Search Results (21)

This research paper investigates the solution of diffusion equations characterized by Third-Kind (Robin) boundary conditions within n-dimensional complex domains. The analysis is conducted in the $L_2$ Hilbert space, which facilitates the substantiation of both the existence and uniqueness of solutions through variational methods. Analytical solutions are derived for multidimensional domains by employing the Fourier method and spectral analysis techniques. Complementing this theoretical framework, a high-accuracy numerical approach based on the Associated Legendre Polynomials Collocation Spectral Method (ALP-CSM) with Chebyshev–Gauss–Lobatto nodes is developed. Rigorous convergence analysis confirms spectral accuracy, with numerical examples in one, two, and three dimensions demonstrating error decay from $\mathcal{O}\left({10}^{-3}\right)$ to machine precision $\mathcal{O}\left({10}^{-15}\right)$. The mathematical impact of Third-Kind boundary conditions on the diffusion rate and the steady state of the system is demonstrated. The obtained results provide a robust tool for modeling physical processes, particularly in systems involving heat exchange on the surfaces of complex-structured domains, offering both theoretical insight and computational efficiency. Full article

Polylogarithm-weighted sequences and $h\left(x\right)$-Fibonacci/Lucas polynomials have each been studied extensively, but a common formulation that incorporates generalized hyperharmonic weights into both these kernels and related Legendre-type kernels has not been formulated in a unified way. In this paper, the classical generating functions are deformed by the factor ${\mathrm{Li}}_p\left(t\right)/{(1-t)}^q$, and the resulting coefficients are derived by Cauchy product arguments. This construction yields the $h\left(x\right)$-Fibonacci–polylogarithm and $h\left(x\right)$-Lucas–polylogarithm polynomials, explicit coefficient formulas, convolution identities, recurrence relations, and parity properties, together with a unified two-parameter family of generalized $h\left(x\right)$-Fibonacci–Lucas–polylogarithm polynomials ${\mathbb{P}}_{h,n}^{a,b,p,q}\left(x\right)$. The same deformation principle also gives rise to Legendre–polylogarithm polynomials and to a $(q,\lambda )$-extension obtained from a weighted Legendre generating kernel. These families provide a natural generating-function setting for models in which cumulative harmonic or hyperharmonic effects are intrinsic, while also making explicit the main analytic restrictions of the deformation, including convergence constraints and the loss of classical orthogonality in the Legendre setting. Full article

In applying a hierarchical mixed model to genome-wide association analysis (GWAS) of longitudinal data, dimensionality reduction through modeling repeated measurements improves both computational efficiency and statistical power. Legendre polynomials can flexibly fit population growth trajectories, but higher orders substantially increase computational complexity. Instead of using Legendre polynomials, we first estimated fewer individual-specific parameters using biologically meaningful non-linear models and then associated these phenotypic regressions with genetic markers using a multivariate linear mixed model (mvLMM). After performing a canonical transformation of the regressions based on the pre-estimated covariance matrices under the null genomic mvLMM, we decomposed the mvLMM into mutually independent univariate models and incorporated EMMAX to enable rapid genome-wide mixed-model associations for each transformed phenotype. Simulations for longitudinal association analysis in maize and GWAS for the growth trajectories of body weights in mice demonstrated the advantages of hierarchical non-linear mixed models in computing efficiency and statistical power for detecting quantitative trait loci (QTL), compared with mvLMM for multiple growth points and the hierarchical random regression model using Legendre polynomials as sub-models. Full article

This paper introduces a fractional generalization of the classical Legendre differential equation based on the Atangana–Baleanu–Caputo (ABC) derivative. A novel fractional Legendre-type operator is rigorously defined within a functional framework of continuously differentiable functions with absolutely continuous derivatives. The associated initial value problem is reformulated as an equivalent Volterra integral equation, and existence and uniqueness of classical solutions are established via the Banach fixed-point theorem, supported by a proved Lipschitz estimate for the ABC derivative. A constructive solution representation is obtained through a Volterra–Neumann series, explicitly revealing the role of Mittag–Leffler functions. We prove that the fractional solutions converge uniformly to the classical Legendre polynomials as the fractional order approaches unity, with a quantitative convergence rate of order $O(1-\alpha )$ under mild regularity assumptions on the Volterra kernel. A fully reproducible quadrature-based numerical scheme is developed, with explicit kernel formulas and implementation algorithms provided in appendices. Numerical experiments for the quadratic Legendre mode confirm the theoretical convergence and illustrate the smooth interpolation between fractional and classical regimes. An application to time-fractional diffusion in spherical coordinates demonstrates that the operator arises naturally in physical models, providing a mathematically consistent tool for extending classical angular analysis to fractional settings with memory. Full article

