finite-difference methods are widely used to solve partial differential equations in diverse practical applications. Despite their prevalence, the computational efficiency of these methods encounters limitations due to the need to solve linear equation systems through matrix inversion or iterative solver, which is particularly challenging in scenarios involving high dimensions. The demand for numerical methods with high accuracy and fast computational speed is steadily increasing. To address this challenge, we present an efficient and accurate algorithm for high-dimensional numerical modeling. This approach combines a central finite-difference method with the discrete Sine transform (DST) scheme to solve the Poisson equation under Dirichlet boundary conditions (DBCs). To balance numerical accuracy and computation, the DST scheme is applied along one direction in the 2D case and two directions in the 3D case. This strategy effectively reduces problem complexity while maintaining low computational cost. The hybrid DST-accelerated finite-difference approach substantially lowers the computational cost associated with solving the Poisson equation on large grids. Comprehensive numerical experiments for 2D and 3D Poisson equations with DBCs have been conducted. The obtained numerical results demonstrate that the proposed hybrid method not only significantly reduces the computational expenses, but also maintains the central finite-difference accuracy. Full article

Geometric image transformations are fundamental to image processing, computer vision and graphics, with critical applications to pattern recognition and facial identification. The splitting-integrating method (SIM) is well suited to the inverse transformation $T^{-1}$ of digital images and patterns, but it encounters difficulties in nonlinear solutions for the forward transformation T. We propose improved techniques that entirely bypass nonlinear solutions for T, simplify numerical algorithms and reduce computational costs. Another significant advantage is the greater flexibility for general and complicated transformations T. In this paper, we apply the improved techniques to the harmonic, Poisson and blending models, which transform the original shapes of images and patterns into arbitrary target shapes. These models are, essentially, the Dirichlet boundary value problems of elliptic equations. In this paper, we choose the simple finite difference method (FDM) to seek their approximate transformations. We focus significantly on analyzing errors of image greyness. Under the improved techniques, we derive the greyness errors of images under T. We obtain the optimal convergence rates $O\left(H^2\right)+O(H/N^2)$ for the piecewise bilinear interpolations ($\mu =1$) and smooth images, where $H(\ll 1)$ denotes the mesh resolution of an optical scanner, and N is the division number of a pixel split into $N^2$ sub-pixels. Beyond smooth images, we address practical challenges posed by discontinuous images. We also derive the error bounds $O\left(H^{\beta }\right)+O(H^{\beta }/N^2)$, $\beta \in (0,1)$ as $\mu =1$. For piecewise continuous images with interior and exterior greyness jumps, we have $O\left(\sqrt{H}\right)+O(\sqrt{H}/N^2)$. Compared with the error analysis in our previous study, where the image greyness is often assumed to be smooth enough, this error analysis is significant for geometric image transformations. Hence, the improved algorithms supported by rigorous error analysis of image greyness may enhance their wide applications in pattern recognition, facial identification and artificial intelligence (AI). Full article

This paper is devoted to the study of stochastic optimal control of averaged stochastic differential delay equations (SDDEs) with semi-Markov switchings and their applications in economics. By using the Dynkin formula and solution of the Dirichlet–Poisson problem, the Hamilton–Jacobi–Bellman (HJB) equation and the inverse HJB equation are derived. Applications are given to a new Ramsey stochastic models in economics, namely the averaged Ramsey diffusion model with semi-Markov switchings. A numerical example is presented as well. Full article

This paper generalizes the efficient matrix decomposition method for solving the finite-difference (FD) discretized three-dimensional (3D) Poisson’s equation using symmetric 27-point, 4th-order accurate stencils to adapt more boundary conditions (BCs), i.e., Dirichlet, Neumann, and Periodic BCs. It employs equivalent Dirichlet nodes to streamline source term computation due to BCs. A generalized eigenvalue formulation is presented to accommodate the flexible 4th-order stencil weights. The proposed method significantly enhances computational speed by reducing the 3D problem to a set of independent 1D problems. As compared to the typical matrix inversion technique, it results in a speed-up ratio proportional to $n^4$, where $n$ is the number of nodes along one side of the cubic domain. Accuracy is validated using Gaussian and sinusoidal source fields, showing 4th-order convergence for Dirichlet and Periodic boundaries, and 2nd-order convergence for Neumann boundaries due to extrapolation limitations—though with lower errors than traditional 2nd-order schemes. The method is also applied to vortex-in-cell flow simulations, demonstrating its capability to handle outer boundaries efficiently and its compatibility with immersed boundary techniques for internal solid obstacles. Full article

