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In this article, we study a $$(2+1)$$ -dimensional non-linear Vakhnenko equation which appears in a relaxing high rate active barothropic medium. The Lie symmetry analysis is applied to study the solution of $$(2+1)$$ -dimensional Vakhnenko equation and also used to find the optimal system of one-dimensional Lie subalgebra. The two-dimensional non-linear Vakhnenko equation is reduced to various ordinary differential equations by applying similarity transformation method, and solving these reduced ordinary differential equations, we find invariant solutions of $$(2+1)$$ -dimensional Vakhnenko equation. Moreover, the adjoint equation of the Vakhnenko equation and its conservation laws are also obtained, which can be used for mathematical analysis.

Field experiences have shown that sub-synchronous oscillation (SSO) can occur in direct-drive wind farms with VSC-HVDC systems. Due to the complexity of the detailed wind farm model, a dynamic equivalent model, with a reasonable order reduction of the detailed model and still reflecting inside-wind-farm and wind-farm-grid SSO characteristics is essential. In this paper, based on the principle that similar matrices have identical eigenvalues, the SSO analysis of an N-machine wind farm with VSC-HVDC system is conducted by simplifying it into two single-machine systems. The modeling method of the two single-machine systems is presented. Four case studies are presented to verify the effectiveness of the proposed model when compared with the detailed model in various scenarios. The proposed model is also benchmarked with the output multiplication-based equivalent model (OMM). Comparison results show that although the system order is reduced significantly, the proposed simplified equivalent model can still reflect inside-wind-farm and wind-farm-grid SSO modes in various scenarios. Meanwhile, the rationality of the OMM in terms of the wind-farm-grid SSO analysis is verified theoretically.

Disorders have a rich influence on topological and localized properties. Here, we explore the effects of different type of disorders (intracell and intercell) on the non-Hermitian system. We first exhibit the phase diagram and find that the intracell disorder and intercell disorder can broaden and narrow the topological region, respectively. Moreover, the skin effect, which is unique in the non-Hermitian system, is broken by disorders. Furthermore, we propose the generalized localization length to settle the issue of how to determine the topological phase boundary explicitly in the disordered non-Hermitian system. Significantly, the rationality of this definition can be verified by similarity transformation, in which we prove that the topological invariant remains invariant. Finally, a byproduct of our definition is that one can analytically get the criticality of topology in the clean-limit non-Hermitian system.

By using scalar similarity transformation, nonlinear model of time-fractional diffusion/Harry Dym equation is transformed to corresponding ordinary fractional differential equations, from which a travelling-wave similarity solution of time-fractional Harry Dym equation is presented. Furthermore, numerical solutions of time-fractional diffusion equation are discussed. Again, through another similarity transformation, nonlinear model of space-fractional diffusion/Harry Dym equation is turned into corresponding ordinary differential equations, whose two similarity solutions are also worked out.

In this investigation, the stagnation point flow of silver water nanofluid over a linear stretching sheet using Buongiorno’s two-component non-homogeneous nanofluid model is studied. The governing equations are reduced to ordinary differential equations by using similarity transformation and solved numerically by using bvp4c function of MATLAB package. The impact of electrification in presence of viscous dissipation on normalized velocity, temperature and nanoparticle concentration is analysed and examined through graphs. The physical parameters like skin friction coefficient, rate of heat transfer and nanoparticle concentration are derived and presented by tables. It is found that the higher electrification parameter reduces the normalized base fluid temperature and enhances the normalized velocity due to Lorentz force. So, it may be concluded that electrification of particles is an important and possible mechanism for enhancement of thermal conductivity of base fluid.

Three-dimensional dusty fluid flow with volume fraction has been investigated in this paper. The presence of magnetic field and non-uniform heat source/sink is considered along with the flow over a stretching sheet enclosed in a porous medium. Using similarity transformation, the governing equations are transformed into ordinary differential equations. Matlab built-in solver Bvp4c has been used to solve the resulting non-linear ordinary differential equations. The effects of various parameters on the flow have been discussed and characteristics of heat transfer are determined. The results are shown using graphs and tables.

