Semi-Analytical Solutions for Fuzzy Caputo–Fabrizio Fractional-Order Two-Dimensional Heat Equation

In the analysis in this article, we developed a scheme for the computation of a semi-analytical solution to a fuzzy fractional-order heat equation of two dimensions having some external diffusion source term. For this, we applied the Laplace transform along with decomposition techniques and the Adomian polynomial under the Caputo–Fabrizio fractional differential operator. Furthermore, for obtaining a semi-analytical series-type solution, the decomposition of the unknown quantity and its addition established the said solution. The obtained series solution was calculated and approached the approximate solution of the proposed equation. For the validation of our scheme, three different examples have been provided, and the solutions were calculated in fuzzy form. All the three illustrations simulated two different fractional orders between 0 and 1 for the upper and lower portions of the fuzzy solution. The said fractional operator is nonsingular and global due to the presence of the exponential function. It globalizes the dynamical behavior of the said equation, which is guaranteed for all types of fuzzy solution lying between 0 and 1 at any fractional order. The fuzziness is also included in the unknown quantity due to the fuzzy number providing the solution in fuzzy form, having upper and lower branches.

Interaction of Reactive Gases with Platinum Aerosol Particles at Room Temperature: Effects on Morphology and Surface Properties

Nanoparticles produced in technical aerosol processes exhibit often dendritic structures, composed of primary particles. Surprisingly, a small but consistent discrepancy was observed between the results of common aggregation models and in situ measurements of structural parameters, such as fractal dimension or mass-mobility exponent. A phenomenon which has received little attention so far is the interaction of agglomerates with admixed gases, which might be responsible for this discrepancy. In this work, we present an analytical series, which underlines the agglomerate morphology depending on the reducing or oxidizing nature of a carrier gas for platinum particles. When hydrogen is added to openly structured particles, as investigated by tandem differential mobility analysis (DMA) and transmission electron microscopy (TEM) analysis, Pt particles compact already at room temperature, resulting in an increased fractal dimension. Aerosol Photoemission Spectroscopy (APES) was also able to demonstrate the interaction of a gas with a nanoscaled platinum surface, resulting in a changed sintering behavior for reducing and oxidizing atmospheres in comparison to nitrogen. The main message of this work is about the structural change of particles exposed to a new environment after complete particle formation. We suspect significant implications for the interpretation of agglomerate formation, as many aerosol processes involve reactive gases or slightly contaminated gases in terms of trace amounts of unintended species.

On the Electromagnetic Field of an Overhead Line Current Source

This work presents an analytical series-form solution for the time-harmonic electromagnetic (EM) field components produced by an overhead current line source. The solution arises from casting the integral term of the complete representation for the generated axial electric field into a form where the non-analytic part of the integrand is expanded into a power series of the vertical propagation coefficient in the air space. This makes it possible to express the electric field as a sum of derivatives of the Sommerfeld integral describing the primary field, whose explicit form is known. As a result, the electric field is given as a sum of cylindrical Hankel functions, with coefficients depending on the position of the field point relative to the line source and its ideal image. Analogous explicit expressions for the magnetic field components are obtained by applying Faraday’s law. The results from numerical simulations show that the derived analytical solution offers advantages in terms of time cost with respect to conventional numerical schemes used for computing Sommerfeld-type integrals.

Electrostatic Interaction of Point Charges in Three-Layer Structures: The Classical Model

Electrostatic interaction energy W between two point charges in a three-layer plane system was calculated on the basis of the Green’s function method in the classical model of constant dielectric permittivities for all media involved. A regular method for the calculation of W ( Z , Z ′ , R ) , where Z and Z ′ are the charge coordinates normal to the interfaces, and R the lateral (along the interfaces) distance between the charges, was proposed. The method consists in substituting the evaluation of integrals of rapidly oscillating functions over the semi-infinite interval by constructing an analytical series of inverse radical functions to a required accuracy. Simple finite-term analytical approximations of the dependence W ( Z , Z ′ , R ) were proposed. Two especially important particular cases of charge configurations were analyzed in more detail: (i) both charges are in the same medium and Z = Z ′ ; and (ii) the charges are located at different interfaces across the slab. It was demonstrated that the W dependence on the charge–charge distance S = R 2 + Z − Z ′ 2 differs from the classical Coulombic one W ∼ S − 1 . This phenomenon occurs due to the appearance of polarization charges at both interfaces, which ascribes a many-body character to the problem from the outset. The results obtained testify, in particular, that the electron–hole interaction in heterostructures leading to the exciton formation is different in the intra-slab and across-slab charge configurations, which is usually overlooked in specific calculations related to the subject concerned. Our consideration clearly demonstrates the origin, the character, and the consequences of the actual difference. The often used Rytova–Keldysh approximation was analyzed. The cause of its relative success was explained, and the applicability limits were determined.

Analysis of Mode Ⅲ Interface Fracture for Magneto-Electro-Elastic Materials

A novel finite element discretized symplectic method is developed for analyzing interface fracture of magneto-electro-elastic (MEE) materials under anti-plane loads. The overall cracked body is meshed by conventional finite elements and divided into a finite size singular region near the crack tip (near field) and a regular region far away from the crack tip (far field). In the near field, a based-Hamiltonian model is introduced to find the analytical series expressions, and the large number nodal unknowns are condensed into a small set of the undetermined coefficients of the symplectic series by a transformation. The nodal unknowns in the far field remain unchanged. The stress, electric and magnetic intensity factors, energy release rates (ERRs) and explicit expressions of singular field variables in the near field are simultaneously obtained without any processing.

Applications of General Residual Power Series Method to Differential Equations with Variable Coefficients

This paper is devoted to studying the analytical series solutions for the differential equations with variable coefficients. By a general residual power series method, we construct the approximate analytical series solutions for differential equations with variable coefficients, including nonhomogeneous parabolic equations, fractional heat equations in 2D, and fractional wave equations in 3D. These applications show that residual power series method is a simple, effective, and powerful method for seeking analytical series solutions of differential equations (especially for fractional differential equations) with variable coefficients.

PERFORMANCE OF POROUS MARINE STRUCTURES OF SINGLE AND DOUBLE PERFORATED SEAWALLS IN REGULAR OBLIQUE WAVES

In the present paper we examine the performance of two very common types of wave absorbing porous marine structures under regular oblique waves. The first structure consists of a single perforated vertical seawall, whereas the second consists of two perforated vertical seawalls creating what is called a chamber system (Jarlan type breakwater). The structures are surface piercing thus eliminating wave overtopping. Two methods are used for the present investigation. In the first method the problem of the interaction of obliquely incident linear waves upon a pair of porous vertical seawalls is first formulated in the context of the linear diffraction theory. The resulting boundary integral equation, which is matched with far-field solutions represented in terms of analytical series with unknown coefficients, and appropriate boundary conditions at the free surface, seabed and seawalls, is then solved numerically using the multi-domain boundary element method. In the second method a semi-analytical solution is developed by means of the eigenfunction expansions and a minimization approach using a least square method. In both methods the dissipation of the wave energy due to the presence of the perforated seawalls is represented by a simple yet effective relation in terms of the porosity parameter appropriate for thin perforated walls. The results are presented in terms of reflection and transmission coefficients, and the wave energy dissipation. Effects of the incident wave angles, porosities and depths of the walls and other major parameters of interest are explored.