Numerical Solutions of Fractional Variable Order Differential Equations via Using Shifted Legendre Polynomials

In this manuscript,an algorithm for the computation of numerical solutions to some variable order fractional differential equations(FDEs)subject to the boundary and initial conditions is developed.We use shifted Legendre polynomials for the required numerical algorithm to develop some operational matrices.Further,operational matrices are constructed using variable order differentiation and integration.We are finding the operationalmatrices of variable order differentiation and integration by omitting the discretization of data.With the help of aforesaid matrices,considered FDEs are converted to algebraic equations of Sylvester type.Finally,the algebraic equations we get are solved with the help of mathematical software like Matlab or Mathematica to compute numerical solutions.Some examples are given to check the proposed method’s accuracy and graphical representations.Exact and numerical solutions are also compared in the paper for some examples.The efficiency of the method can be enhanced further by increasing the scale level.

Characterization of Optical Aberrations Induced by Thermal Gradients and Vibrations via Zernike and Legendre Polynomials

For every astronomical instrument, the operating conditions are undoubtedly different from those defined in a setup experiment. Besides environmental conditions, the drives, the electronic cabinets containing heaters and fans introduce disturbances that must be taken into account already in the preliminary design phase. Such disturbances can be identified as being mostly of two types: heat sources/sinks or cooling systems responsible for heat transfer via conduction, radiation, free and forced convection on one side and random and periodic vibrations on the other. For this reason, a key role already from the very beginning of the design process is played by integrated model merging the outcomes based on a Finite Element Model from thermo-structural and modal analysis into the optical model to estimate the aberrations. The current paper presents the status of such model, capable of analyzing the deformed surfaces deriving from both thermo-structural and vibrational analyses and measuring their effect in terms of optical aberrations by fitting them by Zernike and Legendre polynomial fitting respectively for circular and rectangular apertures. The independent contribution of each aberration is satisfied by the orthogonality of the polynomials and mesh uniformity.

Galerkin approximation with Legendre polynomials for a continuous-time nonlinear optimal control problem

We investigate the use of an approximation method for obtaining near-optimal solutions to a kind of nonlinear continuous-time(CT) system. The approach derived from the Galerkin approximation is used to solve the generalized Hamilton-Jacobi-Bellman(GHJB) equations. The Galerkin approximation with Legendre polynomials(GALP) for GHJB equations has not been applied to nonlinear CT systems. The proposed GALP method solves the GHJB equations in CT systems on some well-defined region of attraction. The integrals that need to be computed are much fewer due to the orthogonal properties of Legendre polynomials, which is a significant advantage of this approach. The stabilization and convergence properties with regard to the iterative variable have been proved.Numerical examples show that the update control laws converge to the optimal control for nonlinear CT systems.

Duality between Bessel Functions and Chebyshev Polynomials in Expansions of Functions

In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.

SOME EXTREMAL PROPERTIES OF THE INTEGRAL OF LEGENDRE POLYNOMIALS

Some extremal properties of the integral of Legendre polynomials are given, which are of independent interest. Meanwhile they show that a conjecture of P. Erdos[1] is plausible and maybe provides some means to prove this conjecture.

Two-Stream Approximation to the Radiative Transfer Equation:A New Improvement and Comparative Accuracy with Existing Methods

Mathematical modeling of the interaction between solar radiation and the Earth's atmosphere is formalized by the radiative transfer equation(RTE), whose resolution calls for two-stream approximations among other methods. This paper proposes a new two-stream approximation of the RTE with the development of the phase function and the intensity into a third-order series of Legendre polynomials. This new approach, which adds one more term in the expression of the intensity and the phase function, allows in the conditions of a plane parallel atmosphere a new mathematical formulation of γparameters. It is then compared to the Eddington, Hemispheric Constant, Quadrature, Combined Delta Function and Modified Eddington, and second-order approximation methods with reference to the Discrete Ordinate(Disort) method(δ –128 streams), considered as the most precise. This work also determines the conversion function of the proposed New Method using the fundamental definition of two-stream approximation(F-TSA) developed in a previous work. Notably,New Method has generally better precision compared to the second-order approximation and Hemispheric Constant methods. Compared to the Quadrature and Eddington methods, New Method shows very good precision for wide domains of the zenith angle μ 0, but tends to deviate from the Disort method with the zenith angle, especially for high values of optical thickness. In spite of this divergence in reflectance for high values of optical thickness, very strong correlation with the Disort method(R ≈ 1) was obtained for most cases of optical thickness in this study. An analysis of the Legendre polynomial series for simple functions shows that the high precision is due to the fact that the approximated functions ameliorate the accuracy when the order of approximation increases, although it has been proven that there is a limit order depending on the function from which the precision is lost. This observation indicates that increasing the order of approximation of the phase function of the RTE leads to a better precision in flux calculations. However, this approach may be limited to a certain order that has not been studied in this paper.