We continue the study of the octagon form factor which helps to evaluate a class of four-point correlation functions in $\mathcal{N}=4$ SYM theory. The octagon is characterised, besides the kinematical parameters, by a "bridge" of $\ell$ propagators connecting two non-adjacent operators. In this paper we construct an operator representation of the octagon with finite bridge as an expectation value in the Fock space of free complex fermions. The bridge $\ell$ appears as the level of filling of the Dirac sea. We obtain determinant identities relating octagons with different bridges, which we derive from the expression of the octagon in terms of discrete fermionic oscillators. The derivation is based on the existence of a previously conjectured similarity transformation, which we find here explicitly., 29 pages, no figures. Minor corrections in v2

How best to measure spatial saliency shift induced by image distortions is an open research question. Our previous study has shown that image distortions cause saliency to deviate from their original places in natural images, and the degree of such distortion-induced saliency variation (DSV) depends on image content and the properties of distortion. Being able to measure DSV benefits the development of saliency-based image quality algorithms. In this article, we first investigate the plausibility of using existing mathematical algorithms for measuring DSV and their potential limitations. We then develop a new algorithm for quantifying DSV based on a deep neural network. In the algorithm, namely, saliency similarity transformation DSV (ST-DSV), we design a coarse-grained to fine-grained saliency similarity transformation approach to achieve DSV measurement. The experimental results show that the proposed ST-DSV algorithm significantly outperforms existing methods in predicting the ground truth DSV.

In this article the problem of strong shock wave moving in a self-gravitating gaseous medium is examined for cylin- drically and spherically symmetric flow. It is assumed that the density distribution ahead of the shock wave is obeying power law. The governing system of quasi-linear partial differential equations are non-dimensionalized using similarity transformation and are expanded in a power series. An approximate analytical solution is constructed for the zeroth order approximation and the effect of flow parameters on the various hydrodynamical quantities are studied in detail and the results are presented for both cylindrical and spherical symmetry.

A study associated with bioconvection flow and heat transfer analysis due to Casson fluid and gyrotactic microorganisms over a linear stretching sheet through a porous medium in the presence of magnetic field is considered. The related governing equations of the physical situation are deformed into a system of nonlinear ordinary differential equations using Oberbeck–Boussinesq approximations and similarity transformation .The obtained coupled equations are solved with the help of differential transform method to get Taylor’s series solutions for the momentum, energy, concentration of nanoparticles, and density of microorganisms. The combined effects of distinct nondimensional parameters on the solutions are represented through graphs.

Symmetry methods for differential equations are a powerful tool for the solutions of differential equations. It linearizes nonlinear differential equations, reduces the order of differential equations, reduces the number of independent variables in partial differential equations, and solves almost all those differential equations for which the other analytic methods fail to solve them. Similarity transformation is a particular case of symmetries, but it is easy and often used to deal with differential equations. The similarity transformation can do all the aforementioned works. In this research, we use the similarity transformation to solve different nonlinear differential equations. Particularly, we will apply this transformation to the nonlinear Navier–Stokes partial differential equations to reduce them to ordinary differential equations. Ordinary differential equations are easy to deal with than partial differential equations. Some nonlinear physical examples of ODEs and PDEs are given to show that the similarity transformation solves those problems where the other analytic methods fail to work.

Using the experimental relations for approximating the effective thermophysical properties of a water/MgO-Ag hybrid nanofluid is the novelty of the present investigation in order to simulate the two dimensional MHD Casson flow past a linearly stretching/shrinking sheet with suction, radiation and convective boundary condition effects. The dual solutions along with its stability analysis are also taken into account. The Tiwari-Das mathematical formulation coupled with mass-based hybrid nanofluid model is implemented to write the governing PDEs which are then transferred to a complicated system of dimensionless ODEs with help of the similarity transformation technique. Then a well-known finite difference method with forth order accuracy in MATLAB is employed to solve them. It is proved that the selective experimental relations for effective thermophysical models must be considered more serious in the numerical modeling of hybrid nanofluids in the future. Moreover, results indicate that the range of the Casson fluid parameter for which the solution exists, enhances with radiation parameter, suction parameter and also second nanoparticle mass. Also it is shown that the magnetic field which is applied normal to the sheet results in boost up the similarity velocity profiles into the hydrodynamic boundary layer.

