Boundary Integral Equations

A shape optimization procedure is developed, using the boundary element method, for two-dimensional anisotropic structures to minimize weight while satisfying certain constraints upon stresses and geometry. A directly differentiated form of boundary integral equation with respect to geometric design variables is used to calculate shape design sensitivities of anisotropic materials. The boundary element method is very suitable for shape optimization and in comparison with the finite element method needs much fewer data, related only to the boundary of the structure being considered. Because a directly differentiated form of the boundary integral equation can be used to determine the derivatives of the objective and constraint functions, the accuracy of computation is very high. Because of the non-linear nature of weight and stresses, the numerical optimization method used in the program is the feasible direction approach, together with the golden section method for the one-dimensional search. Three example problems with anisotropic material properties are presented to demonstrate the applications of this general-purpose program.

Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent?

Explicit high-order energy-preserving methods for general Hamiltonian partial differential equations

In this paper we shall introduce new constructions of approximate solutions of general linear partial differential equations with constant coefficients on the whole spaces, and establish fundamental estimates of the solutions depending on the inhomogeneous terms. This will be done by combining general ideas of the Tikhonov regularization and discretization of bounded linear operator equations on reproducing kernel Hilbert spaces. Furthermore, we will provide approximate solutions for the related inverse source problems.

This paper presents the applicati on of a general purpose partial differential equation solver, called PDEase2D and the associated soil property functions to the analysis of saturated-unsaturated seepage problem. Examples of steady state and transient seepage through an earth dam are used to illustrate the use of PDEase2D in solving saturated-unsaturated seepage problems.

On the conventional boundary integral equation formulation for piezoelectric solids with defects or of thin shapes

usual boundary conditions, and constraints u~U(t, z)~Em, where the control space U = U(t, z) is a given subset of Era. Here z =(z 1 . . . . . : ) are state variables and u =(u 1 .. . . , u m) control variables. As stated by P. K. RASHEVSKY [6], very general partial differential equations and systems can be written locally in the form (2). For instance, second order partial differential equations such as z~,+z~s= 1 u) can be immediately written in the form f ( t , s, z 1, zlt, z~,

Data-driven analysis of parametrized acoustic systems in the frequency domain

Direct transformation of the volume integral in the boundary integral equation for treating three-dimensional steady-state anisotropic thermoelasticity involving volume heat source

A very general partial differential equation in space and time satisfied by the gene frequency in a monoecious population distributed continuously over an arbitrary habitat is derived. The treatment is restricted to a single diallelic locus in the absence of mutation and random drift, and it is supposed that time is continuous, births and deaths occur at random, and migration is independent of genotype. With the further assumptions that migration is isotropic and homogeneous, the population density is constant and uniform (as permitted by the population regulation mechanism included in the formulation), and Hardy-Weinberg proportions obtain locally, this partial differential equation reduces to the simplest multidimensional generalization of the classical Fisher-Haldane cline model. The efficacy of migration and selection in maintaining genetic variability at equilibrium in this model is investigated by deducing conditions for the existence of clines under various circumstances. The effects of the degree of dominance, a neutral belt between the regions where a particular allele is advantageous and deleterious, finiteness of the habitat, and habitat dimensionality are evaluated. Provided at least one of the alleles is favored only in a finite region, excluding the special case in which its total effective selective coefficient is zero, if conditions for supporting a cline are too unfavorable because migration is too strong, selection is too weak, or both, a cline cannot exist at all. Thus, unless there is overdominance, the population must be monomorphic. It is possible for a cline which can barely exist under the prevailing ecological circumstances to show a large amount of variation in gene frequency.

The primary aim is to develop a new class of pseudo-differential operators involving Linear Canonical Transform (LCT) and some of its essential results in Schwartz space and tempered distribution space. The secondary aim is to find applications of the LCT while constructing new general partial differential equations (Heat and Wave) and derive their solutions. Further, we have discussed particular cases of these partial differential equations. Furthermore, we have also shown the resolution of the solution of these equations graphically.

The electric field integral equation (EFIE) is one of simpl formulations of boundary integral equations for Maxwell's equations. It is widely known that the discretised EFIE is ill-conditioned, namely iteration methods to solve the discretised EFIE require a lot of iteration numbers, which results in much computational time. One of effective solutions to this problem is the Caldern preconditioning The Caldern preconditioning constructs a preconditioner based on the so-called Caldern formula, which implies that the square of the electric field integral operator (EFIO) is well-conditioned. Unfortunately simple discretisation of the squared EFIO is known to break down. Hence, in many previous studies of the Caldern preconditioning for the EFIE, the dual basis function is used in addition to the standard basis function, such as the Buffa-Christiansen (BC) basis function Although the Caldern preconditioning can be successfully implemented with the BC and RWG functions, discretisation of the EFIO with the BC basis function takes much more computational time than that with the RWG function. On the other hand our group has found that the dual basis function for the roof-top basis function defined on a quadrilateral mesh is also the roof-top basis function

Formulation of boundary integral equations for three-dimensional elasto-plastic flow

Huygen's principle, in the form of boundary integral equations, is applied to the problem of radio wave propagation. Complex propagation is analyzed by dividing the region between transmitter and receiver into a number of zones and propagating the solution between these zones by means of integral equations with simple approximate kernels. In the limit that the regions become dense, the approach leads to a useful form of the WKB solution, and this is demonstrated through its application to the study of propagation through a disturbed ionosphere.

In this paper, we introduce the operators related to mixed boundary value problems for general linear elliptic partial complex differential equations in the unit disc of the complex plane. The solvability of the relevant boundary value problems will be studied by transforming them into singular integral equations.