Kind Chebyshev Polynomials Research Articles

Piecewise second kind Chebyshev functions for a class of piecewise fractional nonlinear reaction–diffusion equations with variable coefficients

In this paper, a new type of piecewise fractional derivative (PFD) is introduced. The ordinary and distributed-order fractional derivatives in the Caputo sense are used to define this type of PFD. A new version of nonlinear reaction–diffusion equations with variable coefficients is defined using this type of PFD. The orthonormal piecewise second kind Chebyshev functions (CFs), as a new family of basic functions, are generated. An explicit formula is extracted for PFD of these piecewise functions. A hybrid method based on the orthonormal piecewise second kind CFs and orthonormal second kind Chebyshev polynomials is proposed to solve the aforementioned problem. The established approach transforms solving the expressed problem into solving an algebraic system of equations. To illustrate the accuracy of the developed method, some numerical examples are considered.

The Numerical Solutions of weakly singular Fredholm integral equations of the Second kind Using Chebyshev Polynomials of the Second Kind

In this study, the second kind Chebyshev Polynomials were utilized to acquire interpolated solutions for the second kind Fredholm integral equations with weakly singular kernel. To accomplish this, the data, unknown, and kernel functions were converted into matrix form, and consequently we completely isolated the singularity of the kernel. The primary benefit of this method is the ability to change the form of integral equation to an equivalent algebraic system, which is easier to solve. The effectiveness of our technique was evaluated by applying it to three illustrated examples, and it was observed that the solutions obtained exhibit strong convergence towards the exact solutions.

Numerical treatment of the fractional Rayleigh-Stokes problem using some orthogonal combinations of Chebyshev polynomials

<p>This work aims to provide a new Galerkin algorithm for solving the fractional Rayleigh-Stokes equation (FRSE). We select the basis functions for the Galerkin technique to be appropriate orthogonal combinations of the second kind of Chebyshev polynomials (CPs). By implementing the Galerkin approach, the FRSE, with its governing conditions, is converted into a matrix system whose entries can be obtained explicitly. This system can be obtained by expressing the derivatives of the basis functions in terms of the second-kind CPs and after computing some definite integrals based on some properties of CPs of the second kind. A thorough investigation is carried out for the convergence analysis. We demonstrate that the approach is applicable and accurate by providing some numerical examples.</p>

An Adomian decomposition method with some orthogonal polynomials to solve nonhomogeneous fractional differential equations (FDEs)

<p>The present study introduced modifications to the standard Adomian decomposition method (ADM) by combining the Taylor series with orthogonal polynomials, such as Legendre polynomials and the first and second kinds of Chebyshev polynomials. These improvements can be applied to solve fractional differential equations with initial-value problems in the Caputo sense. The approaches are based on the use of orthogonal polynomials, which are essential components in approximation theories. The study carefully analyzed their respective absolute error differences, highlighting the computational benefits of the proposed modifications, which offer improved accuracy and require fewer computational steps. The effectiveness and accuracy of the approach were validated through numerical examples, confirming its efficiency and reliability.</p>

Stationary Stochastic Response Analysis of the FG-GPLRC Irregular Quadrilateral Plate Based on the Chebyshev–Ritz Method

The stationary random response characteristics of irregular quadrilateral plates under the base acceleration excitation are investigated in this paper. The computational model chosen in this paper is functionally graded graphene platelet reinforced composites (FG-GPLRC). Simultaneously, the material parameters are calculated employing the modified Halpin–Tsai model. A unified dynamic model based on the first-order shear deformation theory (FSDT) is proposed for FG-GPLRC irregular plates, where the displacement variables are considered as the first kind Chebyshev polynomials combined with coordinate transformation. The Chebyshev–Ritz method is utilized to derive the kinetic equations of irregular quadrilateral plates, and then the pseudo-excitation method (PEM) is employed to solve the stationary stochastic response of the system under the base acceleration excitation. The free vibration and stationary stochastic response of the plates under various boundaries are validated by a sufficient number of numerical studies, and the computational results display a favorable match with the published literature and finite element method (FEM), which verifies the correctness and reliability of the present method. Finally, a detailed parametric analysis of the stationary stochastic response characteristics of FG-GPLRC irregular quadrilateral plates is presented. The effects of various mass fractions, number of layers, geometric dimensions of GPL, and boundary constraints on the stationary stochastic vibration behaviors of FG-GPLRC irregular quadrilateral plates are considered.

Time-domain multiple traces boundary integral formulation for acoustic wave scattering in 2D

We present a novel computational scheme to solve acoustic wave transmission problems over two-dimensional composite scatterers, i.e. penetrable obstacles possessing junctions or triple points. The continuous problem is cast as a local multiple traces time-domain boundary integral formulation. For discretization in time and space, we resort to convolution quadrature schemes coupled to a non-conforming spatial spectral discretization based on second kind Chebyshev polynomials displaying fast convergence. Computational experiments confirm convergence of multistep and multistage convolution quadrature for a variety of complex domains.

