Search Results (48)

This research investigates the numerical analysis of a two-dimensional (2D) delay mixed integral equation (DMIE) of the second kind with a continuous kernel. The existence, uniqueness, stability, and convergence of the solution are rigorously established in the functional space $C\left(\right[a,b]\times [0,T\left]\right)$, providing a solid theoretical foundation. A collocation method based on Chebyshev and Legendre polynomials is applied to obtain accurate numerical approximations, leading to a system of algebraic equations applicable to both linear and nonlinear cases. Numerical simulations are performed using MATLAB to evaluate the method’s performance and calculate the results, including error behavior, convergence, and stability. Several illustrative applications are presented to demonstrate the method’s numerical stability and versatility in handling both linear and nonlinear problems. The results underscore the efficiency, accuracy, and robustness of the proposed numerical schemes in solving complex 2D DMIE, highlighting the novelty of the framework and its strong potential for broad applications in contemporary scientific and engineering problems. Full article

The bichromatic triangle polynomial $P_G\left(k\right)$ counts vertex k-colorings in which every triangle uses exactly two colors. We develop a transfer matrix framework for three canonical families of 2-trees: book graphs $B_n$, 1-fans $F_n^1$, and triangulated ladders $T{L}_m$. In each case, $P_G\left(k\right)$ satisfies a second-order linear recurrence with an explicit closed form; for $T{L}_m$ this yields a Chebyshev representation, while for $F_n^1$ the binary specialization gives $P_{F_n^1}\left(2\right)=2F_{n+1}$. A spectral identity ${\alpha }^2=r_{+}$ links the dominant characteristic roots of the fan and ladder recurrences, implying identical exponential growth rates when indexed by vertex count, whereas book graphs grow strictly faster for $k\ge 4$. In fact, this correspondence is exact: for all $k\ge 2$, the triangulated ladder polynomial coincides with that of a suitably indexed 1-fan. Passing to line graphs, we interpret $P_{L\left(K_n\right)}\left(k\right)$ as counting edge colorings of $K_n$ that forbid both monochromatic and rainbow triangles, and we identify a sharp obstruction threshold at $n\ge 6$. Full article

This article proposes single-layer neural network algorithms for solving second-order ordinary differential equations, based on the principles of functional connection. According to this principle, the hidden layer of the neural network is replaced by a functional expansion unit to improve input patterns using orthogonal Chebyshev, Legendre, and Laguerre polynomials. The polynomial neural network algorithms were implemented in the Python programming language using the PyCharm environment. The performance of the polynomial neural network algorithms was tested by solving initial-boundary value problems for the nonlinear Lane–Emden equation. The solution results are compared with the exact solution of the problems under consideration, as well as with the solution obtained using a multilayer perceptron. It is shown that polynomial neural networks can perform more efficiently than multilayer neural networks. Furthermore, a neural network based on Laguerre polynomials can, in some cases, perform more accurately and faster than neural networks based on Legendre and Chebyshev polynomials. The issues of overtraining of polynomial neural networks and scenarios for overcoming it are also considered. Full article

The free vibration of stiffened plates analyzed using classical plate–beam theoretical theory (PBM) simplified the vibrations of stiffeners parallel to the plane of the stiffened plate as the first-order torsional vibration of the stiffener cross-section. This simplification introduces errors in both the natural frequencies and mode shapes of the structure for stiffened plates with relatively tall stiffeners. To mitigate the issue previously described, this paper proposes an enhanced plate–beam theoretical model (EPBM). The EBPM decouples stiffener deformation into two components: (1) bending deformation along the transverse direction of the stiffened plate, governed by Euler–Bernoulli beam theory, and (2) transverse deformation of the stiffeners, modeled using thin plate theory. Virtual torsional springs are introduced at the stiffener–plate and stiffener–stiffener interfaces via penalty function method to enforce rotational continuity. These constraints are transformed into energy functionals and integrated into the system’s total energy. Displacement trial functions constructed from Chebyshev polynomials of the first kind are solved using the Ritz method. Numerical validation demonstrates that the EBPM significantly improves accuracy over the BPM: errors in free-vibration frequency decrease from 2.42% to 0.63% for the first mode and from 9.79% to 1.34% for the second mode. For constrained vibration, the second-mode error is reduced from 4.22% to 0.03%. This approach provides an effective theoretical framework for the vibration analysis of structures with high stiffeners. Full article

In this paper, we establish new hypergeometric representations for two classical integer sequences, namely the Pell and Jacobsthal sequences. Motivated by Dilcher’s hypergeometric formulations of the Fibonacci sequence, we extend this framework to other second-order linear recurrence sequences with distinct characteristic structures. By employing Binet-type formulas, recurrence relations, Chebyshev polynomial connections, and classical transformation properties of Gauss hypergeometric functions, we derive several explicit and alternative representations for the Pell and Jacobsthal numbers. These representations unify known identities, yield new closed-form expressions, and reveal deeper structural parallels between hypergeometric functions and linear recurrence sequences. The results demonstrate that hypergeometric functions provide a systematic and versatile analytical tool for studying special number sequences beyond the Fibonacci case, and they suggest potential extensions to broader families such as Horadam-type sequences and their generalizations. Full article

