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Fredholm Theory
Published in Eberhard Malkowsky, Vladimir Rakočević, Advanced Functional Analysis, 2019
Eberhard Malkowsky, Vladimir Rakočević
Fredholm operators, the basic objects in Fredholm operator theory, are generalizations of the operators of the form A = I − K, where K∈K(X) and X is a Banach space. They are very important for the spectral theory of operators. In this chapter we present some elements from Fredholm theory. We refer the interested readers to the textbooks and monographs [2,3,15,38,101,112,142] for further reading.
Three-dimensional interfacial mechanical-thermal fracture analysis of a two-dimensional hexagonal quasicrystal coating structure
Published in Mechanics of Advanced Materials and Structures, 2023
Xin Zhang, MingHao Zhao, CuiYing Fan, Chunsheng Lu, HuaYang Dang
In the present work, the problem is considered to be quasi-static, with one-side coupling relation. This implies that the temperature field is not influenced by mechanical loads. For brevity, the basic equations of 2D hexagonal QCs and cubic crystals are given in Appendix A. Then, based on the strict differential operator theory and generalized Almansi’s theorem, the 3D thermoelastic general solution of a 2D hexagonal QC can be expressed by six potential functions [37, 41], i.e. wherein and are material-related constants, and with and the roots of positive real part [37].
Approximation properties of bivariate α-fractal functions and dimension results
Published in Applicable Analysis, 2021
Sangita Jha, A. K. B. Chand, M. A. Navascués, Abhilash Sahu
The most essential question in classical approximation theory is how to represent an arbitrary function or a dataset concerning traditional (piecewise) smooth functions. In most physical situations, the original data-generating functions are non-smooth in nature, and we cannot use classical approximation techniques to approximate them. Barnsley [1,2] introduced the concept of fractal interpolation functions (FIFs) that are constructed as attractors of suitable iterated function systems (IFSs) on complete metric spaces. In [3], authors have studied the attractors of IFSs in uniform spaces. Fractal functions constitute advancement in the technique of approximation, since all the traditional functions of real-world data interpolation, can be generalized through fractal methods. For a given continuous function f, a family of smooth or non-smooth α-fractal functions can be obtained depending on the choice of scaling parameters. Wang and Yu [4] introduced FIF with variable scaling functions and Serpa and Buescu [5] gave an explicit representation of FIF with this setting. The notion of FIF provides a bounded linear operator, termed the α-fractal operator. Navascués studied these operators in details in earlier works [6,7]. The concept of α-fractal operator links the theory of fractal function to the area Functional analysis, Operator theory, Harmonic analysis, and Approximation theory.
In-plane and out-of-plane free vibrations of functionally graded composite arches with graphene reinforcements
Published in Mechanics of Advanced Materials and Structures, 2021
Zhicheng Yang, Shaoyu Zhao, Jie Yang, Jiangen Lv, Airong Liu, Jiyang Fu
Arch structures are widely used in modern engineering design, including mechanical, civil, and aerospace engineering. For example, arch is used as the basic structural element in an aircraft to support the skeleton. Due to the graceful shape and excellent mechanical characteristics, more and more arch bridges have also been built in cities. Therefore, there is an urgent need for an in-depth understanding of the mechanical behaviors of arches. Extensive research works on the conventional arch structures have been carried out. Liu et al. [24–27] conducted a series of investigations on the lateral-torsional buckling and dynamic instability of conventional arch structures. Xu et al. [28–32] experimentally studied the deformation of composite arch under static loading conditions by using terrestrial laser scanning (TLS) technology. Pi et al. [33, 34] analytically studied the nonlinear equilibrium and buckling of an isotropic arch with fixed and pinned ends under an arbitrary radial point load. Lee et al. [35] derived the governing equations for out-of-plane vibrations of curved non-uniform beams and discussed the effects of taper ratio, center angle and arc length on the first two natural frequencies. Chang et al. [36] obtained the upper and lower bounds of the first four natural frequencies of elastic clamped arches based on differential operator theory. Irie et al. [37] studied the steady state of out-of-plane vibration of curved beams with internal damping. It should be noted that the above studies are for isotropic homogeneous arches only.