Applications of the Second Kind Chebyshev Polynomials of Bi-Starlike and Bi-Convex λ-Pseudo Functions Associated with Sakaguchi Type Functions

The purpose of this paper is to use the second kind Chebyshev polynomials to introduce a new class of analytic and bi-univalent functions associating bi-starlike and biconvex $\lambda$-pseudo functions with Sakaguchi type functions defined in the open unit disk. We determinate upper bounds for the initial Taylor-Maclaurin coefficients $\left|a_{2}\right|$ and $\left|a_{3}\right|$ for functions in this class.

A hybrid spectral approach based on 2D cardinal and classical second kind Chebyshev polynomials for time fractional 3D Sobolev equation

In this work, the Caputo fractional derivative defines the time fractional 3D Sobolev equation. The 2D shifted second kind Chebyshev cardinal polynomials (SSKCCPs) and 2D shifted second kind Chebyshev polynomials (SSKCPs) (as two well‐known classes of basis functions) are utilized to establish a hybrid technique for this new problem. First, the problem solution is approximated simultaneously using the 2D SSKCCPs (relative to ) and 2D SSKCCPs (relative to ). Next, the classical and fractional operational matrices of these polynomials are achieved and applied to make the hybrid algorithm. With a combination of derived operational matrices and the collocation approach, solving the expressed fractional 3D problem turns into solving an equivalent system of algebraic equations. The proposed algorithm's accuracy is checked using four numerical examples.

Computational Algorithm for Fractional Fredholm Integro-Differential Equations

Fractional calculus is a fascinating field of mathematics that focuses on the study of integrals and derivatives of arbitrary orders, extending the principles of basic calculus. Its applications span across various scientific, engineering, and other disciplines. In this study, the collocation method, in conjunction with the utilization of the fourth kind Chebyshev polynomials, is employed to explore solutions for fractional integro-differential equations of Fredholm type. By applying the collocation method, the problem at hand is transformed into a system of linear algebraic equations. These equations are subsequently solved by employing matrix inversion techniques to determine the unknown constants. To provide a comprehensive understanding and visualization of the results, the research incorporates tables and figures, which present numerical examples and comparisons. These comparisons serve to highlight the superior performance of the proposed method in terms of efficiency and convenience when compared to traditional methods. By showcasing the advantages of the collocation method and the utilization of fourth kind Chebyshev polynomials, the research underscores the potential of these approaches in solving fractional integro-differential equations.

On projection method for numerical solution of hypersingular integral equations of the first kind combined with quadrature methods

The Fredholm integral equations of the first kind with two regular and hypersingular kernels are considered on [−1, 1]. The hypersingular kernel is considered smooth enough with no additional conditions. A projection method based on second kind Chebyshev polynomials approximation, combined with quadrature integration method, is developed to obtain high accurate approximations. The proposed method reduces the underlying integral equation to a system of algebraic equations. Several illustrative examples are provided to show the efficiency of the method.

A fully adaptive explicit stabilized integrator for advection–diffusion–reaction problems

A novel second order family of explicit stabilized Runge–Kutta–Chebyshev methods for advection–diffusion–reaction equations is introduced. The new methods outperform existing schemes for relatively high Peclet number due to their favorable stability properties and explicitly available coefficients. The construction of the new schemes is based on stabilization using second kind Chebyshev polynomials first used in the construction of the stochastic integrator SK-ROCK. An adaptive algorithm to implement the new scheme is proposed. This algorithm is able to automatically select the suitable step size, number of stages, and damping parameter at each integration step. Numerical experiments that illustrate the efficiency of the new algorithm are presented.

Some Novel Formulas of Lucas Polynomials via Different Approaches

Some new formulas related to the well-known symmetric Lucas polynomials are the primary focus of this article. Different approaches are used for establishing these formulas. A matrix approach to Lucas polynomials is followed in order to obtain some fundamental properties. Particularly, some recurrence relations and determinant forms are determined by suitable Hessenberg matrices. Conjugate Lucas polynomials and generating functions are derived and examined. Several connection problems between the Lucas polynomials and other celebrated symmetric and non-symmetric orthogonal polynomials such as the first and second kinds of Chebyshev polynomials and their shifted counterparts are solved. We prove that several argument-type hypergeometric functions are involved in the connection coefficients. In addition, we construct new formulas for high-order derivatives of Lucas polynomials in terms of their original polynomials, as well as formulas for repeated integrals of Lucas polynomials.

Spectral treatment for the fractional-order wave equation using shifted Chebyshev orthogonal polynomials

This paper will examine the approximate solution for the fractional-order wave equation. The method used for this purpose fundamentally is based on the second kind of Chebyshev polynomials besides the Chebyshev collocation mechanism in addition to the forward finite difference scheme. The fractional-order terms are expressed through Caputo’s fractional-order derivative definition. With these tools, the problem will be converted into an algebraic system of equations that are solved numerically. The accuracy and applicability of the proposed technique are demonstrated through some given numerical applications.

