In this paper, fractal calculus, which is called Fα-calculus, is reviewed. Fractal calculus is implemented on fractal interpolation functions and Weierstrass functions, which may be ...non-differentiable and non-integrable in the sense of ordinary calculus. Graphical representations of fractal calculus of fractal interpolation functions and Weierstrass functions are presented.
Materials science is highly significant in space program investigation, energy production and others. Therefore, designing, improving and predicting advanced material properties is a crucial ...necessity. The high temperature creep and corrosion resistance of Ni-based superalloys makes them important materials for turbine blades in aircraft engines and land-based power plants. The investment casting process of turbine blades is costly and time consuming, which makes process simulations a necessity. These simulations require fundamental models for the microstructure formation. In this paper, we present advanced analytical techniques in describing the microstructures obtained experimentally and analyzed on different sample’s cross-sectional images. The samples have been processed on board the International Space Station using the MSL-EML device based on electromagnetic levitation principles. We applied several aspects of fractal analysis and obtained important results regarding fractals and Hausdorff dimensions related to the surface and structural characteristics of CMSX-10 samples. Using scanning electron microscopy (SEM), Zeiss LEO 1550, we analyzed the microstructure of samples solidified in space and successfully performed the fractal reconstruction of the sample’s morphology. We extended the fractal analysis on the microscopic images based on samples solidified on earth and established new frontiers on the advanced structures prediction.
The microstructure–property relationship in poly(methyl methacrylate) PMMA composites is very important for understanding interface phenomena and the future prediction of properties that further help ...in designing improved materials. In this research, field emission scanning electron microscopy (FESEM) images of denture PMMA composites with SrTiO3, MnO2 and SrTiO3/MnO2 were used for fractal reconstructions of particle agglomerates in the polymer matrix. Fractal analysis represents a valuable mathematical tool for the characterization of the microstructure and finding correlation between microstructural features and mechanical properties. Utilizing the mathematical affine fractal regression model, the Fractal Real Finder software was employed to reconstruct agglomerate shapes and estimate the Hausdorff dimensions (HD). Controlled energy impact and tensile tests were used to evaluate the mechanical performance of PMMA-MnO2, PMMA-SrTiO3 and PMMA-SrTiO3/MnO2 composites. It was determined that PMMA-SrTiO3/MnO2 had the highest total absorbed energy value (Etot), corresponding to the lowest HD value of 1.03637 calculated for SrTiO3/MnO2 agglomerates. On the other hand, the highest HD value of 1.21521 was calculated for MnO2 agglomerates, while the PMMA-MnO2 showed the lowest Etot. The linear correlation between the total absorbed impact energy of composites and the HD of the corresponding agglomerates was determined, with an R2 value of 0.99486, showing the potential use of this approach in the optimization of composite materials’ microstructure–property relationship.
Polymers and polymer matrix composites are commonly used materials with applications extending from packaging materials to delicate electronic devices. Epoxy resins and fiber-reinforced epoxy-based ...composites have been used as adhesives and construction parts. Fractal analysis has been recognized in materials science as a valuable tool for the microstructural characterization of composites by connecting fractal characteristics with composites’ functional properties. In this study, fractal reconstructions of different microstructural shapes in an epoxy-based composite were performed on field emission scanning electron microscopy (FESEM) images. These images were of glass fiber reinforced epoxy as well as a hybrid composite containing both glass and electrospun polystyrene fibers in an epoxy matrix. Fractal reconstruction enables the identification of self-similarity in the fractal structure, which represents a novelty in analyzing the fractal properties of materials. Fractal Real Finder software, based on the mathematical affine fractal regression model, was employed to reconstruct different microstructure shapes and calculate fractal dimensions to develop a method of predicting the optimal structure–property relations in composite materials in the future.
We consider systems of non-affine iterative functional equations. From the constructive form of the solutions, recently established by the authors, representations of these systems in terms of ...symbolic spaces as well as associated fractal structures are constructed. These results are then used to derive upper bounds both for the appropriate fractal dimension and the corresponding Hausdorff dimension of solutions. Using the formalism of iterated function systems, we obtain a sharp result on the Hausdorff dimension in terms of the corresponding fractal structures. The connections of our results with related objects known in the literature, including Girgensohn functions, fractal interpolation functions and Weierstrass functions, are established.
The challenges in productivity of satellite mobile devices are growing rapidly to overcome the question of miniaturization. The intention is to supply the electrical and microwave properties of ...materials by discovering their outstanding electronic properties. Neodymium Zinc Titanate (NZT) can be a promising ferroelectric material due to its stable dielectric and microwave properties. The grain size and shape of NZT have a strong influence on overall material performances. Therefore, shape, reconstruction and property of the coming compound take an important part and can be predicted before being utilized in the devices. The significant of this research is to define ferroelectric properties of NZT and to characterize it by using Fractal Nature Analysis (FNA). FNA is a powerful mathematical technique that could be applied to improve the grain shape and interface reconstruction. The fractal structure is identified by its self-similarity. The self-similarity of an object means a repetition of shapes in smaller scales. A measure of this structure is computed using the Hausdorff dimension. It is for the first time in this investigation the Fractal analysis method is applied for the microwave materials microstructure reconstruction which makes this research an innovative work and will open the door for Curie–Weiss law fractal correction. In connection to our previous research for dielectric properties fractalization, we had some characterization and reconstruction data which include the Hausdorff dimension (HD).
We construct an explicit formula for the fractal interpolation function associated to an IFS with variable parameters. The solution is given in terms of the base p representation of numbers. This ...construction is a consequence of the formulation of the problem in a general functional equation setting. We introduce compatibility conditions as essential hypotheses to ensure problems in the functional system form are well-defined.
Fractal analysis was used to estimate the fractal dimensions and further define the structure of lanthanum magnesium titanium oxide. The microstructures and space groups of the materials were ...determined. This is relevant to the electrical properties at microwave frequencies, which are important for miniaturising electronic components. Fractal analysis or scanning mathematical algorithms of the microstructure were used for the first time in this study to determine the microstructure of the material, which demonstrates the novelty of this research. The shape, side, and angle calculations do not match those of the cubic structure. Based on the Bravais lattices in the cubic structure, the lattice parameters are equal, and all angles must be equal to 90°, whereas our results proved that the side parameters are not equal and at least one angle differs. The fractal regression analysis results showed that lanthanum magnesium titanate has a monoclinic structure with a P21/n space group.
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Systems of iterative functional equations with a non-trivial set of contact points are not necessarily solvable, as the resulting intersections may lead to an overdetermination of the system. To ...obtain existence and uniqueness results additional conditions must be imposed on the system. These are the compatibility conditions, which we define and study in a general setting. An application to the affine and doubly affine cases allows us to solve an open problem in the theory of functional equations. In the last section we consider a special problem in a different perspective, showing that a complex compatibility condition may result in an elegant and simple property of the solution.
We formulate a general theoretical framework for systems of iterative functional equations between general spaces
X
and
Y
. We find general necessary conditions for the existence of solutions. When
X
...and
Y
are topological spaces, we characterize continuity of solutions; when
X
,
Y
are metric spaces, we find sufficient conditions for existence and uniqueness. For finite-order systems, we construct explicit formulae for the solution. We provide an extended list of examples, including fractal interpolation functions, which are covered by our general framework.
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