In this paper, a new approach for solving the system of fractional integro-differential equation with weakly singular kernels is introduced. The method based on class symmetric orthogonal polynomials called shifted sixth-kind Chebyshev polynomials. First, operational matrices are constructed, and after that, described. This reduces equations (WSFIDEs) by collocation into algebraic equations. Th...

We show that the lamplighter group L has a system of generators for which spectrum discrete Laplacian on Cayley graph is union an interval and countable set isolated points accumulating to point outside this interval. This first example with infinitely many gaps in its graph. The result obtained by careful study spectral properties one-parametric family convolution operators L. Our results pure...

We reveal the relationship between a Petrov–Galerkin method and a spectral collocation method at the Chebyshev points of the second kind (±1 and zeros of Uk) for the two-point boundary value problem. Derivative superconvergence points are identified as the Chebyshev points of the first kind (Zeros of Tk). Super-geometric convergent rate is established for a special class of solutions.

We provide a detailed description of numerical approach that makes use the shifted Chebyshev polynomials sixth kind to approximate solution some fractional order differential equations. Specifically, we choose Fisher–Kolmogorov–Petrovskii–Piskunov equation (FFKPPE) describe this method. write our in product form, which consists unknown coefficients and polynomials. To compute values coefficient...

The main aim of this article is to start with an expository introduction the trigonometric ratios and then proceed latest results in field. Historically, exact were obtained using geometric constructions. methods have their own limitations arising from certain theorems. In view methods, we shall focus on powerful techniques equations deriving surds. cubic higher-order naturally arise while rati...

The purpose of this note is to characterize all the sequences orthogonal polynomials \((P_n)_{n\ge 0}\) such that $$\begin{aligned} \frac{\triangle }{\mathbf{\triangle } x(s-1/2)}P_{n+1}(x(s-1/2))=c_n(\triangle +2\,\mathrm {I})P_n(x(s-1/2)), \end{aligned}$$where \(\,\mathrm {I}\) identity operator, x defines a class lattices with, generally, nonuniform step-size, and \(\triangle f(s)=f(s+1)-f(s...

Remark: A similar argument shows that any secondorder linear recurrent sequence also satisfies a quadratic second-order recurrence relation. A familiar example is the identity Fn−1Fn+1 − F2 n = (−1)n for Fn the nth Fibonacci number. More examples come from various classes of orthogonal polynomials, including the Chebyshev polynomials mentioned below. Second solution. We establish directly that ...