Solution of the population balance equation for wet granulation using second kind Chebyshev polynomials

Wet granulation is used in many industrial fields, such as pharmaceutical and nutraceutical, to improve compressibility properties of powders. The comprehension of the physical phenomena of the granulation process, such as simultaneous particle breakage, aggregation, and nucleation, is fundamental to optimize the operating conditions. In this work the population balance equation (PBE), used to mathematically describe these physical phenomena, has been numerical solved with a new finite element method with expansion coefficients in terms of a truncated series expansion with the orthogonal second kind Chebyshev basis polynomials. Such a numerical method is a straightforward and effective method, which has the advantage of concurrently giving the distribution and the different required moments. Moreover, the proposed model accounting for the breakage, aggregation, and nucleation phenomena was used in an ad-hoc optimization procedure to describe the particle size distribution of experimental results of a wet granulation process reaching satisfactorily RMS values.

A New Numerical Approach Using Chebyshev Third Kind Polynomial for Solving Integrodifferential Equations of Higher Order

There are several classifications of linear Integral Equations. Some of them include; Voltera Integral Equations, Fredholm Linear Integral Equations, Fredholm-Voltera Integrodifferential. In the past, solutions of higher-order Fredholm-Volterra Integrodifferential Equations [FVIE] have been presented. However, this work uses a computational techniques premised on the third kind Chebyshev polynomials method. The performance of the results for distinctive degrees of approximation (M) of the trial solution is cautiously studied and comparisons have been additionally made between the approximate/estimated and exact/definite solution at different intervals of the problems under consideration. Modelled Problems have been provided to illustrate the performance and relevance of the techniques. However, it turned out that as M increases, the outcomes received after every iteration get closer to the exact solution in all of the problems considered. The results of the experiments are therefore visible from the tables of errors and the graphical representation presented in this work.

A numerical algorithm for solving index-1 weakly singular integral-algebraic equations with non-smooth solutions

This paper aims to present a product integration approach based on the first kind of Chebyshev polynomials to solve the weakly singular integral-algebraic equations of index-1. These are mixed systems of weakly singular Volterra integral equations of the second and first kind. The solutions of these equations have a singularity at the lower bound of the domain of integration. To deal with this non-smooth behavior of solutions, we apply the suitable transformations to get a new system with regular solutions. We use the differentiation index in the theoretical part. One of the advantages of selecting the weights of the presented numerical method is that when we determine the error bounds, some expressions will be equal to zero speeding up the process of achieving the error bounds. Finally, by providing some numerical examples, we will observe that the numerical results confirm the validity of convergence analysis.

New Results of the Fifth-Kind Orthogonal Chebyshev Polynomials

The principal objective of this article is to develop new formulas of the so-called Chebyshev polynomials of the fifth-kind. Some fundamental properties and relations concerned with these polynomials are proposed. New moments formulas of these polynomials are obtained. Linearization formulas for these polynomials are derived using the moments formulas. Connection problems between the fifth-kind Chebyshev polynomials and some other orthogonal polynomials are explicitly solved. The linking coefficients are given in forms involving certain generalized hypergeometric functions. As special cases, the connection formulas between Chebyshev polynomials of the fifth-kind and the well-known four kinds of Chebyshev polynomials are shown. The linking coefficients are all free of hypergeometric functions.

Time fractional advection-dispersion model to study transportation of particles with time-memory for unsteady nonequilibrium suspension in open-channel turbulent flows

In this study, the effects of time-memory on the mixing and nonequilibrium transportation of particles in an unsteady turbulent flow are investigated. The memory effect of particles is captured through a time-fractional advection-dispersion equation rather than a traditional advection-dispersion equation. The time-fractional derivative is considered in Caputo sense which includes a power-law memory kernel that captures the power-law jumps of particles. The time-fractional model is solved using the Chebyshev collocation method. To make the solution procedure more robust three different kinds of Chebyshev polynomials are considered. The time-fractional derivative is approximated using the finite difference method at small time intervals and numerical solutions are obtained in terms of Chebyshev polynomials. The model solutions are compared with existing experimental data of traditional conditions and satisfactory results are obtained. Apart from this, the effects of time-memory are analyzed for bottom concentration and transient concentration distribution of particles. The results show that for uniform initial conditions, bottom concentration increases with time as the order of fractional derivative decreases. In the case of transient concentration, the value of concentration initially decreases when T < 1 and thereafter increases throughout the flow depth. The effects of time-memory are also analyzed under steady flow conditions. Results show that under steady conditions, transient concentration is more sensitive for linear, parabolic, and parabolic-constant models of sediment diffusivity rather than the constant model.

The Characteristics of the First Kind of Chebyshev Polynomials and its Relationship to the Ordinary Polynomials

The Characteristics of the First Kind of Chebyshev Polynomials and its Relationship to the Ordinary Polynomials

Adomian Decomposition Method with Orthogonal Polynomials: Laguerre Polynomials and the Second Kind of Chebyshev Polynomials

In this paper, a new efficient and practical modification of the Adomian decomposition method is proposed with Laguerre polynomials and the second kind of Chebyshev polynomials which has not been introduced in other articles to the best of our knowledge. This approach can be utilized to approximately solve linear and nonlinear differential equations. The proposed formulations are examined by a representative example and the numerical results confirm their efficiency and accuracy.