Solving nonlinear equations in Banach spaces (real or complex nonlinear equations, nonlinear systems, and nonlinear matrix equations, among others), is a non-trivial task that involves many areas of ...science and technology. Usually the solution is not directly affordable and require an approach using iterative algorithms. This Special Issue focuses mainly on the design, analysis of convergence, and stability of new schemes for solving nonlinear problems and their application to practical problems. Included papers study the following topics: Methods for finding simple or multiple roots either with or without derivatives, iterative methods for approximating different generalized inverses, real or complex dynamics associated to the rational functions resulting from the application of an iterative method on a polynomial. Additionally, the analysis of the convergence has been carried out by means of different sufficient conditions assuring the local, semilocal, or global convergence. This Special issue has allowed us to present the latest research results in the area of iterative processes for solving nonlinear equations as well as systems and matrix equations. In addition to the theoretical papers, several manuscripts on signal processing, nonlinear integral equations, or partial differential equations, reveal the connection between iterative methods and other branches of science and engineering.
Hairpin-containing pre-miRNAs, produced from pri-miRNAs, are precursors of miRNAs (microRNAs) that play essential roles in gene expression and various human diseases. Current qPCR-based methods used ...to quantify pre-miRNAs are not effective to discriminate between pre-miRNAs and their parental pri-miRNAs. Here, we developed the intramolecular ligation qPCR method (iLIME) to quantify and sequence pre-miRNAs specifically. This method utilizes T4 RNA ligase 1 to convert pre-miRNAs into circularized RNAs, allowing us to design PCR primers to quantify pre-miRNAs, but not their parental pri-miRNAs. In addition, the iLIME also enables us to sequence the ends of pre-miRNAs using next-generation sequencing. Therefore, this method offers a simple and effective way to quantify and sequence pre-miRNAs, so it will be highly beneficial for investigating pre-miRNAs when addressing research questions and medical applications.
Exact solutions of nonlinear differential equations are of great importance to the theory and practice of complex systems. The main point of this review article is to discuss a specific methodology ...for obtaining such exact solutions. The methodology is called the SEsM, or the Simple Equations Method. The article begins with a short overview of the literature connected to the methodology for obtaining exact solutions of nonlinear differential equations. This overview includes research on nonlinear waves, research on the methodology of the Inverse Scattering Transform method, and the method of Hirota, as well as some of the nonlinear equations studied by these methods. The overview continues with articles devoted to the phenomena described by the exact solutions of the nonlinear differential equations and articles about mathematical results connected to the methodology for obtaining such exact solutions. Several articles devoted to the numerical study of nonlinear waves are mentioned. Then, the approach to the SEsM is described starting from the Hopf–Cole transformation, the research of Kudryashov on the Method of the Simplest Equation, the approach to the Modified Method of the Simplest Equation, and the development of this methodology towards the SEsM. The description of the algorithm of the SEsM begins with the transformations that convert the nonlinearity of the solved complicated equation into a treatable kind of nonlinearity. Next, we discuss the use of composite functions in the steps of the algorithms. Special attention is given to the role of the simple equation in the SEsM. The connection of the methodology with other methods for obtaining exact multisoliton solutions of nonlinear differential equations is discussed. These methods are the Inverse Scattering Transform method and the Hirota method. Numerous examples of the application of the SEsM for obtaining exact solutions of nonlinear differential equations are demonstrated. One of the examples is connected to the exact solution of an equation that occurs in the SIR model of epidemic spreading. The solution of this equation can be used for modeling epidemic waves, for example, COVID-19 epidemic waves. Other examples of the application of the SEsM methodology are connected to the use of the differential equation of Bernoulli and Riccati as simple equations for obtaining exact solutions of more complicated nonlinear differential equations. The SEsM leads to a definition of a specific special function through a simple equation containing polynomial nonlinearities. The special function contains specific cases of numerous well-known functions such as the trigonometric and hyperbolic functions and the elliptic functions of Jacobi, Weierstrass, etc. Among the examples are the solutions of the differential equations of Fisher, equation of Burgers–Huxley, generalized equation of Camassa–Holm, generalized equation of Swift–Hohenberg, generalized Rayleigh equation, etc. Finally, we discuss the connection between the SEsM and the other methods for obtaining exact solutions of nonintegrable nonlinear differential equations. We present a conjecture about the relationship of the SEsM with these methods.
The Special Issue entitled “Advances in Rock Mechanics and Geotechnical Engineering” is devoted to the publication of the latest research, field works, and laboratory investigations in the area of ...rock mechanics and geotechnical engineering. This Special Issue has published novel contributions in different areas of geotechnical and geomechanical engineering such as slope and embankment, tunneling and underground space technologies, pile and foundation, rock mechanics and rock blasting, excavation and leveling projects, ground improvement techniques, unsaturated soil, practical issues in soft soil, mining technology, geo-environmental engineering, new laboratory testing, applied geology for construction, and novel geotechnical construction methods. The focus of this reprint is on the development of computational methods for solving problems in the fields of rock mechanics and geotechnical engineering.
