The bidomain equations have been widely used to mathematically model the electrical activity of the cardiac tissue. In this work, we present a potential theory-based Cartesian grid method which is referred as the kernel-free boundary integral (KFBI) method which works well on complex domains to efficiently simulate the linear diffusion part of the bidomain equation. After a proper temporal discretization, the KFBI method is applied to solve the resulting homogeneous Neumann boundary value problems with a second-order accuracy. According to the potential theory, the boundary integral equations reformulated from the boundary value problems can be solved iteratively with the simple Richardson iteration or the Krylov subspace iteration method. During the iteration, the boundary and volume integrals are evaluated by limiting the structured grid-based discrete solutions of the equivalent interface problems at quasi-uniform interface nodes without the need to know the analytical expression of Green's functions. In particular, the discrete linear system of the equivalent interface problem obtained from the standard finite difference schemes or the finite element schemes can be efficiently solved by fast elliptic solvers such as the fast Fourier transform based solvers or those based on geometric multigrid iterations after an appropriate modification at the irregular grid nodes. Numerical results for solving the FitzHugh-Nagumo bidomain equations in both two- and three-dimensional spaces are presented to demonstrate the numerical performance of the KFBI method such as the second-order accuracy and the propagation and scroll wave of the voltage simulated on the real human left ventricle model.
We study in this work an integral formulation for the radiative transfer equation (RTE) in anisotropic media with truncated approximation to the scattering phase function. The integral formulation consists of a coupled system of integral equations for the angular moments of the transport solution. We analyze the approximate separability of the kernel functions in these integral formulations, deriving asymptotic lower and upper bounds on the number of terms needed in a separable approximation of the kernel functions as the moment grows. Our analysis provides the mathematical understanding on wh…
The paper is devoted to the approximate solutions of the Fredholm integral equations of the second kind with the weak singular kernel that can have additional singularity in the numerator. We describe two problems that lead to such equations. They are the problem of minimization of small deviation and the entropy minimization problem. Both of them appear when considering dynamical system involving mixed fractional Brownian motion. In order to deal with the kernel with additional singularity applying well-known methods for weakly singular kernels, we prove the theorem on the approximation of so…
Consider a Urysohn integral equation $x - \mathcal{K} (x) = f$, where $f$ and the integral operator $\mathcal{K}$ with kernel of the type of Green's function are given. In the computation of approximate solutions of the given integral equation by Galerkin method, all the integrals are needed to be evaluated by some numerical integration formula. This gives rise to the discrete version of the Galerkin method. For $r \geq 1$, a space of piecewise polynomials of degree $\leq r-1$ with respect to a uniform partition is chosen to be the approximating space. For the appropriate choice of a numerical…
A method for analytical computation of the double surface integrals for all layer potential kernels associated with the Laplace Green's function, in the Galerkin boundary element method (BEM) in $\mathbb{R}^3$ using piecewise constant flat elements is presented. The method uses recursive dimensionality reduction from 4D ($\mathbb{R}^2\times\mathbb{R}^2$) based on Gauss' divergence theorem. Computable analytical expressions for all cases of relative location of the source and receiver triangles are covered for the single and double layer potentials and their gradients with analytical treatment …
The systems of nonlinear Volterra integral equations of the first kind with jump discontinuous kernels are studied. The iterative numerical method for such nonlinear systems is proposed. Proposed method employs the modified Newton-Kantorovich iterative process for the integral operators linearization. On each step of the iterative process the linear system of integral equations is obtained and resolved using the discontinuity driven piecewise constant approximation of the exact solution. The convergence theorem is proved. The polynomial collocation technique is also applied to solve such syste…
We consider a Urysohn integral operator $\mathcal{K}$ with kernel of the type of Green's function. For $r \geq 1$, a space of piecewise polynomials of degree $\leq r-1 $ with respect to a uniform partition is chosen to be the approximating space and the projection is chosen to be the orthogonal projection. Iterated Galerkin method is applied to the integral equation $x - \mathcal{K}(x) = f$. It is known that the order of convergence of the iterated Galerkin solution is $r+2$ and, at the above partition points it is $2r$. We obtain an asymptotic expansion of the iterated Galerkin solution at th…
The Riesz transform of $u$ : $\mathcal{S}(\mathbb{R}^n) \rightarrow \mathcal{S'}(\mathbb{R}^n)$ is defined as a convolution by a singular kernel, and can be conveniently expressed using the Fourier Transform and a simple multiplier. We extend this analysis to higher order Riesz transforms, i.e. some type of singular integrals that contain tensorial polyadic kernels and define an integral transform for functions $\mathcal{S}(\mathbb{R}^n) \rightarrow \mathcal{S'}(\mathbb{R}^{ n \times n \times \dots n})$. We show that the transformed kernel is also a polyadic tensor, and propose a general metho…