This paper proposes a calibration method for redundant robot arms with highly elastic joints. The method uses the second-order Chebyshev polynomial to characterize the variation in the error with the poses of all joints. This error model is consistent with the variation in the gravitational torque on each joint and demonstrates good generalization. Based on this, the calibration model includes both kinematic errors and non-kinematic errors. For this high-dimensional model, an adaptive iterative identification algorithm is proposed for a large number of small error parameters of various types. The algorithm sets specific iteration rules for different types of error parameters and adjusts the convergence amplitude in each iteration, ensuring that the iterative algorithm converges to the global optimum. The simulation results show that for a redundant robot arm with 12 highly elastic joints, even with large linearization modeling errors, the new identification algorithm can gradually eliminate them during iteration, achieving an identification accuracy higher than 99.975% for all of the error parameters. The experimental results indicate that on a redundant robot arm with eight cable-driven elastic joints, the new model and identification algorithm reduce the 96.6% absolute positioning errors of the robot arm, enabling it to perform precise and flexible operations. It takes 40.534 s and 29.077 s to run the identification algorithm on MATLAB (R2023b, 2.10 GHz CPU) in the simulation and experiment, respectively. Full article

This paper introduces an efficient numerical method based on applying the typical Petrov–Galerkin approach ($\mathrm{PGA}$) to solve the time fractional diffusion wave equation ($\mathcal{TFDWE}$). The method utilises asymmetric polynomials, namely, shifted second-kind Chebyshev polynomials ($\mathrm{SSKCPs}$). New derivative formulas are derived and used for these polynomials to establish the operational matrices of their derivatives. The paper presents rigorous error bounds for the proposed method in Chebyshev-weighted Sobolev space and demonstrates its accuracy and efficiency through several illustrative numerical examples. The results reveal that the method achieves high accuracy with relatively low polynomial degrees. Full article

In this study, we define a new family of Gaussian polynomials, called Gaussian Chebyshev polynomials, by extending classical Chebyshev polynomials into the complex domain. These polynomials are characterized by second-order linear recurrence relations, and their connections with the Chebyshev polynomials are established. We also examine properties such as Binet-type formulas and generating functions. Moreover, we characterize some relationships between Gaussian and classical Chebyshev polynomials for the first and second kinds. We obtain some well-known theorems, such as Cassini, Catalan, and d’Ocagne’s theorems, for the first and second kinds. Furthermore, we present important connections among four types of these new polynomials. In the proofs of our results, we utilize the symmetric and antisymmetric properties of the Chebyshev polynomials. Finally, it is shown that Gaussian Chebyshev polynomials are closely related to well-known special sequences such as the Fibonacci, Lucas, Gaussian Fibonacci, and Gaussian Lucas numbers for some specific values of variables. Full article

The main goal of this paper is to present a new numerical algorithm for solving two models of one-dimensional fractional partial differential equations (FPDEs) subject to initial conditions (ICs) and integral boundary conditions (IBCs). This paper builds a modified shifted Chebyshev polynomial of the second kind (MSC2Ps) basis function that meets homogeneous IBCs, named IMSC2Ps. We also introduce two types of MSC2Ps that satisfy the given ICs. We create two operational matrices (OMs) for both ordinary derivatives (ODs) and Caputo fractional derivatives (CFDs) connected to these basis functions. By employing the spectral collocation method (SCM), we convert the FPDEs into a system of algebraic equations, which can be solved using any suitable numerical solvers. We validate the efficacy of our approach through convergence and error analyses, supported by numerical examples that demonstrate the method’s accuracy and effectiveness. Comparisons with existing methodologies further illustrate the advantages of our proposed technique, showcasing its high accuracy in approximating solutions. Full article

Trajectory planning and tracking control strategies have a significant impact on the fast and stable operation of high-precision position servo permanent magnet synchronous motors (PMSMs). Therefore, this paper proposes an active disturbance rejection control (ADRC) strategy for high-precision position servo PMSMs based on jerk- and time-optimal trajectory planning. Firstly, in order to meet the requirement of continuous jerk in the positioning process of precision loads, the seventh-degree Chebyshev polynomial is adopted to establish the point-to-point trajectory planning function. Based on the dynamic boundary conditions under the short-term overload of PMSMs, and with the positioning time as the optimization objective, the optimal coefficient of the polynomial is solved through the fast particle swarm optimization (FPSO) algorithm to obtain the trajectory planning function that takes into account both jerk and time performance. Then, the trajectory plan is used as the position loop reference signal to construct a non-cascade second-order ADRC strategy, leading to a position servo PMSM control strategy that combines a second-order disturbance observer and feedback control law. Finally, the experimental platform is set up to verify the proposed method. The results show that, compared with the traditional control methods, the steady-state positioning time of the control strategy proposed under typical working conditions is reduced by 12.5%, and the jerk continuity during the positioning process has also been significantly improved. Full article