Legendre expansion method for Helmholz equations

A spectral method based on the Legendre polynomials for solving Helmholz equations was proposed. With an explicit formula for the Legendre polynomials in terms of arbitrary order of their derivatives, the successive integration of the Legendre polynomials was represented by the Legendre polynomials. Then the method was formulized for secondorder differential equations in one dimension and two dimensions. Numerical results indicate that the suggested method is significantly accurate and in satisfactory agreement with the exact solution.

Operator Methods and SU(1,1) Symmetry in the Theory of Jacobi and of Ultraspherical Polynomials

Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting properties. It leads to an alternative definition of the Ultraspherical polynomials by a fixed integral operator in application to powers of the variable u in an analogous way as it is possible for Hermite polynomials. From this follows a generating function which is apparently known only for the Legendre and Chebyshev polynomials as their special case. Furthermore, we show that the Ultraspherical polynomials form a realization of the SU(1,1) Lie algebra with lowering and raising operators which we explicitly determine. By reordering of multiplication and differentiation operators we derive new operator identities for the whole set of Jacobi polynomials which may be applied to arbitrary functions and provide then function identities. In this way we derive a new “convolution identity” for Jacobi polynomials and compare it with a known convolution identity of different structure for Gegenbauer polynomials. In short form we establish the connection of Jacobi polynomials and their related orthonormalized functions to the eigensolution of the Schrödinger equation to Pöschl-Teller potentials.

Legendre Polynomial Kernel: Application in SVM

In machines learning problems, Support Vector Machine is a method of classification. For non-linearly separable data, kernel functions are a basic ingredient in the SVM technic. In this paper, we briefly recall some useful results on decomposition of RKHS. Based on orthogonal polynomial theory and Mercer theorem, we construct the high power Legendre polynomial kernel on the cube [-1,1]d. Following presentation of the theoretical background of SVM, we evaluate the performance of this kernel on some illustrative examples in comparison with Rbf, linear and polynomial kernels.

An analysis of the characteristics of crustal magnetic anomaly in China based on CHAMP satellite data

Based on the observation data of CHAMP satellite from 2006 to 2009, a 2D crustal magnetic anomaly model in China is established to study the distribution characteristics of crustal magnetic anomaly. In this paper, the 2D anomaly model is derived from the Legendre polynomial expansion of harmonic term N =6-50. The result shows that many elaborate structures reflected in magnetic anomaly map well correspond to the geologic structures in China and its adjacent area. The magnetic anomaly at low satellite height behaves complexly, which is mainly caused by the magnetic disturbance of shallow rocks.In contrast, the magnetic field isolines at high satellite height are relatively sparse and only magnetic anomalies of deep crust are reflected. This fact implies that the 2D model of crustal magnetic anomaly provides an important method of the space prolongation of geomagnetic field, and is of theoretical and practice importance in geologic structure analysis and geophysical prospecting.

A Remark on Pl Type Interpolation on Non-Uniformly Distributed Nodes on the Unit Circle

In this paper we study the problem of explicit representation and convergence of Pal type （0;1） interpolation and its converse, with some additional conditions, on the non-uniformly distributed nodes on the unit circle obtaIned by projecting the interlaced zeros of Pn （x） and Pn′ （x） on the unit circle. The motivation to this problem can be traced to the recent studies on the regularity of Birkhoff interpolation and Pal type interpolations on non-uniformly distributed zeros on the unit circle.

A Unified Petrov-Galerkin Spectral Method and Fast Solver for Distributed-Order Partial Differential Equations

Fractional calculus and fractional-order modeling provide effective tools for modeling and simulation of anomalous diffusion with power-law scalings.In complex multi-fractal anomalous transport phenomena,distributed-order partial differential equations appear as tractable mathematical models,where the underlying derivative orders are distributed over a range of values,hence taking into account a wide range of multi-physics from ultraslow-to-standard-to-superdiffusion/wave dynamics.We develop a unified,fast,and stable Petrov–Galerkin spectral method for such models by employing Jacobi poly-fractonomials and Legendre polynomials as temporal and spatial basis/test functions,respectively.By defining the proper underlying distributed Sobolev spaces and their equivalent norms,we rigorously prove the well-posedness of the weak formulation,and thereby,we carry out the corresponding stability and error analysis.We finally provide several numerical simulations to study the performance and convergence of proposed scheme.

