Search Results (38)

This study is devoted to solving the nonlinear inhomogeneous time-fractional FitzHugh–Nagumo differential problem ($\mathsf{TFFNDP}$) using the spectral modified collocation method. The proposed algorithm uses non-symmetric polynomials, namely shifted Legendre polynomials ($\mathsf{SLPs}$), which are orthogonal. The orthogonality property of $\mathsf{SLPs}$ and certain relations facilitate the acquisition of accurate spectral approximations. Comprehensive convergence and error studies are conducted to validate the accuracy of the suggested shifted Legendre expansion. Several numerical examples are presented to demonstrate the method’s effectiveness and accuracy. The proposed scheme is benchmarked against known analytical solutions and compared with other algorithms to ensure the applicability and efficiency of the proposed algorithm. Full article

Algorithms for jointly obtaining projection estimates of the density and distribution function of a random variable using Legendre polynomials are proposed. For these algorithms, a problem of conditional optimization is solved. Such optimization allows one to increase the approximation accuracy with minimum computational costs. The proposed algorithms are tested on examples with different degrees of smoothness of the density. A projection estimate of the density is compared to a histogram that is often used in applications to estimate distributions. Full article

This paper addresses the numerical evaluation of highly oscillatory integrals by developing a spectral Galerkin–Levin approach that efficiently solves Levin’s differential equation formulation for such integrals. The method employs Legendre polynomials as basis functions to approximate the solution, leveraging their orthogonality and favorable numerical properties. A key finding is that the Galerkin–Levin formulation is invariant with respect to the choice of polynomial basis—be it monomials or classical orthogonal polynomials—although the use of Legendre polynomials leads to a more straightforward derivation of practical quadrature rules. Building on this, this paper derives a simple and efficient numerical quadrature for both scalar and matrix-valued highly oscillatory integrals. The proposed approach is computationally stable and well-conditioned, overcoming the limitations of collocation-based methods. Several numerical examples validate the method’s high accuracy, stability, and computational efficiency. Full article

This study focuses on the numerical solution and dynamics analysis of fractional governing equations related to double-layer nanoplates based on the shifted Legendre polynomials algorithm. Firstly, the fractional governing equations of the complicated mechanical behavior for bilayer nanoplates are constructed by combining the Fractional Kelvin–Voigt (FKV) model with the Caputo fractional derivative and the theory of nonlocal elasticity. Then, the shifted Legendre polynomial is used to approximate the displacement function, and the governing equations are transformed into algebraic equations to facilitate the numerical solution in the time domain. Moreover, the systematic convergence analysis is carried out to verify the convergence of the ternary displacement function and its fractional derivatives in the equation, ensuring the rigor of the mathematical model. Finally, a dimensionless numerical example is given to verify the feasibility of the proposed algorithm, and the effects of material parameters on plate displacement are analyzed for double-layer plates with different materials. Full article

We introduce a novel family of compactly supported basis functions, termed Legendre Delta-Shaped Functions (LDSFs), constructed using the eigenfunctions of the Legendre differential equation. We begin by proving that LDSFs form a basis for a Haar space. We then demonstrate that interpolation using classical Legendre polynomials is a special case of interpolation with the proposed Legendre Delta-Shaped Basis Functions (LDSBFs). To illustrate the potential of LDSBFs, we apply the corresponding series to approximate a rectangular pulse. The results reveal that Gibbs oscillations decay rapidly, resulting in significantly improved accuracy across smooth regions. This example underscores the effectiveness and novelty of our approach. Furthermore, LDSBFs are employed within the collocation framework to solve Poisson-type equations and systems of nonlinear differential equations arising in energy transfer problems. We also derive new error bounds for interpolation polynomials in a special case, expressed in both the discrete ${(L}_2)$ norm and the Sobolev $\left(H_p\right)$ norm. To validate the proposed method, we compare our results with those obtained using the Legendre pseudospectral method. Numerical experiments confirm that our approach is accurate, efficient, and highly competitive with existing techniques. Full article

This paper gives a new representation for the fractional Brownian motion that can be applied to simulate this self-similar random process in continuous time. Such a representation is based on the spectral form of mathematical description and the spectral method. The Legendre polynomials are used as the orthonormal basis. The paper contains all the necessary algorithms and their theoretical foundation, as well as the results of numerical experiments. Full article