The aim of this work is to model the nonlinear dynamics of conservative oscillators, with restoring force originating from even-order potentials. In particular, we extend our previous findings on inverting the time-integral equation that arises in the solution of such dynamical systems, a task that is almost always intractable in exact form. This is faced and solved by approximating the restoring force with its Chebyshev series truncated to order five; such a quintication approach yields a quinticate oscillator, whose associated time-integral can be inverted in closed form. Our solution procedure is based on the quinticate oscillator coefficients, upon which a second-order polynomial is constructed, which appears in the time-integrand of the quinticate problem, and whose roots determine the expression of the closed-form solution, as well as that of its period. The presented algorithm is implemented in the Mathematica software and validated on some conservative nonlinear oscillators taken from the relevant literature. Full article

The current article introduces a Petrov–Galerkin method (PGM) to address the fourth-order uniform Euler–Bernoulli pinned–pinned beam equation. Utilizing a suitable combination of second-kind Chebyshev polynomials as a basis in spatial variables, the proposed method elegantly and simultaneously satisfies pinned–pinned and clamped–clamped boundary conditions. To make PGM application easier, explicit formulas for the inner product between these basis functions and their derivatives with second-kind Chebyshev polynomials are derived. This leads to a simplified system of algebraic equations with a recognizable pattern that facilitates effective inversion to produce an approximate spectral solution. Presentations are made regarding the method’s convergence analysis and the computational cost of matrix inversion. The efficiency of the method described in precisely solving the Euler–Bernoulli beam equation under different scenarios has been validated by numerical testing. Additionally, the procedure proposed in this paper is more effective compared to other existing techniques. Full article

This article proposes numerical algorithms for solving second-order and telegraph linear partial differential equations using a matrix approach that employs certain generalized Chebyshev polynomials as basis functions. This approach uses the operational matrix of derivatives of the generalized Chebyshev polynomials and applies the collocation method to convert the equations with their underlying conditions into algebraic systems of equations that can be numerically treated. The convergence and error bounds are examined deeply. Some numerical examples are shown to demonstrate the efficiency and applicability of the proposed algorithms. Full article

In this paper, we have established a numerical method for a class of time-fractional and Riesz space distributed-order advection–diffusion equation with time-delay. Firstly, we transform the Riesz space distributed-order derivative term of the diffusion equation into multi-term fractional derivatives by using the Gauss quadrature formula. Secondly, we discretize time by using second-order finite differences, discretize space by using second kind Chebyshev polynomials, and convert the multi-term fractional equation to a system of algebraic equations. Finally, we solve the algebraic equations by the iterative method, and prove the stability and convergence. Moreover, relevant examples are shown to verify the validity of our algorithm. Full article

In this paper, the finite element method (FEM) is integrated with orthogonal polynomial approximation in high-dimensional spaces to innovatively model the Moon’s surface gravity anomaly. The aim is to approximate solutions to Laplace’s classical differential equations of gravity, employing classical Chebyshev polynomials as basis functions. Using classical Chebyshev polynomials as basis functions, the least-squares approximation was used to approximate discrete samples of the approximation function. These test functions provide an understanding of errors in approximation and corresponding errors due to differentiation and integration. These test functions provide an understanding of errors in approximation and corresponding errors due to differentiation and integration. The first application of this project is to substitute the globally valid classical spherical harmonic series of approximations with locally valid series of orthogonal polynomial approximations (i.e., using the FEM approach). With an error tolerance set at ${10}^{-9}$$\mathrm{m}{\mathrm{s}}^{-2}$, this method is used to adapt the gravity model radially upwards from the lunar surface. The results showcase a need for a higher degree of approximation on and near the lunar surface, with the necessity decreasing as the radius increases. Notably, this method achieves a computational speedup of five orders of magnitude when applying the method to radial adaptation. More intrinsically, the second application involves using the methodology as an effective tool in solving boundary value problems. Specifically, this approach is implemented to solve classical differential equations involved with high-precision, long-term orbit propagation. This application provides a four-order-of-magnitude speedup in computational time while maintaining an error within the ${10}^{-10}$$\mathrm{m}{\mathrm{s}}^{-2}$ error range for various orbit propagation tests. Alongside the advancements in orthogonal approximation theory, the FEM enables revolutionary speedups in orbit propagation without compromising accuracy. Full article