Two issues that arise when solving Stokes flow problems with the hydrodynamical single-layer potential are addressed. First, the resulting boundary integral equation is singular, and second, discretizations lead to dense matrices. We discuss a well-posed modified equation which is equivalent for zero net-flux. Furthermore, we describe a multiscale basis that lead to sparse stiffness matrices. T...

This setting can be generalized by replacing E by a differential 1-form ω, subject to the Maxwell-type equation (δd− k)ω = 0 [7]. Herein, d denotes the exterior derivative, and δ the co-derivative, its formal L adjoint [2,8]. The latter equation seamlessly extends to dimensions n other than 3. Moreover, it can be posed in Riemannian manifolds, while the original double curl equation relies on f...

Since the classical iterative methods for solving nonlinear ill-posed problems are locally convergent, this paper constructs a robust and widely convergent method for identifying parameter based on homotopy algorithm, and investigates this method’s convergence in the light of Lyapunov theory. Furthermore, we consider 1-D elliptic type equation to testify that the homotopy regularization can ide...

In many applications, such as the heat conduction and hydrology, there is a need to recover the (possibly discontinuous) diffusion coefficient a from boundary measurements of solutions of a parabolic equation. The complete inverse problem is ill posed and nonlinear, so numerical solution is quite difficult, and we linearize the problem around constant a. We study and solve numerically the linea...

In this paper we study finite dimensional approximations to Boussinesq type equations. Our methods are based on infinite dimensional center manifold theory. The main advantage of our approach is that we can handle both well-posed and ill-posed versions of the Boussinesq equation. We show that for suitable initial conditions, our approximations describe the dynamics accurately for long enough ti...

Many physical phenomena can be described by a partial differential equation Pu=0. Here P denotes some differential operator or system of such operators, and u, the unknown function, is either a scalar or a vector. The differential equation connects the derivatives of u at each point of its domain D. The mathematician is interested in the global consequences of this local constraint, especially ...

We propose a fuzzy-based approach aiming at finding numerical solutions to some classical problems. We use the technique of F-transform to solve a second-order ordinary differential equation with boundary conditions. We reduce the problem to a system of linear equations and make experiments that demonstrate applicability of the proposed method. We estimate the order of accuracy of the proposed ...

The Perona-Malik equation is an ill-posed forward-backward parabolic equation with some application in image processing. In this paper, we study the Perona-Malik type equation on a ball in an arbitrary dimension n and show that there exist infinitely many radial weak solutions to the homogeneous Neumann boundary problem for smooth nonconstant radially symmetric initial data. Our approach is to ...

A one-way wave equation is a partial differential equation which, in some approximate sense, behaves like the wave equation in one direction but permits no propagation in the opposite one. Such equations are used in geophysics, in underwater acoustics, and as numerical "absorbing boundary conditions". Their construction can be reduced to the approximation of vl s2 on [-1,1] by a rational functi...