The Morris-Shore (MS) transformation is a powerful tool for decomposition of the dynamics of multistate quantum systems to a set of two-state systems and uncoupled single states. It assumes two sets of states wherein any state in the first set can be coupled to any state in the second set, but the states within each set are not coupled between themselves. Another important condition is the degeneracy of the states in each set, although all couplings between the states from different sets can be detuned from resonance by the same detuning. The degeneracy condition limits the application of the MS transformation in various physically interesting situations, e.g., in the presence of electric and/or magnetic fields or light shifts, which lift the degeneracy in each set of states, e.g., when these sets comprise the magnetic sublevels of levels with nonzero angular momentum. This paper extends the MS transformation to such situations, in which the states in each of the two sets are nondegenerate. To this end, we develop an alternative way for the derivation of Morris-Shore transformation, which can be applied to nondegenerate sets of states. We present an approximated eigenvalue approach, by which, in the limit of small detunings from degeneracy, we are able to generate an effective Hamiltonian that is dynamically equivalent to the nondegenerate Hamiltonian. The effective Hamiltonian can be mapped to the Morris-Shore basis with a two-step similarity transformation. After the derivation of the general framework, we demonstrate the application of this technique to the popular $\mathrm{\ensuremath{\Lambda}}$ three-state system, and the four-state tripod, double-$\mathrm{\ensuremath{\Lambda}}$, and diamond systems. In all of these systems, our formalism allows us to reduce their quantum dynamics to simpler two-state systems even in the presence of various detunings, e.g., generated by external fields of frequency drifts.

The equation of motion of linear systems is one of the most commonly used equations in science and engineering. It has long been recognized that coordinate coupling in linear systems is a considerable barrier to analysis and design. In this context, it is common for engineers to seek coordinate decoupling techniques. Coordinate decoupling is the process of simultaneously diagonalizing the coefficient matrices of a dynamical system. The main objective of this chapter is to provide an overview of configuration-space decoupling techniques. Three decoupling algorithms are provided and eleven examples are supplemented.

A general method based on a similarity transformation is presented, which treats many-fermion and many-boson systems in an exact fashion. It can be shown that under such a similarity transformation, the many-electron problem will split into an interacting condensate-like bosonic reference system, and a non-Hermitian eigenvalue problem of independent (quasi-)electrons. We discuss several elementary examples to show that the present approach is compatible with the canonical quantum mechanical formalism and that the resulting splitting offers several useful simplifications as compared to other quantum chemical approaches.

Jordan decomposition is a special form of matrix decomposition. As a method applicable to general square matrix, Jordan decomposition is capable of finding a similar matrix that is diagonal or almost diagonal, called the Jordan normal form. The problem of Jordan decomposition can be solved by computer with certain algorithms. Some of them are classified as numerical methods, which usually perform on constant matrices. Others are symbolic computation methods, which can perform Jordan decomposition on matrices with variables. Both of them have computational errors, which are designated as static errors. Additionally, in time-varying cases, using traditional methods above causes not only static errors, but also lag errors (i.e., dynamic errors). In this paper, numerical experiments conducted on some time-varying matrices are introduced. Then, the results are shown and analyzed to reach a conclusion.

Consequences of aligned and non-aligned magnetic field combined with inclined outer velocity in a Casson fluid towards a stretching surface have been analysed numerically. The reduced mathematical equation of heat and flow transportation has been solved using appropriate similarity transformation. The computed outcomes of the moulded equations have been figure out by the Runge–Kutta Fehlberg method with shooting technique. Numerical conclusions for various fluid parameters like outer velocity, aligned angle of magnetism, magnetic and Casson fluid have been investigated. The behaviours of emerging fluid parameters on heat and flow are interpreted graphically. Endorsement of the current investigation is accessible by the correlated current outcomes with the extant outcomes in the literature.

The purpose of audio monitoring can be achieved through audio comparison technology. This article mainly studies the audio comparison algorithm based on feature extraction. Aiming at the problem of audio similarity calculation, a weighted comprehensive similarity algorithm was proposed, considering the structural similarity of audio. The short time average magnitude and Mel frequency cepstrum coefficient(MFCC) of physical characteristics and pitch of the perceptual features were extracted. Analytic hierarchy process was used to perform weight distribution, and the final audio similarity was calculated. Experimental results show that compared with the dynamic time warping (DTW) algorithm and the matrix similarity algorithm based only on MFCC parameters, the weighted comprehensive similarity calculation method has better accuracy and stronger anti-interference ability.

