A generalization of the Mittag–Leffler function and solution of system of fractional differential equations

Abstract The solutions of system of linear fractional differential equations of incommensurate orders are considered and analytic expressions for the solutions are given by using the Laplace transform and multi-variable Mittag–Leffler functions of matrix arguments. We verify the result with numeric solutions of an example. The results show that the Mittag–Leffler functions are important tools for analysis of a fractional system. The analytic solutions obtained are easy to program and are approximated by symbolic computation software such as MATHEMATICA

Relaxation Functions Interpolating the Cole–Cole and Kohlrausch–Williams–Watts Dielectric Relaxation Models

To describe non-Debye relaxation phenomena observed in dielectric materials, the Cole–Cole (CC) relaxation model in the frequency domain and the Kohlrausch–Williams–Watts (KWW) relaxation model in the time domain were introduced in the physics of dielectrics. In this paper, we propose a new relaxation model with two parameters besides a relaxation time by expressing the relaxation function in the time domain in terms of the Mittag–Leffler functions. The proposed model represents a group of non-Debye relaxation phenomena and shows a transition between the CC and the KWW models. The relaxation properties described by the new model are analyzed, including the response function, the normalized complex dielectric permittivity, dielectric storage and loss factors as well as the relaxation frequency and time spectral functions. The presented relaxation function has a concise form and is expected to be applied to more complex relaxation phenomena

Solution of Fractional Differential Equation Systems and Computation of Matrix Mittag–Leffler Functions

In this paper, solutions for systems of linear fractional differential equations are considered. For the commensurate order case, solutions in terms of matrix Mittag–Leffler functions were derived by the Picard iterative process. For the incommensurate order case, the system was converted to a commensurate order case by newly introducing unknown functions. Computation of matrix Mittag–Leffler functions was considered using the methods of the Jordan canonical matrix and minimal polynomial or eigenpolynomial, respectively. Finally, numerical examples were solved using the proposed methods

Generalized Path Optimization Problem for a Weighted Digraph over an Additively Idempotent Semiring

In this paper, a generalized path optimization problem for a weighted digraph (i.e., directed graph) over an additively idempotent semiring was considered. First, the conditions for power convergence of a matrix over an additively idempotent semiring were investigated. Then we proved that the path optimization problem is associated with powers of the adjacency matrix of the weighted digraph. The classical matrix power method for the shortest path problem on the min-plus algebra was generalized to the generalized path optimization problem. The proposed generalized path optimization model encompasses different path optimization problems, including the longest path problem, the shortest path problem, the maximum reliability path problem, and the maximum capacity path problem. Finally, for the four special cases, we illustrate the pictorial representations of the graphs with example data and the proposed method

Multiple mixed solutions of the nonlocal sine-Gordon equation

Abstract In this paper, we propose the nonlocal sine-Gordon equation from the AKNS system by the Parity and the Time symmetries. Moreover, the main work is to construct the Darboux transformation for the nonlocal sine-Gordon equation. In particular, various types of solutions are obtained by taking a seed solution, such as, soliton solutions, kink solutions, mixed solutions

Influence of carbon fiber preform structure on microstructure and mechanical properties of C/C-SiC composites fabricated by combined CVI-GSI method

Using carbon fiber cloth with different areal densities and tow sizes, two kinds of carbon fiber preforms with same carbon cloth laminated structure were produced through different z-direction stitching methods. Then, C/C-SiC composites were prepared by combining chemical vapor infiltration (CVI) with gas silicon infiltration (GSI). The influence of carbon fiber preform structure on the microstructure and mechanical properties of CVI-GSI C/C-SiC composites was studied. The results show that the density, phase composition, structure, and properties of the two composites prepared from preforms with the same fiber volume fraction and C/C preform density are significantly different. The smaller carbon fiber tow (1K) and carbon cloth surface density (92 g/m2), as well as the larger voids left by lock stitching, provide more sufficient channels for the infiltration of Si vapor in the GSI reaction process. Thus T1 composite finally prepared has low porosity, uniform structure, and higher performance, with bending strength, modulus, and fracture toughness of 300.97 MPa, 51.75 GPa, and 11.32 MPa·m1/2, respectively. The comprehensive control of the initial preform structure and the C/C intermediate structure is the key to the preparation of high performance C/C-SiC composites by the CVI-GSI process

Preparation of C/Zr0.5Hf0.5C-SiC composite by PIP process and its microstructure and flexural properties

Based on the self-made Zr0.5Hf0.5C precursor and commercial liquid polycarbosilane, C/Zr0.5Hf0.5C-SiC composite was successfully prepared by the precursor impregnation and pyrolysis(PIP) process. The influence of the thickness of pyrolytic C coating on the structure and bending properties of composite materials was studied. The results show that the self-made Zr0.5Hf0.5C precursor can be converted into Zr0.5Hf0.5C solid solution at a relatively low temperature of 1400 ℃. Because of its good permeability, the transformed Zr0.5Hf0.5C matrix exists in both the inter-bundle and intra-bundle regions of the C/Zr0.5Hf0.5C-SiC composite, which presents as a layered structure on SiC matrix. The phase composition of C/Zr0.5Hf0.5C-SiC composite mainly includes C, SiC and Zr0.5Hf0.5C. The densities of three groups of composites with different thicknesses of pyrolytic C coating (0.67, 0.84, 1.36 μm) are 2.07, 1.99, 1.98 g/cm3, respectively. SiC content in the composite decreases with the increase of the thickness of pyrolytic C coating. The three groups of composites with different thicknesses show pseudoplastic fracture mode during bending loading tests, bending strength, bending modulus and fracture toughness are above 410 MPa, 60 GPa and 15.6 MPa·m1/2, respectively. Good interface bonding and pre-introduced SiC matrix are the keys to obtaining excellent bending properties of C/Zr0.5Hf0.5C-SiC composites

Synthesis of a Novel P/N/S-Containing Flame Retardant and Its Application in Epoxy Resin: Thermal Property, Flame Retardance, and Pyrolysis Behavior

The combination of DOPO and 2-aminobenzothiazole (ABZ) was designed to develop P/N/S-containing flame retardant DOPO-ABZ, and its chemical structure was confirmed by HRMS, FTIR, and ¹H and ³¹P NMR. The reduced thermal stability of EP/DOPO-ABZ formulations was found through DSC and TGA, as compared to that of EP. Fire properties were evaluated by LOI, UL-94, and cone calorimeter tests, respectively. The results indicated that DOPO-ABZ imparted flame retardance to EP, and that EP/7.5 wt % DOPO-ABZ passed the V-0 rating, and acquired a LOI value of 33.5%; moreover, when the loading of DOPO-ABZ increased to 10 wt %, it could further suppress the heat release and smoke release of the curved epoxy resin. Finally, the flame-retardant mechanism was studied by TG-FTIR and py-GC/MS, disclosing that DOPO-ABZ exerted predominant gaseous-phase activity of fire inhibition via generating phosphorus-containing free radicals and nitrogen/sulfur-containing volatiles