In this paper, we propose an improved Nested Error Regression model in which the random effects for each area are given a prior distribution using the Poisson–Dirichlet Process. Based on this model, we mainly investigate the construction of the parameter estimation using the Empirical Bayesian(EB) estimation method, and we adopt various methods such as the Maximum Likelihood Estimation(MLE) method and the Markov chain Monte Carlo algorithm to solve the model parameter estimation jointly. The viability of the model is verified using numerical simulation, and the proposed model is applied to an actual small area estimation problem. Compared to the conventional normal random effects linear model, the proposed model is more accurate for the estimation of complex real-world application data, which makes it suitable for a broader range of application contexts. Full article

Cell migration in a biological medium towards a blood vessel is modeled, as a random process, sucessively inside an annulus (two-dimensional domain) and an annular cylinder (three-dimensional domain). The conditional probability function u for the cell moving inside such domains (tissue) fulfills by assumption a diffusion–advection equation that is subject to a Dirichlet boundary condition on the outer boundary and a Robin boundary condition on the inner boundary. The mean first-passage time (MFPT) function determined by u estimates the average time for the travelling cell to reach various interesting targets. The MFPT function fulfills a Poisson equation inside a domain with suitable boundary conditions, which give rise to various mathematical problems. The main novelty of this study is the characterization of such an MFPT function inside an annulus and an annular cylinder, which is subject to a Robin boundary condition on the inner boundary and a Dirichlet boundary condition on the outer one, and these are integral functions whose densities are the solution of an inhomogeneous system of linear integral equations. Full article

The role played by explicit formulas for solving boundary value problems for elliptic equations and systems is well known. In this paper, explicit formulas for a general solution of the Dirichlet problem for second-order elliptic systems in the unit disk are given. In addition, an iterative method for solving this problem for systems with respect to two unknown functions is described, and an integral representation of the Poisson type is obtained by applying this method. Full article

We investigate the half-space Dirichlet problem with summable boundary-value functions for an elliptic equation with an arbitrary amount of potentials undergoing translations in arbitrary directions. In the classical case of partial differential equations, the half-space Dirichlet problem for elliptic equations attracts great interest from researchers due to the following phenomenon: the solutions acquire qualitative properties specific for nonstationary (more exactly, parabolic) equations. In this paper, such a phenomenon is studied for nonlocal generalizations of elliptic differential equations, more exactly, for elliptic differential-difference equations with nonlocal potentials arising in various applications not covered by the classical theory. We find a Poisson-like kernel such that its convolution with the boundary-value function satisfies the investigated problem, prove that the constructed solution is infinitely smooth outside the boundary hyperplane, and prove its uniform power-like decay as the timelike independent variable tends to infinity. Full article

A general approach to solving the Dirichlet problem, both for bounded $3D$ domains and for their unbounded complements, in terms of the fractional ($3D$) Poisson equation, is presented. Lauren Schwartz class solutions are sought for tempered distributions. The solutions found are represented by a formula that contains the volume Riesz potential and the one-layer potential, the latter depending on the boundary data. Infinite regularity of fractional harmonic functions, analogous to the infinite smoothness of the classical harmonic functions, is also proved in the respective domain, no matter what the boundary conditions are. Other properties of the solutions, that are presumably of interest to mathematical physics, are also investigated. In particular, an intrinsic decay property, valid far from the common boundary, is shown. Full article