The order of the detailed model of doubly-fed induction generator (DFIG) wind farms are too high and the simulation is difficult. Most of the existing research has used a single-machine equivalent model and clustering aggregation model for equivalence and few papers have explored the principles and equivalent conditions of the single-machine equivalent model under sub-synchronous resonance (SSR). Due to this reason, this paper equates DFIG wind farms connected with series compensated transmission network to two separate units based on the principle of matrix similarity transformation and the mathematical model and physical model of each unit has been studied. Then, the DFIG wind farm equivalent model’s validity is analyzed in number difference and collecting line difference based on linearization analysis. Finally, the system model is built in EMTDC/PSCAD, the damping analysis method is used to test the equivalent model’s validity and further reveal the mechanism of the system’s unstable operation. The results show that: the equivalent model can effectively reduce the system dimension and accurately reflect the dominant oscillation characteristics of DFIG wind farm under SSR; when SSR occurs, the damping coefficient of DFIG wind farm is negative under the oscillation frequency.

In this paper, the famous Burgers’ equation investigated using Lie group method for solving the problem of one-dimensional unsteady adiabatic flow of radiative ideal magnetogasdynamics. Extremely complicated wave motions for systems have been studied in one dimension by reducing them into simpler one. We provide a numerical solution of one-dimensional magnetogasdynamics system for Burgers equation which provides an asymptotic description.

In the present letter, some invariant solutions of (2+1)-dimensional extended shallow water wave equation are studied. We reduce the equation to an ordinary differential equation(ODE) via the Lie symmetry method. The similarity transformation method is used to determine the group-invariant solutions for the equation. The method plays an important role to reduce the number of independent variables by one in each stage and finally forms an ODE. On the other hand, solutions provide rich physical configuration due to the presence of some distinct arbitrary constants and functions. Furthermore, numerical simulation has been done to trace out the derived results which illustrate the dynamic behavior of these obtained solutions.

The mechanical fault diagnosis of transformer is mostly determined by independent analysis of a vibration signal, which is too one-sided to reflect the mechanical state of transformer. Therefore, firstly, the vibration signals of multiple positions of the transformer are measured, and the proportion of each frequency amplitude of several groups of signals is used as the improved eigenvalue to form the characteristic matrix. Then, the matrix similarity of the characteristic matrix between the identification signal and the normal signal is optimized. Finally, the effects of load rate, power factor and imbalance on the matrix similarity of vibration signals are studied by experiments. The results show that the improvement made in this paper can effectively reduce the misjudgment caused by the change of load conditions in fault diagnosis, and can accurately diagnose the existence of faults.

In this analysis, we aim to examine the heat transfer and flow characteristics of a copper-aluminum/water hybrid nanofluid in the presence of viscous dissipation, magnetohydrodynamic (MHD), and porous medium effect over the shrinking sheet. The governing equations of the fluid model have been acquired by employment of the model of Tiwari and Das, with additional properties of the hybrid nanofluid. The system of partial differential equations (PDEs) has been converted into ordinary differential equations (ODEs) by adopting the exponential similarity transformation. Similarity transformation is an essential class of phenomenon where the symmetry of the scale helps to reduce the number of independent variables. Note that ODE solutions demonstrate the PDEs symmetrical behavior for the velocity and temperature profiles. With BVP4C solver in the MATLAB program, the system of resulting equations has been solved. We have compared the present results with the published results and found in excellent agreements. The findings of the analysis are also displayed and discussed in depth graphically and numerically. It is discovered that two solutions occur in definite ranges of suction and magnetic parameters. Dual (no) similarity solutions can be found in the range of Sc&le, S&nbsp, and&nbsp, Mc&le, M (Sc&gt, Mc&gt, M). By performing stability analysis, the smallest values of eigenvalue are obtained, suggesting that a stable solution is the first one. Furthermore, the graph of the smallest eigenvalue shows symmetrical behavior. By enhancing the Eckert number values the temperature of the fluid is raised.