The Special Issue entitled “Advances in Rock Mechanics and Geotechnical Engineering” is devoted to the publication of the latest research, field works, and laboratory investigations in the area of ...rock mechanics and geotechnical engineering. This Special Issue has published novel contributions in different areas of geotechnical and geomechanical engineering such as slope and embankment, tunneling and underground space technologies, pile and foundation, rock mechanics and rock blasting, excavation and leveling projects, ground improvement techniques, unsaturated soil, practical issues in soft soil, mining technology, geo-environmental engineering, new laboratory testing, applied geology for construction, and novel geotechnical construction methods. The focus of this reprint is on the development of computational methods for solving problems in the fields of rock mechanics and geotechnical engineering.
Wavelet analysis is a new method called ‘numerical microscope’ in signal and image processing. It has the desirable advantages of multi-resolution properties and various basis functions, which ...fulfill an enormous potential for solving partial differential equations (PDEs). The numerical analysis with wavelet received its first attention in 1992, since then researchers have shown growing interest in it. Various methods including wavelet weighted residual method (WWRM), wavelet finite element method (WFEM), wavelet boundary method (WBM), wavelet meshless method (WMM) and wavelet-optimized finite difference method (WOFD), etc. have acquired an important role in recent years. This paper aims to make a comprehensive review and classification on wavelet-based numerical analysis and to note their merits, drawbacks, and future directions. And thus the present review helps readers identify research starting points in wavelet-based numerical analysis and guides researchers and practitioners.
•Wavelet analysis is a method called numerical microscope in signal processing and numerical analysis.•A comprehensive review on wavelet-based numerical analysis is made and their merits, drawbacks, and future directions are noted.•From the aspect of algorithm construction, the wavelet-based numerical analysis methods are categorized into five types.
We have assembled de novo the Escherichia coli K-12 MG1655 chromosome in a single 4.6-Mb contig using only nanopore data. Our method has three stages: (i) overlaps are detected between reads and then ...corrected by a multiple-alignment process; (ii) corrected reads are assembled using the Celera Assembler; and (iii) the assembly is polished using a probabilistic model of the signal-level data. The assembly reconstructs gene order and has 99.5% nucleotide identity.
A nonconformal twofold domain decomposition method (TDDM) based on the hybrid finite element method-boundary element method (FEM-BEM) is proposed for analyzing 3-D multiscale composite structures. ...The proposed TDDM starts by partitioning the composite object into a closed exterior boundary domain and an interior volume domain. The interior and exterior boundary value problems are coupled to each other through the Robin transmission conditions (RTCs). Both domains are then independently decomposed into subregions to facilitate computation. Specifically, FEM-DDM with the second order transmission conditions (SOTCs) is employed for the interior domain, and BEM-discontinuous Galerkin (BEM-DG) based on the combined field integral equation (CFIE) is applied for the exterior boundary domain. The proposed TDDM allows for nonconformal discretization between any touching subdomains. Without the introduction of a stabilization term that relies on a line integral over intersection of nonmatching meshes and relevant terms involving surface-line integrals, the proposed TDDM provides an effective domain decomposition (DD) preconditioner for the global system. Numerical examples are presented, and the comparisons of the simulation results with FEM-BEM confirm the validity and accuracy of TDDM. Moreover, its ability to model practical large-scale and multiscale targets is also demonstrated.
The goal of this article is to discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations and to show that several well-known methods for ...obtaining exact solutions of such equations are connected to SEsM. In more detail, we show that the Hirota method is connected to a particular case of SEsM for a specific form of the function from Step 2 of SEsM and for simple equations of the kinds of differential equations for exponential functions. We illustrate this particular case of SEsM by obtaining the three- soliton solution of the Korteweg-de Vries equation, two-soliton solution of the nonlinear Schrödinger equation, and the soliton solution of the Ishimori equation for the spin dynamics of ferromagnetic materials. Then we show that a particular case of SEsM can be used in order to reproduce the methodology of the inverse scattering transform method for the case of the Burgers equation and Korteweg-de Vries equation. This particular case is connected to use of a specific case of Step 2 of SEsM. This step is connected to: (i) representation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a "small" parameter ϵ; (ii) solving the differential equations arising from this representation by means of Fourier series, and (iii) transition from the obtained solution for small values of ϵ to solution for arbitrary finite values of ϵ. Finally, we show that the much-used homogeneous balance method, extended homogeneous balance method, auxiliary equation method, Jacobi elliptic function expansion method, F-expansion method, modified simple equation method, trial function method and first integral method are connected to particular cases of SEsM.
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