We present an analytical theory of electrostatic interactions of two spherical dielectric particles of arbitrary radii and dielectric constants, immersed into a polarizable ionic solvent (assuming that the linearized Poisson–Boltzmann framework holds) and bearing arbitrary charge distributions expanded in multipolar terms. The presented development entails a novel two-center re-expansion analytical theory that expands upon and improves the existing ones, bypassing the conventional expansions in modified Bessel functions. On this basis, we develop a specific matrix formalism that facilitates the construction of asymptotic expansions in ascending order of Debye screening terms of potential coefficients, which are then employed to find exact closed-form expressions for the total electrostatic energy. In particular, this work allows us to explicitly and precisely quantify the k-screened terms of the potential coefficients and mutual interaction energy. Specific cases of monopolar and dipolar distributions are described in particular detail. Comprehensive numerical examples and tests of series convergence and the relative balance of leading and higher-order terms of the mutual interaction energy are presented depending on the inter-particle distance and particles’ radii. The results of this work find application in soft matter modeling and, in particular, in computational biophysics and colloid science, where the availability of increasingly larger experimental structures at the atomic-level resolution makes numerical treatment challenging and calls for more efficient expressions and an increased range of validity.
We analyze the inverse boundary value-problem to determine the fractional order ν of nonautonomous semilinear subdiffusion equations with memory terms from observations of their solutions during small time. We obtain an explicit formula reconstructing the order. Based on the Tikhonov regularization scheme and the quasi-optimality criterion, we construct the computational algorithm to find the order ν from noisy discrete measurements. We present several numerical tests illustrating the algorithm in action.
An equivalent definition of the fractional Caputo derivative D ν t g, for ν ∈ (0, 1), is found, within suitable assumptions on g. Some applications to the fractional calculus and to the theory of fractional partial differential equations are then discussed. In particular, this alternative definition is used to prove the maximum principle for the classical solutions to the linear subdiffusion equation subject to nonhomogeneous boundary conditions. This approach also allows to construct numerical solutions to the initial-boundary value problem for the subdiffusion equation with memory. Contents 1. Introduction 383 2. Function Spaces and Notation 385 3. Equivalence of (1.1) and (1.2) and Applications 386 4. The Maximum Principle 388 5. Some Technical Results 390 6. Proof of Theorem 3.1 391 7. Proof of Theorems 4.1-4.3 393 Appendix: Numerical Simulations 397 References 401
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