During switching in electrical systems, transient electromagnetic processes occur. The resulting dangerous current surges are best studied by computer simulation. However, the time required for computer simulation of such processes is significant for complex electromagnetic devices, which is undesirable. The use of spectral methods can significantly speed up the calculation of transient processes and ensure high accuracy. At present, we are not aware of publications showing the use of spectral methods for calculating transient processes in electromagnetic devices containing ferromagnetic cores. The purpose of the work: The objective of this work is to develop a highly effective method for calculating electromagnetic transient processes in a coil with a ferromagnetic magnetic core connected to a voltage source. The method involves the use of nonlinear magnetoelectric substitution circuits for electromagnetic devices and a spectral method for representing solution functions using orthogonal polynomials. Additionally, a schematic model for applying the spectral method is developed. Obtained Results: A method for calculating transients in magnetoelectric circuits based on approximating solution functions with algebraic orthogonal polynomial series is proposed and studied. This helps to transform integro-differential state equations into linear algebraic equations for the representations of the solution functions. The developed schematic model simplifies the use of the calculation method. Representations of true electric and magnetic current functions are interpreted as direct currents in the proposed substitution circuit. Based on these methods, a computer program is created to simulate transient processes in a magnetoelectric circuit. Comparing the application of various polynomials enables the selection of the optimal polynomial type. The proposed method has advantages over other known methods. These advantages include reducing the simulation time for electromagnetic transient processes (in the examples considered, by more than 12 times than calculations using the implicit Euler method) while ensuring the same level of accuracy. The simulation of processes over a long time interval demonstrate error reduction and stabilization. This indicates the potential of the proposed method for simulating processes in more complex electromagnetic devices, (for example, transformers). Full article

Borehole pulsed eddy-current (PEC) systems based on uniform linear multicoil arrays (ULMAs) perform efficient nondestructive evaluations (NDEs) of metal casings. However, the limited physical space of the borehole restricts the degrees of freedom (DoFs) of ULMAs to be less than the number of constraints, which leads to the difficulty of compensating for the differences in signals acquired by different receivers with different transmitting-to-receiving distances (TRDs), and thus limits the effectiveness of the ULMA system. To solve this problem, this paper proposes sparse linear constraint minimum variance (S-LCMV) for NDEs of downhole casings employing ULMAs. By transforming and characterizing the original PEC signal, it was observed that the signal power dramatically decreased with increasing Legendre polynomial stage, confirming that the signal was sparsely distributed over the Gauss–Legendre stages. Using this property, the S-LCMV cost function with reduced constraints was constructed to provide enough DoFs to accurately calculate the weight coefficients, thus improving the detection performance. The effectiveness of the proposed method was verified through field experiments on an 8-element ULMA installed in a borehole PEC system for NDEs of oil-well casings. The results demonstrate that the proposed method could improve the weighting effect by reducing the number of constraints by 70% while ensuring the approximation accuracy, which effectively improved the signal-to-noise ratio of the measured signals and reduced the computational cost by about 87.9%. Full article

This study investigates viscoelastic guided wave properties (e.g., complex–wavenumber–, phase–velocity–, and attenuation–frequency relations) for multiple modes, including different orders of antisymmetric, symmetric, and shear horizontal modes in viscoelastic anisotropic laminated composites. To obtain those frequency–dependent relations, a guided wave characteristic equation is formulated based on a Legendre orthogonal polynomials expansion (LOPE)–assisted viscoelastodynamic model, which fuses the hysteretic viscoelastic model–based wave dynamics and the LOPE–based mode shape approximation. Then, the complex–wavenumber–frequency solutions are obtained by solving the characteristic equation using an improved root–finding algorithm, which leverages coefficient matrix determinant ratios and our proposed local tracking windows. To trace the solutions on the dispersion curves of different wave modes and avoid curve–tracing misalignment in regions with phase–velocity curve crossing, we presented a curve–tracing strategy considering wave attenuation. With the LOPE–assisted viscoelastodynamic model, the effects of material viscosity and fiber orientation on different guided wave modes are investigated for unidirectional carbon–fiber–reinforced composites. The results show that the viscosity in the hysteresis model mainly affects the frequency–dependent attenuation of viscoelastic guided waves, while the fiber orientation influences both the phase–velocity and attenuation curves. We expect the theoretical work in this study to facilitate the development of guided wave–based techniques for the NDT and SHM of viscoelastic anisotropic laminated composites. Full article