Image enhancement is a necessary and indispensable technique for increasing the quality of digitalimages. The main task is to generate a new intensity value for each pixel in the image using a transformationfunction after the input image receives the intensity value of each pixel. The proposed transfer function in thisstudy is called the Regional Similarity Transfer Function (RSTF) that considers the density distribution similaritybetween adjoining pixels. Dragonfly Algorithm (DA) intuitive optimization technique, which is preferred inengineering applications, has been used to optimize the parameter values of the proposed transfer function.Image quality evaluation is performed with six criteria between the improved and original images. Ourexperimental results show that the intensity distribution between adjoining pixels show an increase in contrastand brightness over the similarity degree. Excessive brightness, blur, and deterioration in the images is resolvedwith the proposed method. Görüntü iyileştirme, dijital görüntülerde görüntü kalitesini artırmak için gerekli ve vazgeçilmez bir tekniktir. Görüntü iyileştirme de temel görev, giriş görüntüsünün her bir pikselin yoğunluk değeri alındıktan sonra, dönüşüm fonksiyonu kullanarak, görüntü içerisinde karşılık gelen her piksel değeri için yeni bir yoğunluk değeri üretmektir. Çalışma da önerilen dönüşüm fonksiyonu Bölgesel Benzerlik Transfer Fonksiyonu olarak adlandırılmakta olup, komşu pikseller arasındaki yoğunluk dağılım benzerliğini dikkate alarak, dönüşüm uygulanmaktadır. Son zamanlarda mühendislik uygulamalarında çok tercih edilen Yusufçuk Algoritması sezgisel optimizasyon tekniği, önerilen dönüşüm fonksiyonunun parametre değerlerini optimize etmek amacıyla kullanılmıştır. Görüntü kalitesi değerlendirme işlemi, iyileştirilmiş ve orijinal görüntüler arasında altı kriterle gerçekleştirilmiştir. Deneysel sonuçlar, bitişik pikseller arasındaki yoğunluk dağılımının benzerlik derecesi üzerinde kontrast ve parlaklıkta bir artış gösterdiği tespit edilmiştir. Görüntülerdeki aşırı parlaklık, bulanıklık ve bozulma gibi istenmeyen durumlar, önerilen yöntemle çözülmektedir. 2-s2.0-85095945144

We investigate non-autonomous solitons in a general coherently coupled nonlinear Schr\"odinger (CCNLS) system with temporally modulated nonlinearities and with an external harmonic oscillator potential. This general CCNLS system encompasses three distinct types of CCNLS equations that describe the dynamics of beam propagation in an inhomogeneous Kerr-like nonlinear optical medium for different choices of nonlinear polarizations owing to the anisotropy of the medium. We identify a generalized similarity transformation to relate the considered model into the standard integrable homogeneous coupled nonlinear evolution equations with constant nonlinearities, accompanied by a constraint relation expressed in the form of the Riccati equation. With the help of a non-standard Hirota's bilinearization method and exact soliton solutions, we explore the impact of varying nonlinearities and refractive index in the propagation and collisions analytically by reverse engineering. Interestingly, we show the emergence of several modulated solitonic phenomena such as periodic oscillation, amplification, compression, tunneling/cross-over, excitons, as well as their combined effect in the single-soliton propagation and two-soliton collisions with appropriate forms of nonlinearity. Notably, we identify a tool to transform the nature of soliton collisions with certain type of inhomogeneous nonlinearities. The results could be of significant interest to the studies on management of nonlinear waves in the contexts like nonlinear optics and can also be extended to Bose-Einstein condensates and super-fluids., Comment: 26 pages and 16 figures. Accepted for Publication

The basis elements spanning the Sato Grassmannian element corresponding to the KP Open image in new window -function that serves as generating function for rationally weighted Hurwitz numbers are shown to be Meijer G-functions. Using their Mellin-Barnes integral representation the Open image in new window -function, evaluated at the trace invariants of an externally coupled matrix, is expressed as a matrix integral. Using the Mellin-Barnes integral transform of an infinite product of Γ functions, a similar matrix integral representation is given for the KP Open image in new window -function that serves as generating function for quantum weighted Hurwitz numbers.

The fractional diffusion equations can be discretized by applying the implicit finite difference scheme and the unconditionally stable shifted Grünwald formula. Hence, the generating linear system has a real Toeplitz structure when the two diffusion coefficients are non-negative constants. Through a similarity transformation, the Toeplitz linear system can be converted to a generalized saddle point problem. We use the generalization of a parameterized inexact Uzawa (GPIU) method to solve such a kind of saddle point problem and give a new algorithm based on the GPIU method. Numerical results show the effectiveness and accuracy for the new algorithm.

J. Badekas reinterpreted M. S. Molodensky’s three-dimensional similarity transformation as a vector solution using a centroid. The solution has since been (mis)interpreted by some others wi...

