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This paper presents the modified boundary knot method (MBKM) for solving the homogeneous harmonic type boundary value problems (BVPs). Since no non-singular general solutions are applicable for harmonic-type equations, the general solutions of Helmholtz-type operator with a free parameter λ can be used to approximate the solutions of these problems by adjusting the λ. Compared with the classical boundary knot method (BKM) where the source nodes are the same as the boundary collocation nodes, the MBKM employs the ghost points method, which resets the source points to a disk-like region covering the primary problem area. This modification results in better accuracy without any increase in the computational cost. On the other hand, as the accuracy of the MBKM depends heavily on the parameter λ , the effective condition number (ECN) is first employed to find a proper λ in MBKM. Several 2D and 3D numerical examples are listed to illustrate the superior performance of the MBKM in solving harmonic-type BVPs. The accuracy of MBKM is improved by one to two orders of magnitude compared to the classical BKM. Meanwhile, the validity of the ECN for obtaining a suitable λ for problems under complex geometric regions is also demonstrated. [ABSTRACT FROM AUTHOR]

This manuscript presents a comprehensive exploration of the nonlinear charged particle oscillation equation, employing the Laplace–Adomian decomposition method (LDM) to obtain approximate analytical solutions. The investigation leads to the formulation of five initial equations governing the oscillatory behavior of a charged particle, which are visually represented and analyzed with insightful interpretations. Notably, the existing literature lacks an exact solution to this problem. However, this paper fills this gap by presenting an approximate analytical solution utilizing the LDM. The solution is carefully studied and analyzed, contributing to a deeper understanding of the complex behavior of charged particle oscillation. [ABSTRACT FROM AUTHOR]

Upon the problem of transforming triangular mesh into quadrilateral mesh, gradient field via harmonic equation is created, integral flow is tracked, and parameterized mesh is reconstructed. First, gradient field construction theory, data structure model and solution scheme of sparse matrix are constructed. Then, one uniform algorithm for solving flow line node by integration of local coordinate transformation and parameterized equation is advanced, and schemes like gradient convergence, shortest distance and extreme parameter are optimally occupied upon special cases of no intersection or multiple intersections when tracing flow line. Finally, the algorithm is verified via case studies. The result shows that flow lines are characterized as iso-parameterized and closed, mesh reconstruction of complicate models has no bifurcation, and mesh quality is increased with its density. The result proves that mathematical method has robustness and uniqueness for mesh reconstruction representation compared with traditional geometry method, and it will have more application scenarios.

In various fields of applied mathematics, e.g., electrostatics, heat conduction, fluid mechanics, elastostatics, etc., boundary-value problems involving regions described in spheroidal, toroidal, and bispherical coordinate systems reduce to a system of second-order difference equations, whose solution, {xn}T0 with xn E should vanish asymptotically, i.e., limn--xn = 0. There are several methods for constructing such {xn}T0. However, in general, those methods do not guarantee limn--xn = 0. Moreover, in actual computations, they yield an approximate solution {xn}=0 different from the truncated true solution {xn}=0 and coinciding with the solution of the system being truncated at N with xjv+i set to 0. This work establishes sufficient conditions for the existence of an asymptotically vanishing solution to the system and provides the rate of convergence of the solution to the truncated system. Those results are used to analyze systems of second-order difference equations arising in the boundary-value problems in electrostatics, heat conduction, fluid mechanics, and elastostatics when a medium contains an inhomogeneity having the shape of either a torus or two unequal spheres. In particular, when m = 1 in those problems, ||xn -- xn|| decays exponentially as N oc. [ABSTRACT FROM AUTHOR]

The model of elastic porous solid medium for simulation of fluid penetration in an elastic porous body has been investigated. Similar processes can occur as a result of exposure of, for example, hard coal seams or deep solid type concrete, glass, etc. to fluid under high pressure. Assuming that an elastic body is a bundle of capillaries, a linear relationship between the filtration coefficient and the first invariant of the stress tensor has been revealed. Therefore, the problem of filtration through a deformable elastic medium has been reduced to two tasks: finding the stress tensor and solving the problem of filtration with the known filtration coefficient. In the general case, both tasks are complicated for analytical studies, but they can be solved numerically, for example, by the method of finite element analysis. The problem is significantly simplified for infinitely long cylindrical bodies. In the present work, the simplest mathematical model of plane elastic stress state in the transverse weightness field within both Hooke's law and Darcy's law with a constant filtration coefficient has been considered. In this case, the elastic deformation is described by the Airy biharmonic function, while filtration by the harmonic function. Based on Green's integral formula, integral relations combined into a single system have been found for the desired functions. The numerical solution has been performed using the method of boundary elements, with the help of which the problem has been reduced to a system of linear equations. Using a round tube, comparative analysis of the numerical and analytical solutions has been carried out. Numerical values have been obtained for the required parameters of an elliptical tube.

Roughly speaking a regular Dirichlet subspace of a Dirichlet form is a subspace which is also a regular Dirichlet form on the same state space. In particular, the domain of regular Dirichlet subspace is a closed subspace of the Hilbert space induced by the domain and α-inner product of original Dirichlet form. We investigate the orthogonal complement of regular Dirichlet subspace for one-dimensional Brownian motion in this paper. Our main results indicate that this orthogonal complement has a very close connection with the α-harmonic equation under Neumann type condition. [ABSTRACT FROM AUTHOR]

In this paper, we study the harmonic equation involving subcritical exponent $$(P_{\varepsilon })$$ : $$ \Delta u = 0 $$ , in $$\mathbb {B}^n$$ and $$\displaystyle \frac{\partial u}{\partial \nu } + \displaystyle \frac{n-2}{2}u = \displaystyle \frac{n-2}{2} K u^{\frac{n}{n-2}-\varepsilon }$$ on $$ \mathbb {S}^{n-1}$$ where $$\mathbb {B}^n $$ is the unit ball in $$\mathbb {R}^n$$ , $$n\ge 5$$ with Euclidean metric $$g_0$$ , $$\partial \mathbb {B}^n = \mathbb {S}^{n-1}$$ is its boundary, K is a function on $$\mathbb {S}^{n-1}$$ and $$\varepsilon $$ is a small positive parameter. We construct solutions of the subcritical equation $$(P_{\varepsilon })$$ which blow up at two different critical points of K. Furthermore, we construct solutions of $$(P_{\varepsilon })$$ which have two bubbles and blow up at the same critical point of K. [ABSTRACT FROM AUTHOR]

In geometry processing and shape analysis, several applications have been addressed through the properties of the spectral kernels and distances, such as commute-time, biharmonic, diffusion, and wave distances. Our survey is intended to provide a background on the properties, discretization, computation, and main applications of the Laplace-Beltrami operator, the associated differential equations (e.g., harmonic equation, Laplacian eigenproblem, diffusion and wave equations), Laplacian spectral kernels and distances (e.g., commute-time, biharmonic, wave, diffusion distances). While previous work has been focused mainly on specific applications of the aforementioned topics on surface meshes, we propose a general approach that allows us to review Laplacian kernels and distances on surfaces and volumes, and for any choice of the Laplacian weights. All the reviewed numerical schemes for the computation of the Laplacian spectral kernels and distances are discussed in terms of robustness, approximation accuracy, and computational cost, thus supporting the reader in the selection of the most appropriate method with respect to shape representation, computational resources, and target application. [ABSTRACT FROM AUTHOR]