The Bessel function of the first kind $J_N\left(kx\right)$ is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind $I_N\left(kx\right)$. The purpose of these expansions in Legendre polynomials was not an attempt to rival established numerical methods for calculating Bessel functions but to provide a form for $J_N\left(kx\right)$ useful for analytical work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, one can easily truncate the series at 21 terms to provide 33-digit accuracy that matches the IEEE extended precision in some compilers. The analytical theme is furthered by showing that infinite series of like-powered contributors (involving ${ }_1{F}_2$ hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes $1/\left(2^i{3}^j{5}^k{7}^l{11}^m{13}^n{17}^o{19}^p\right)$ multiplying powers of the coefficient k. Full article

The purpose of this paper is to leverage the advantages of physics-informed neural network (PINN) and convolutional neural network (CNN) by using Legendre multiwavelets (LMWs) as basis functions to approximate partial differential equations (PDEs). We call this method Physics-Informed Legendre Multiwavelets CNN (PiLMWs-CNN), which can continuously approximate a grid-based state representation that can be handled by a CNN. PiLMWs-CNN enable us to train our models using only physics-informed loss functions without any precomputed training data, simultaneously providing fast and continuous solutions that generalize to previously unknown domains. In particular, the LMWs can simultaneously possess compact support, orthogonality, symmetry, high smoothness, and high approximation order. Compared to orthonormal polynomial (OP) bases, the approximation accuracy can be greatly increased and computation costs can be significantly reduced by using LMWs. We applied PiLMWs-CNN to approximate the damped wave equation, the incompressible Navier–Stokes (N-S) equation, and the two-dimensional heat conduction equation. The experimental results show that this method provides more accurate, efficient, and fast convergence with better stability when approximating the solution of PDEs. Full article

Wavelet neural networks have been widely applied to dynamical system identification fields. The most difficult issue lies in selecting the optimal control parameters (the wavelet base type and corresponding resolution level) of the network structure. This paper utilizes the advantages of Legendre multiwavelet (LW) bases to construct a Legendre multiwavelet neural network (LWNN), whose simple structure consists of an input layer, hidden layer, and output layer. It is noted that the activation functions in the hidden layer are adopted as LW bases. This selection if based on the its rich properties of LW bases, such as piecewise polynomials, orthogonality, various regularities, and more. These properties contribute to making LWNNs more effective in approximating the complex characteristics exhibited by uncertainties, step, nonlinear, and ramp in the dynamical systems compared to traditional wavelet neural networks. Then, the number of selection LW bases and the corresponding resolution level are effectively optimized by the simple Genetic Algorithm, and the improved gradient descent algorithm is implemented to learn the weight coefficients of LWNN. Finally, four nonlinear dynamical system identification problems are applied to validate the efficiency and feasibility of the proposed LWNN-GA method. The experiment results indicate that the LWNN-GA method achieves better identification accuracy with a simpler network structure than other existing methods. Full article

One of the issues in numerical solution analysis is the non-linear distributed-order fractional Bagley–Torvik differential equation (DO-FBTE) with boundary and initial conditions. We solve the problem by proposing a numerical solution based on the shifted Legendre Gauss–Lobatto (SL-GL) collocation technique. The solution of the DO-FBTE is approximated by a truncated series of shifted Legendre polynomials, and the SL-GL collocation points are employed as interpolation nodes. At the SL-GL quadrature points, the residuals are computed. The DO-FBTE is transformed into a system of algebraic equations that can be solved using any conventional method. A set of numerical examples is used to verify the proposed scheme’s accuracy and compare it to existing findings. Full article

The fractional Legendre polynomials (FLPs) that we present as an effective method for solving fractional delay differential equations (FDDEs) are used in this work. The Liouville–Caputo sense is used to characterize fractional derivatives. This method uses the spectral collocation technique based on FLPs. The proposed method converts FDDEs into a set of algebraic equations. We lay out a study of the convergence analysis and figure out the upper bound on error for the approximate solution. Examples are provided to demonstrate the precision of the suggested approach. Full article

Under simply supported plate (SSP) boundary conditions, a numerical method based on the higher-order Legendre polynomial approximation was studied and developed for fourth-order problems with variable coefficients. We first divide the SSP boundary conditions into two types, namely, forced boundary conditions and natural boundary conditions. According to the forced boundary conditions, an appropriate Sobolev space is defined, and a variational formulation and a discrete scheme associated with the original problem are established. Then, the existence and uniqueness of this weak solution and approximate solution are proved. By using the Céa lemma and the tensor Jacobian polynomial approximation, we further obtain the error estimation for the numerical solutions. In addition, we use the orthogonality of Legendre polynomials to construct a set of effective basis functions and derive the equivalent tensor product linear system associated with the discrete scheme, respectively. Finally, some numerical tests were carried out to validate our algorithm and theoretical analysis. Full article