Regularized Collocation in Distribution of Diffusion Times Applied to Electrochemical Impedance Spectroscopy

This paper is inspired by recently proposed approach for interpreting data of Electrochemical Impedance Spectroscopy (EIS) in terms of Distribution of Diffusion Times (DDT). Such an interpretation requires to solve a Fredholm integral equation of the first kind, which may have a non-square-integrable kernel. We consider a class of equations with above-mentioned peculiarity and propose to regularize them in weighted functional spaces. One more issue associated with DDT-problem is that EIS data are available only for a finite number of frequencies. Therefore, a regularization should unavoidably be combined with a collocation. In this paper we analyze a regularized collocation in weighted spaces and propose a scheme for its numerical implementation. The performance of the proposed scheme is illustrated by numerical experiments with synthetic data mimicking EIS measurements.

Nonlocal Symmetries for Time-Dependent Order Differential Equations

A new type of ordinary differential equation is introduced and discussed, namely, the time-dependent order ordinary differential equations. These equations can be solved via fractional calculus and are mapped into Volterra integral equations of second kind with singular integrable kernel. The solutions of the time-dependent order differential equations smoothly deforms solutions of the classical integer order ordinary differential equations into one-another, and can generate or remove singularities. An interesting symmetry of the solution in relation to the Riemann zeta function and Harmonic numbers was also proved.

Correlation Between Mesh Geometry and Stiffness Matrix Conditioning for Nonlocal Diffusion Models

Nonlocal diffusion models involve integral equations that account for nonlocal interactions and do not explicitly employ differential operators in the space variables. Due to the nonlocality they might look different from classical partial differential equation (PDE) models, but their local limit reduces to partial differential equations. The effect of mesh element anisotropy mesh refinement and kernel functions on the conditioning of the stiffness matrix for a nonlocal diffusion model on 2D geometric domains is considered, and the results compared with those obtained from typical local PDE models. Numerical experiments show that the condition number is bounded by (where c is a constant) for an integrable kernel function, and is not affected by the choice of the basis function. In contrast to local PDE models, mesh anisotropy and refinement affect the condition number very little.

Integral equations of the first kind of Sonine type

A Volterra integral equation of the first kindKφ(x):≡∫−∞xk(x−t)φ(t)dt=f(x)with a locally integrable kernelk(x)∈L1loc(ℝ+1)is called Sonine equation if there exists another locally integrable kernelℓ(x)such that∫0xk(x−t)ℓ(t)dt≡1(locally integrable divisors of the unit, with respect to the operation of convolution). The formal inversionφ(x)=(d/dx)∫0xℓ(x−t)f(t)dtis well known, but it does not work, for example, on solutions in the spacesX=Lp(ℝ1)and is not defined on the whole rangeK(X). We develop many properties of Sonine kernels which allow us—in a very general case—to construct the real inverse operator, within the framework of the spacesLp(ℝ1), in Marchaud form:K−1f(x)=ℓ(∞)f(x)+∫0∞ℓ′(t)[f(x−t)−f(x)]dtwith the interpretation of the convergence of this “hypersingular” integral inLp-norm. The description of the rangeK(X)is given; it already requires the language of Orlicz spaces even in the case whenXis the Lebesgue spaceLp(ℝ1).

Eigenvalues of smooth kernels

Suppose is a symmetric square integrable kernel on the unit square [0, 1]2. Thenis a compact symmetric operator on the Hilbert space L2[0, 1]. H. Weyl (see [2]) has shown that, if then the eigenvaluesof T satisfy as n → ∞. We prove a related result. Let W12[0, 1]2 denote the space of all K(x, t) ε L2[0, 1]2 which are absolutely continuous in x for each t and absolutely continuous in t for each x, and the partial derivatives ∂K/∂x(x, t), ∂K/∂t(x, t) are both in L2[0, 1]2. We slow that the eigenvalues of any satisfy .