In this study, the effect of heat and mass transfer on the MHD flow of a non-Newtonian nanofluid in a permeable channel using Buongiorno’s model is analyzed. The UCM fluid model is used to characterize the non-Newtonian fluid behavior. A uniform magnetic field is applied in the transverse direction to the UCM nanofluid flow. Employing similarity transformation, the governing nonlinear PDEs are reduced to non-linear coupled ODEs. The resulting ODEs are solved numerically via Keller-box method. The computed numerical values are shown graphically in order to predict the features of the involved key parameters into the mathematical model. The dual nature of velocity is noticed with references to Reynolds number and Hartmann number.

Present results show the effect of magnetic field-dependent viscosity on the flow of magnetic fluid due to a non-conducting rotating disk. Shliomis model has been used in the problem formulation. A similarity transformation has been used to reduce a set of partial differential equations into nonlinear coupled differential equations. The nonlinear coupled differential equations involved in the problem are solved numerically through mathematical modeling in COMSOL Multiphysics. The radial, tangential, and axial velocity distribution is presented for different values of the magnetic field-dependent viscosity parameter. Some theoretical results used in the model have verified mathematically. These theoretical results are useful to validate the Shliomis model for rotational motion. The results indicate that the Shiliomis model is more appropriate to study the ferrofluid flow due to an infinite rotating disk in comparison to the Rosensweig model.

An analytical consideration of a nonlocal nonlinear Schrodinger equation with distributed coefficients is presented. The analysis is twofold: first, exact solutions of the nonlocal equation with constant coefficients are found along with the appropriate conditions for their existence, and second, using a similarity transformation the distributed equation, i.e. the equations with varying coefficients, is reduced to the constant coefficients case; the solutions of the latter will be utilized in the construction of the solutions for the general case. This similarity transformation is also physically relevant as it allows for the physical parameters of the solution (amplitude, width, centre and phase) to vary in a specific way so that the constant coefficient system is a natural similarity reduction of the distributed case. Certain compatibility conditions needed for the consistency of this reduction are also deduced.

This paper introduces a mixed-mode matrix representation of scattering parameters that is suitable for the investigation of wireless communication based on orbital angular momentum (OAM). For this purpose, a unitary transformation matrix is defined, which contains the transmitted and received amplitudes as basis vectors corresponding to OAM-based communication between two antenna arrays. The mixed-mode matrix is then obtained from the corresponding similarity transformation of the standard scattering matrix available, e.g., from numerical simulations or measurements. The mixed-mode matrix allows better physical insight into OAM-based communication by clearly separating all modes and is more efficient than postprocessing of single-antenna results in a system simulator. Mode conversion/isolation and proper mode termination can be easily quantified. Also, it allows generalizing the interpretation of OAM-based communication by focusing on the property of a constant phase difference within each antenna array. The usefulness of this approach is demonstrated using method of moments (MoM) simulations of dipole radiators in various arrangements of a transmitting and a receiving array. Systematic parameter studies that reveal dependencies of OAM-based communication are possible by using the mixed-mode matrix. As an application, communication patterns are studied, in order to obtain the suitable position and relative orientation in space for good mode isolation or communication.

This study investigates the hydromagnetic stagnation point flow to Casson fluid towards a vertically stretching sheet with convective surface boundary condition. Casson fluid model is used to characterize the non-Newtonian fluid behavior. Using similarity transformation, the governing nonlinear partial differential equations are transformed into coupled nonlinear ordinary differential equations which are solved numerically by employing Runge-Kutta Fehlberg integration scheme with shooting technique. Graphical results showing the effects of various thermophysical parameters on the velocity and temperature profiles are presented and discussed quantitatively.

The rotating boundary layer flow over a shrinking permeable surface in nanofluid is numerically studied. The similarity transformation is used to transform the partial differential equations into nonlinear ordinary differential equations. Later, these equations are determined by using bvp4c package in the MATLAB software. The numerical results reveal that there is more than one solution called dual solutions obtained for a certain range of the rotation and suction parameters. A stability analysis is performed to determine which solution is stable by depending on the sign of the eigenvalues. Based on this analysis, the results indicate that the upper branch solution (first solution) is linearly stable, while the lower branch solution (second solution) is linearly unstable.

The similarity transformation is introduced for studying free boundary value problems for a class of generalized convection-diffusion equation. A class of singular nonlinear boundary value problems are obtained and solved by using Adomian decomposition method (ADM). The approximate solution can be expressed in terms of a rapid convergent power series with easily computable terms. The efficiency and reliability of the approximate solution are verified by numerical ones. The effects of the variable thermal conduction k(z), convection functional coefficient h(z), power law exponent n, and parameter α on the flux transport characteristics are presented graphically and analyzed in detail.

