Chebyshev Polynomials Of Fourth Kind Research Articles

Study of unsteady nonequilibrium stratified suspended sediment distribution in open-channel turbulent flows using shifted Chebyshev polynomials

ABSTRACT In this paper the effect of stratification on vertical variation of sediment concentration in the suspended-load layer in turbulent flows through open channels under unsteady and nonequilibrium condition is investigated. Unlike previous studies, the model includes the effect of stratification in terms of reduction of the eddy diffusivity. A new approach for solving the governing advection-diffusion equation using orthogonal shifted Chebyshev polynomials of fourth kind is presented. In this approach, the solution of the problem is obtained using a finite degree Chebyshev polynomial. The distribution of the sediment diffusivity is considered including the effect of stratification. Three different types (constant, linear and parabolic) of sediment diffusivity distribution is considered both in neutral and stratified flow condition. The final transport equation is solved under both neutral and stratified condition and appropriate bottom boundary condition using the conventional backward Euler scheme. The method is unconditionally convergent and converges more rapidly than previous methods as Laplace transformation and generalized integral transformation technique. The solutions are compared with previous methods and satisfactory results are obtained. The proposed solutions are also validated with experimental data under steady flow conditions. Results show that under stratified flow condition, sediment concentration also decreases when flow is still unsteady.

Fourth Kind Shifted Chebyshev Polynomials for Solving Space Fractional Order Advection–Dispersion Equation Based on Collocation Method and Finite Difference Approximation

In this paper, we study the numerical solution of one-dimensional space fractional advection–dispersion equations (FADE). The integer order space derivatives are replaced by fractional derivatives in the Caputo sense. The proposed scheme is based on the collocation method and the finite difference method (FDM). An effective method is applied for solving FADE by using the shifted Chebyshev polynomials of fourth kind (SCPFK) to reduce FADE to a system of ordinary differential equations, which can be solved by FDM, then apply an iteration method to solve this system of equations. This method is unconditionally stable, consistent and convergent. Also we are discussed about error analysis, convergence and error bound. The efficiency of the proposed method is tested through number of examples and compared with previous work in the literature for this we needed small number of SCPFK.

Connection between orthogonal polynomials on the unit circle and bounded interval

In this paper we establish the connection between measures on a bounded interval and on the unit circle by a transformation related with the Szegő classical transformation. We transform a measure on the interval [ - 1 , 1 ] into a measure on [ 0 , 2 π ] in the same form in which the orthogonality measure of the Chebyshev polynomials of fourth kind becomes the Lebesgue measure. We relate the sequences of orthogonal polynomials with respect to both measures and we also relate the coefficients of the three-term recurrence relation with the Schur parameters. When the measures belong to the Szegő class, we study the asymptotic behavior of the orthogonal polynomials on the interval, outside the support of the measure, as well as inside. We also transform the generalized polynomials and we study the orthogonality properties of the new polynomials, obtaining new interesting results, and finally we solve two inverse problems connected with the transformation studied.