Model of the Nerve Impulse with Account of Mechanosensory Processes: Stationary Solutions.

Mechanotransduction refers to a physiological process by which mechanical forces, such as pressures exerted by ionized fluids on cell membranes and tissues, can trigger excitations of electrical natures that play important role in the control of various sensory (i.e. stimuli-responsive) organs and homeostasis of living organisms. In this work, the influence of mechanotransduction processes on the generic mechanism of the action potential is investigated analytically, by considering a mathematical model that consists of two coupled nonlinear partial differential equations. One of these two equations is the Korteweg-de Vries equation governing the spatio-temporal evolution of the density difference between intracellular and extracellular fluids across the nerve membrane, and the other is Hodgkin-Huxley cable equation for the transmembrane voltage with a self-regulatory (i.e. diode-type) membrane capacitance. The self-regulatory feature here refers to the assumption that membrane capacitance varies with the difference in density of ion-carrying intracellular and extracellular fluids, thus ensuring an electromechanical feedback mechanism and consequently an effective coupling of the two nonlinear equations. The exact one-soliton solution to the density-difference equation is obtained in terms of a pulse excitation. With the help of this exact pulse solution the Hodgkin-Huxley cable equation is shown to transform, in steady state, to a linear eigenvalue problem some bound states of which can be obtained exactly. Few of such bound-state solutions are found analytically.

Geometric Covariance Modeling for Surface Variation of Compliant Parts Based on Hybrid Polynomial Approximation and Spectrum Analysis

Part variation characterization is essential to analyze the variation propagation in flexible assemblies. Aiming at two governing types of surface variation,warping and waviness,a comprehensive approach of geometric covariance modeling based on hybrid polynomial approximation and spectrum analysis is proposed,which can formulate the level and the correlation of surface variations accurately. Firstly,the form error data of compliant part is acquired by CMM. Thereafter,a Fourier-Legendre polynomial decomposition is conducted and the error data are approximated by a Legendre polynomial series. The weighting coefficient of each component is decided by least square method for extracting the warping from the surface variation. Consequently,a geometrical covariance expression for warping deformation is established. Secondly,a Fourier-sinusoidal decomposition is utilized to approximate the waviness from the residual error data. The spectrum is analyzed is to identify the frequency and the amplitude of error data. Thus,a geometrical covariance expression for the waviness is deduced. Thirdly,a comprehensive geometric covariance model for surface variation is developed by the combination the Legendre polynomials with the sinusoidal polynomials. Finally,a group of L-shape sheet metals is measured along a specific contour,and the covariance of the profile errors is modeled by the proposed method. Thereafter,the result is compared with the covariance from two other methods and the real data. The result shows that the proposed covariance model can match the real surface error effectively and represents a tighter approximation error compared with the referred methods.

Critical velocity of sandwich cylindrical shell under moving internal pressure

Critical velocity of an infinite long sandwich shell under moving internal pressure is studied using the sandwich shell theory and elastodynamics theory. Propagation of axisymmetric free harmonic waves in the sandwich shell is studied using the sandwich shell theory by considering compressibility and transverse shear deformation of the core, and transverse shear deformation of face sheets. Based on the elastodynamics theory, displacement components expanded by Legendre polynomials, and position-dependent elastic constants and densities are introduced into the equations of motion. Critical velocity is the minimum phase velocity on the desperation relation curve obtained by using the two methods. Numerical examples and the finite element （FE） simulations are presented. The results show that the two critical velocities agree well with each other, and two desperation relation curves agree well with each other when the wave number k is relatively small. However, two limit phase velocities approach to the shear wave velocities of the face sheet and the core respectively when k limits to infinite. The two methods are efficient in the investigation of wave propagation in a sandwich cylindrical shell when k is relatively small. The critical velocity predicted in the FE simulations agrees with theoretical prediction.