Many anchor-graph-based spectral clustering methods have been proposed to accelerate spectral clustering for large scale problems. In this paper, we revisit the popular large-scale spectral clustering method based on the anchor graph which is equivalent to the spectral decomposition on a similar matrix obtained using a second-order transition probability. However, due to the special structure of the bipartite graph, there is no stable distribution of the random walk process. The even-order transition probabilities may only a side view of the bipartite structure, resulting in breaking the independence of data points and leading to undesired artifacts for boundary samples. Therefore, we propose a Fast Spectral Clustering based on the Random Walk Laplacian (FRWL) method. The random walk Laplacian balances explicitly the popularity of anchors and the independence of data points, which keeps the structure of boundary samples. The experimental results demonstrate the efficiency and effectiveness of our method.

PurposeThe purpose of this paper is to explore the effects of binary chemical reaction and activation energy on nano Casson liquid flow past a stretched plate with non-linear radiative heat, and also, the effect of a novel exponential space-dependent heat source (ESHS) aspect along with thermal-dependent heat source (THS) effect in the analysis of heat transfer in nanofluid. Comparative analysis is carried out between the flows with linear radiative heat process and non-linear radiative heat process.Design/methodology/approachA similarity transformation technique is utilised to access the ODEs from the governed PDEs. The manipulation of subsequent non-linear equations is carried out by a well-known numerical approach called Runge–Kutta–Fehlberg scheme. Obtained solutions are briefly discussed with the help of graphical and tabular illustrations.FindingsThe effects of various physical parameters on temperature, nanoparticles volume fraction and velocity fields within the boundary layer are discussed for two different flow situations, namely, flow with linear radiative heat and flow with non-linear radiative heat. It is found that an irregular heat source/sink (ESHS and THS) and non-linear solar radiation play a vital role in the enhancement of the temperature distributions.Originality/valueThe problem is relatively original to study the effects of activation energy and binary chemical reaction along with a novel exponential space-based heat source on laminar boundary flow past a stretched plate in the presence of non-linear Rosseland radiative heat.

The motive of this paper is to explore the influence of unsteady flow near a stagnation-point comprising Eyring-Powell nanoliquid over a convectively-heated stretched surface. The highly non-linear PDEs are transmuted into ODEs through similarity transformation before being settled numerically via the shooting technique. The numerical outcomes of prominent parameters are examined by a graphical representation and in tabular form. It is investigated that velocity profile shows an increasing behaviour due to the stretched parameter, while the conflicting effect is examined on temperature distribution and concentration of nanoparticle. Finally, we compare our results with the available literature and found this to be in excellent agreement.

In the present paper a steady two dimensional radiative MHD free convection flow over a moving vertical porous plate with heat generation, viscous dissipation and chemical reaction is considered. Using a similarity transformation the governing equations of the problem are converted into simultaneous linear differential equations of first order. The governing boundary layer equations are solved numerically by using shooting technique. Solutions are obtained for various values of the parameters of the problem and certain important results are presented in form of graphs.

In view of the urgent need for high-resolution and high-precision infrared image data for geological environmental monitoring, we have developed a lightweight optical butting infrared imaging system for unmanned aerial vehicles (UAVs) to obtain infrared image data. In this paper, we proposed a robust mosaicking method combination of as-projective-as-possible warps and similarity transformation, which uses multiple images to generate a wide viewing field panorama. It is developed based on an analysis of projective transformation and similarity transformation. The proposed method simultaneously takes into consideration the flexibility of alignment and the preservation of image shapes. On the overlapping regions, it performs as-projective-as-possible warps. On the non-overlapping regions, it gradually becomes a similarity warp. The results show that the proposed method weakens the “ghosting,” shape distortion, and area distortion.