In this paper, in the class of smooth functions, integration and differentiation operators connected with fractional conformable derivatives are introduced. The mutual reversibility of these operators is proved, and the properties of these operators in the class of smooth functions are studied. Using transformations generalizing involutive transformations, a nonlocal analogue of the Laplace operator is introduced. For the corresponding nonlocal analogue of the Poisson equation, the solvability of some boundary value problems with fractional conformable derivatives is studied. For the problems under consideration, theorems on the existence and uniqueness of solutions are proved. Necessary and sufficient conditions for solvability of the studied problems are obtained, and integral representations of solutions are given. Full article

Assume that $(X,d,\mu )$ is a metric measure space that satisfies a Q-doubling condition with $Q>1$ and supports an $L^2$-Poincaré inequality. Let $𝓛$ be a nonnegative operator generalized by a Dirichlet form $\mathcal{E}$ and V be a Muckenhoupt weight belonging to a reverse Hölder class $R{H}_q\left(X\right)$ for some $q\ge (Q+1)/2$. In this paper, we consider the Dirichlet problem for the Schrödinger equation $-{\partial }_t^{2}u+𝓛u+Vu=0$ on the upper half-space $X\times {\mathbb{R}}_{+}$, which has f as its the boundary value on X. We show that a solution u of the Schrödinger equation satisfies the Carleson type condition if and only if there exists a square Morrey function f such that u can be expressed by the Poisson integral of f. This extends the results of Song-Tian-Yan [Acta Math. Sin. (Engl. Ser.) 34 (2018), 787-800] from the Euclidean space ${\mathbb{R}}^Q$ to the metric measure space X and improves the reverse Hölder index from $q\ge Q$ to $q\ge (Q+1)/2$. Full article

In this paper, we establish the unique existence and some decay properties of a global solution of a free boundary problem of the incompressible Navier–Stokes equations in $L_p$ in time and $L_q$ in space framework in a uniformly $H_{\infty }^2$ domain $\Omega \subset {\mathbb{R}}^N$ for $N\ge 4$. We assume the unique solvability of the weak Dirichlet problem for the Poisson equation and the $L_q$-$L_r$ estimates for the Stokes semigroup. The novelty of this paper is that we do not assume the compactness of the boundary, which is essentially used in the case of exterior domains proved by Shibata. The restriction $N\ge 4$ is required to deduce an estimate for the nonlinear term $\mathbf{G}\left(\mathbf{u}\right)$ arising from ${\mathrm{div}}_{}\mathbf{v}=0$. However, we establish the results in the half space ${\mathbb{R}}_{+}^N$ for $N\ge 3$ by reducing the linearized problem to the problem with $\mathbf{G}=\mathbf{0}$, where $\mathbf{G}$ is the right member corresponding to $\mathbf{G}\left(\mathbf{u}\right)$. Full article

The article is devoted to the development and creation of a multiphysics simulator that can, on the one hand, simulate the most significant physical processes in the IPMC actuator, and on the other hand, unlike commercial products such as COMSOL, can use computing resources economically. The developed mathematical model is an adjoint differential equation describing the transport of charged particles and water molecules in the ion-exchange membrane, the electrostatic field inside, and the mechanical deformation of the actuator. The distribution of the electrostatic potential in the interelectrode space is located by means of the solution of the Poisson equation with the Dirichlet boundary conditions, where the charge density is a function of the concentration of cations inside the membrane. The cation distribution was obtained by means of the solution of the equation system, in which the fluxes of ions and water molecules are described by the modified Nernst-Planck equations with boundary conditions of the third kind (the Robin problem). The cantilever beam forced oscillation equation in the presence of resistance (allowing for dissipative processes) with assumptions of elasticity theory was used to describe the actuator motion. A combination of the following computational methods was used as a numerical algorithm for the solution: the Poisson equation was solved by a direct method, the modified Nernst-Planck equations were solved by the Newton-Raphson method, and the mechanical oscillation equation was solved using an explicit scheme. For this model, a difference scheme has been created and an algorithm has been described, which can be implemented in any programming language and allows for fast computational experiments. On the basis of the created algorithm and with the help of the obtained experimental data, a program has been created and the verification of the difference scheme and the algorithm has been performed. Model parameters have been determined, and recommendations on the ranges of applicability of the algorithm and the program have been given. Full article