Kernel polynomial representation for imaginary-time Green's functions in continuous-time quantum Monte Carlo impurity solver

Inspired by the recently proposed Legendre orthogonal polynomial representation for imaginary-time Green s functions G（τ）,we develop an alternate and superior representation for G（τ） and implement it in the hybridization expansion continuous-time quantum Monte Carlo impurity solver.This representation is based on the kernel polynomial method,which introduces some integral kernel functions to filter the numerical fluctuations caused by the explicit truncations of polynomial expansion series and can improve the computational precision significantly.As an illustration of the new representation,we re-examine the imaginary-time Green＇s functions of the single-band Hubbard model in the framework of dynamical mean-field theory.The calculated results suggest that with carefully chosen integral kernel functions,whether the system is metallic or insulating,the Gibbs oscillations found in the previous Legendre orthogonal polynomial representation have been vastly suppressed and remarkable corrections to the measured Green＇s functions have been obtained.

A new quantum mechanical photon counting distribution formula

By virtue of the density operator＇s P-representation in the coherent state representation, we derive a new quantum mechanical photon counting distribution formula. As its application, we calculate photon counting distributions for some given light fields. It is found that the pure squeezed state＇s photon counting distribution is related to the Legendre function, which is a new result.

A two-dimensional lithospheric magnetic anomaly field model of Egypt using the measurements from Swarm satellites

We use magnetic field data observed by the Swarm mission from 2014 to 2020 to construct,for the first time,a two-dimensional(2 D)lithospheric magnetic anomaly model of Egypt and its surrounding area.Nighttime data during quiet geomagnetic conditions has been expanded in terms of the Legendre polynomial in harmonic terms N=6-50.The damped least square method has been used to estimate the model coefficients based on the lithospheric magnetic data.Modeled data at two different altitudes(438-448 km and 503-511 km)were compared with the CHAOS model.Results exhibit that the 2 D model is superior to the CHAOS model in the capability of extracting more information about small-scale crustal anomaly field.At low altitudes(438-448 km),the strength of the anomaly field increases,but the noise of the external fields has greatly reduced at high altitudes(503-511 km).Besides,the magnetic anomaly field at low altitudes has illuminated short-scale anomalies that didn’t appear at high altitudes.Both the total and vertical magnetic anomaly vectors showed their ability to reveal tectonic structures compared with Moho depth map and the geological maps.

Rotation Scaling and Translation Invariants of 3D Radial Shifted Legendre Moments

This paper proposes a new set of 3D rotation scaling and translation invariants of 3D radially shifted Legendre moments. We aim to develop two kinds of transformed shifted Legendre moments： a 3D substituted radial shifted Legendre moments （3DSRSLMs） and a 3D weighted radial one （3DWRSLMs）. Both are centered on two types of polynomials. In the first case, a new 3D ra- dial complex moment is proposed. In the second case, new 3D substituted/weighted radial shifted Legendremoments （3DSRSLMs/3DWRSLMs） are introduced using a spherical representation of volumetric image. 3D invariants as derived from the sug- gested 3D radial shifted Legendre moments will appear in the third case. To confirm the proposed approach, we have resolved three is- sues. To confirm the proposed approach, we have resolved three issues： rotation, scaling and translation invariants. The result of experi- ments shows that the 3DSRSLMs and 3DWRSLMs have done better than the 3D radial complex moments with and without noise. Sim- ultaneously, the reconstruction converges rapidly to the original image using 3D radial 3DSRSLMs and 3DWRSLMs, and the test of 3D images are clearly recognized from a set of images that are available in Princeton shape benchmark （PSB） database for 3D image.

A Chebyshev/Legendre polynomial interpolation approach for fingerprint orientation estimation smoothing and prediction

We introduce a novel coarse ridge orientation smoothing algorithm based on orthogonal polynomials, which can be used to estimate the orientation field (OF) for fingerprint areas of no ridge information. This method does not need any base information of singular points (SPs). The algorithm uses a consecutive application of filtering-and model-based orientation smoothing methods. A Gaussian filter has been employed for the former. The latter conditionally employs one of the orthogonal polynomials such as Legendre and Chebyshev type I or II, based on the results obtained at the filtering-based stage. To evaluate our proposed method, a variety of exclusive fingerprint classification and minutiae-based matching experiments have been conducted on the fingerprint images of FVC2000 DB2, FVC2004 DB3 and DB4 databases. Results showed that our proposed method has achieved higher SP detection, classification, and verification performance as compared to competing methods.