Global geomagnetic field models typically have low spatial resolution, whereas regional models are constrained by boundary effects and limited truncation levels. To address these limitations, this study introduces a novel regional geomagnetic anomaly field model called the regional associated Legendre polynomials magnetic model (R−ALPOLM). This model employs the associated Legendre polynomials method, which combines the QR decomposition approach and a comprehensive evaluation index formula to enhance the computational efficiency of parameter estimation. In addition, it allows for scientific and intuitive determination of the optimal truncation level of the model. The overall prediction accuracy of the model is significantly enhanced by identifying and re-predicting outliers using the exponential moving average approach. The results indicate that the degree 83 R−ALPOLM achieves a root mean square error (RMSE) of 3.21 nT. Compared to traditional models, the proposed model exhibits lower error rates, highlighting its superior efficiency and predictive accuracy. This underscores the potential value of the proposed model in both scientific research and practical applications. Full article

The analysis of resilience indicators was based on daily milk yields recorded from 3347 lactations of 3080 Holstein cows located on 10 farms between 2022 and 2024. Six farms used an automatic milking system. A random regression function with a fourth-degree Legendre polynomial was used to predict the lactation curve. The indicators were the natural log-transformed variance (LnVar), lag-1 autocorrelation (r-auto), and skewness (skew) of daily milk yield (DMY) deviations from the predicted lactation curve, as well as the log-transformed variance of DMY (Var). The single-step genomic prediction method (ssGBLUP) was used for genomic evaluation. A total of 9845 genotyped animals and 36,839 SNPs were included. Heritability estimates were low (0.02–0.13). The strongest genetic correlation (0.87) was found between LnVar and Var. The genetic correlation between r-auto and skew was also strong but negative (−0.73). Resilience indicators showed a negative correlation with milk yield per lactation and a positive correlation with fat and protein contents. The negative correlation between fertility and two resilience indicators may be due to the evaluation period (50th–150th day of lactation) being when cows are most often bred after calving, and a decrease in production may accompany a significant oestrus. The associations between resilience indicators and health traits (clinical mastitis, claw health) were weak but mostly favourable. Full article

In a recent paper, Littlejohn and Quintero studied the orthogonal polynomials ${\left\{K_n\right\}}_{n=0}^{\infty }$—which they named Krein–Sobolev polynomials—that are orthogonal in the classical Sobolev space $H^1[-1,1]$ with respect to the (positive-definite) inner product ${(f,g)}_{1,c}:=-{\displaystyle \frac{\left(f\left(1\right)-f(-1)\right)\left(\overline{g}\left(1\right)-\overline{g}(-1)\right)}{2}}+{\int }_{-1}^1(f^{\prime }\left(x\right){\overline{g}}^{\prime }\left(x\right)+cf\left(x\right)\overline{g}\left(x\right))dx,$ where c is a fixed, positive constant. These polynomials generalize the Althammer (or Legendre–Sobolev) polynomials first studied by Althammer and Schäfke. The Krein–Sobolev polynomials were found as a result of a left-definite spectral study of the self-adjoint Krein Laplacian operator ${\mathcal{K}}_c$$(c>0)$ in $L^2(-1,1).$ Other than $K_0$ and $K_1,$ these polynomials are not eigenfunctions of ${\mathcal{K}}_c.$ As shown by Littlejohn and Quintero, the sequence ${\left\{K_n\right\}}_{n=0}^{\infty }$ forms a complete orthogonal set in the first left-definite space $(H^1[-1,1],{(·,·)}_{1,c})$ associated with $({\mathcal{K}}_c,$$L^2(-1,1))$. Furthermore, they show that, for $n\ge 1,$$K_n\left(x\right)$ has n distinct zeros in $(-1,1).$ In this note, we find an explicit formula for Krein–Sobolev polynomials ${\left\{K_n\right\}}_{n=0}^{\infty }$. Full article

Developing spherical sector harmonics (SSHs) benefits sound field decomposition and analysis over spherical sector regions. Although SSHs demonstrate potential in the field of spatial audio, a comprehensive investigation into their properties and performance is absent. This paper seeks to close this gap by revealing three key limitations of SSHs and exploring their performance in two aspects: sector sound field radial extrapolation and sector sound field decomposition and reconstruction. First, SSHs are not solutions to the Helmholtz equation, which is their main limitation. Then, due to the violation of the Helmholtz equation, SSHs lack the ability to conduct sound field radial extrapolation, especially for interior cases. Third, when using SSHs to decompose and reconstruct a sound field, the shifted associated Legendre polynomials and scaled exponential function in SSHs result in severe distortion around the edge of the sector region. In light of these three limitations, the future implementation of SSHs should focus on processing and analyzing the measurement sector region without any extrapolation process, and the measurement region should be larger than the target sector region. Full article