PurposeIn multi-echo fMRI (ME-fMRI), various weighting schemes have been proposed for the combination of the data across echoes. Here we introduce a framework that facilitates a deeper understanding of the weight dependence of temporal SNR measures in ME-fMRI.Theory and MethodsWe examine two metrics that have been used to characterize ME-fMRI performance: temporal SNR (tSNR) and multi-echo temporal (metSNR). Both metrics can be described using the generalized Rayleigh quotient (GRQ) and are predicted to be relatively insensitive to the weights when there is a high degree of similarity between a metric-specific matrix in the GRQ numerator and a metricindependent covariance matrix in the GRQ denominator. The application of the GRQ framework to experimental data is demonstrated using a resting-state fMRI dataset acquired with a multi-echo multi-band EPI sequence.ResultsIn the example dataset, similarities between the covariance matrix and the metSNR and tSNR numerator matrices are highest in grey matter (GM) and cerebrospinal fluid (CSF) voxels, respectively. For representative GM and CSF voxels that exhibit high matrix similarity values, the metSNR and tSNR values, respectively, are both within 4% of their optimal values across a range of weighting schemes. However, there is a fundamental tradeoff, with a high degree of weight sensitivity in the tSNR and metSNR metrics for the representative GM and CSF voxels, respectively. Geometric insight into the observed weight dependencies is provided through a graphical interpretation of the GRQ.ConclusionA GRQ framework can provide insight into the factors that determine the weight sensitivity of tSNR and metSNR measures in ME-fMRI.

Existing Gauss-Newton methods for solving the problem of 3D similarity transformation are all based on the method of numerical partial derivatives, which can linearize the 3D similarity transformation model by using Taylor expansion for approximated variables to the first order derivative. And by using the linearized 3D transformation model, all of the unknown variables can be obtained by an iterative process. In existing models, the model considering random errors of design matrix is called errors-in-variables (EIV) model. In contrast to the Gauss-Newton methods, a Gauss-quasi-Newton method for solving the problem of 3D similarity transformation is proposed in this paper. In the Gauss-quasi-Newton method, 3D similarity transformation model is linearized by a differential method, and based on the differential linearization model, an iterative robust estimation can be carried out by considering both random errors and gross errors in 3D coordinates of common points. The new method proposed in this paper can be applied to 3D similarity transformation of arbitrary rotational angles. Finally, the new method is validated by three cases, and results showed that the new method is effective and feasible.

A 2-D boundary-layer flow induced by non-linear (quadratic) stretching of a curved surface of an incompressible MHD viscous fluid is investigated. Heat transfer analysis is presented including viscous dissipation and thermal radiation. A radially variable magnetic field is applied that satisfies Maxwell's equation and incorporates the curvature effects. A new similarity variable and similarity transformation are introduced to reduce the governing PDE's into ODE's. A numerical procedure is adopted to find the solution of momentum and energy equations. The numerical scheme is validated with the existing data. The results are illustrated graphically and discussed physically. Comparison with the literature shows a significant improvement compared to existing studies.

The aim of this paper is to study a two-dimensional free convective flow of Eyring-Powell Magneto nanofluid involving collective effects of thermal and mass diffusion with Soret-Dufour effects. The governing equations of the linear momentum, energy equation, and concentration are converted into non-dimensional non-linear ordinary differential equations with the facilitation of suitable group of similarity transformation. The transformed non-linear ordinary differential equations become coupled and numerically solved using the fifth-order Runge-Kutta-Fehlberg method in conjunction with the shooting technique by fitting proper boundary conditions. Computations are performed for many values of different governing parameters influencing the velocity, temperature, and concentration distributions, and obtained results are comprehensively analyzed.

The combined effects of thermal and species concentration on the flow, heat and mass transfer of a non-Newtonian Casson fluid at a vertical permeable stretching sheet under the influence of transverse magnetic field with variable fluid properties are investigated. The fluid properties are assumed to be the function of temperature and species concentration. The governing nonlinear PDEs are transformed into a set of highly non-linear coupled ODEs using a suitable similarity transformation. The resulting system is then solved via OHAM for various physical parameters and the results are analyzed graphically. OHAM provides the control and adjustment of the convergence region, its solutions agrees well with the exact ones for large values of the independent variable. The accuracy of the analytical solution is verified by plotting the residual errors and comparing with available results in the literature for some special cases and is found to be in excellent agreement. Keywords: MHD flow, Heat and mass transfer, Variable fluid properties, Prandtl number, Mixed convection, Non-Newtonian Casson fluid, Optimal homotopy analysis

A (2+1)-dimensional nonlinear Schrodinger equation is under investigation, which can be used to describes optical pulse propagation in a nonlocal optical fibers. The analytical first and second order optical rogue wave solutions were constructed via the similarity transformation. Through choosing the free function in the solutions as hyperbolic secant function, some novel optical rogue waves are excited. The characteristics of the rogue waves are graphically demonstrated. The parameters in the solutions can be used to control the shape, amplitude and scale of the rogue waves.