Improving reproductive traits, particularly semen quality and quantity, is crucial for optimizing poultry production and addressing the current limitations in native chicken reproduction. The aim of this study was to develop a genetic model to estimate genetic parameters guiding the selection of individual Thai native roosters. Using data collected from 3475 records of 242 Thai native grandparent roosters aged 1–4 years, we evaluated semen traits (mass movement, semen volume, and sperm concentration) over 54 weeks. A random regression test–day model incorporating five covariance functions, including a linear spline function with four, five, six, and eight knots (SP4, SP5, SP6, and SP8) and second-order Legendre polynomial function (LG2), was used to estimate genetic parameters. The results showed that the SP8 model consistently outperformed the other models across all traits, with the lowest mean square error, highest coefficient of determination, and superior predictive ability. Heritability estimates for mass movement, semen volume, and sperm concentration ranged from 0.10 to 0.25, 0.22 to 0.25, and 0.11 to 0.24, respectively, indicating moderate genetic influence on these traits. Genetic correlations between semen volume and sperm concentration were highest in the SP8 model, highlighting a strong genetic association between these traits. The SP8 model also revealed a high genetic correlation between mass movement and semen volume, supporting the potential for selecting mass movement as a predictor of semen volume. In conclusion, this study highlights the effectiveness of random regression models with linear spline functions to evaluate the genetic parameters of semen traits in native Thai roosters. The SP8 model is a robust tool for breeders to enhance the reproductive performance of native Thai chickens, contributing to sustainable poultry production systems. Full article

The aim of this paper is to derive formulae for the generating functions of the Bernstein type polynomials. We give a PDE equation for this generating function. By using this equation, we give recurrence relations for the Bernstein polynomials. Using generating functions, we also derive some identities including a symmetry property for the Bernstein type polynomials. We give some relations among the Bernstein type polynomials, Bernoulli numbers, Stirling numbers, Dahee numbers, the Legendre polynomials, and the coefficients of the classical superoscillatory function associated with the weak measurements. We introduce some integral formulae for these polynomials. By using these integral formulae, we derive some new combinatorial sums involving the Bernoulli numbers and the combinatorial numbers. Moreover, we define Bezier type curves in terms of these polynomials. Full article

To solve the fractional gas dynamic equation, this paper presents an effective algorithm using the collocation method and Müntz-Legendre (M-L) polynomials. The approach chooses a solution of a finite-dimensional space that satisfies the desired equation at a set of collocation points. The collocation points in this study are selected to be uniformly spaced meshes or the roots of shifted Legendre and Chebyshev polynomials. Müntz-Legendre polynomials have the interesting property that their fractional derivative is also a Müntz-Legendre polynomial. This property ensures that these bases do not face the problems associated with using the classical orthogonal polynomials when solving fractional equations using the collocation method. The numerical simulations illustrate the method’s effectiveness and accuracy. Full article

The notion of k-harmonic curves is associated with the kth-order variational problem defined by the k-energy functional. The present paper gives a geometric formulation of this higher-order variational problem on a Riemannian manifold M and describes a generalized Legendre transformation defined from the kth-order tangent bundle $T^{k}M$ to the cotangent bundle $T^{*}{T}^{k-1}M$. The intrinsic version of the Euler–Lagrange equation and the corresponding Hamiltonian equation obtained via the Legendre transformation are achieved. Geodesic and cubic polynomial interpolation is covered by this study, being explored here as harmonic and biharmonic curves. The relationship of the variational problem with the optimal control problem is also presented for the case of biharmonic curves. 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

The widespread of composite structures demands efficient numerical methods for the simulation dynamic behaviour of elastic laminates with interface delaminations with interacting faces. An advanced boundary integral equation method employing the Hankel transform of Green’s matrices is proposed for modelling wave scattering and analysis of the eigenfrequencies of interface circular partially closed delaminations between dissimilar media. A more general case of partially closed circular delamination is introduced using the spring boundary conditions with non-uniform spring stiffness distribution. The unknown crack opening displacement is expanded as Fourier series with respect to the angular coordinate and in terms of associated Legendre polynomials of the first kind via the radial coordinate. The problem is decomposed into a system of boundary integral equations and solved using the Bubnov-Galerkin method. The boundary integral equation method is compared with the meshless method and the published works for a homogeneous space with a circular open crack. The results of the numerical analysis showing the efficiency and the convergence of the method are demonstrated. The proposed method might be useful for damage identification employing the information on the eigenfrequencies estimated experimentally. Also, it can be extended for multi-layered composites with imperfect contact between sub-layers and multiple circular delaminations. Full article

In this article, two new subclasses of the bi-univalent function class $\sigma$ related with Legendre polynomials are presented. Additionally, the first two Taylor–Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$ for the functions belonging to these new subclasses are estimated. Full article