Finite Chebyshev Series Research Articles

A Numerical Application of the Chebyshev Operational Matrix Method for HIV Infection of CD4+T-cells

In this research study, we aim to approximate a solution for the mathematical model of the Human Immunodeficiency Virus (HIV) infection of CD4+T-cells. An operational matrix method based on Chebyshev orthogonal polynomials has been adapted to obtain numerical solutions for the model of HIV infection of CD4+T-cells. The proposed numerical scheme is built on a system of a nonlinear algebraic equation, including coefficients of a finite Chebyshev series that are represented the approximate solutions of the model. Results are compared to existing methods to verify the accuracy of the numerical scheme.

Chebyshev polynomial solutions of systems of linear integral equations

A Chebyshev collocation method has been presented to solve systems of linear integral equations in terms of Chebyshev polynomials. This method transforms the integral system into the matrix equation with the help of Chebyshev collocation points and the unknown of this equation is a Chebyshev coefficient matrix. They are easily acquired by using the last matrix equation, which corresponds to the system of linear algebraic equations. As a result, the finite Chebyshev series approach is obtained.

Algorithms for the integration and derivation of Chebyshev series

General formulas for the mth integral and derivative of a Chebyshev polynomial of the first or second kind are presented. The result is expressed as a finite series of the same kind of Chebyshev polynomials. These formulas permit to accelerate the determination of such integrals or derivatives. Besides, it is presented formulas for the mth integral and derivative of finite Chebyshev series and a numerical algorithm for the direct evaluation of the mth derivative of such a series.

Chebyshev polynomial solutions of systems of high-order linear differential equations with variable coefficients

A Chebyshev collocation method has been presented for numerically solving systems of high-order linear ordinary differential equations with variable coefficients. Using the Chebyshev collocation points, this method transforms the ODE system and the given conditions to matrix equations with unknown Chebyshev coefficients. By means of the obtained matrix equations, a new system of equations which corresponds to the system of linear algebraic equations is gained. Hence, by finding the Chebyshev coefficients easily, the finite Chebyshev series approach is obtained.

Spectral method for constrained linear–quadratic optimal control

A computational method based on Chebyshev spectral method is presented to solve the linear–quadratic optimal control problem subject to terminal state equality constraints and state-control inequality constraints. The method approximates each of the system state variables and each of the control variables by a finite Chebyshev series of unknown parameters. The method converts the optimal control problem into a quadratic programming problem which can be solved more easily than the original problem. This paper gives explicit results that simplify the implementation of the method. To show the numerical behavior of the proposed method, the simulation results of an example are presented.

Solution of the nerve cable equation using Chebyshev approximations

The propagation of excitation along the dendrites and the axon of a neurone is described by a partial differential equation which is nonlinear when voltage-gated conductances are present. In this case, numerical methods are employed to obtain a solution: the evolution of the membrane potential in space and time. Even when the membrane is passive (linear), numerical methods might still be preferred to analytical ones that are often too cumbersome to obtain. In this paper, we present the Chebyshev pseudospectral or collocation method as an alternative to the hitherto commonly used finite difference schemes (compartmental models) that are based on sufficiently fine equidistant subdivisions of the spatial structure (dendrites or axon). In the Chebyshev method, solutions are approximated by finite Chebyshev series. The solutions have uniform, usually high, numerical accuracy at any spatial point, not only at the original collocation points. Often, truncation errors become negligible, hence, the total error is essentially the rounding error of the computations. Furthermore, quantities involving spatial derivatives, and in particular the axial current, can be computed exactly from the solution, i.e. the membrane potential. Space-dependent parameter distributions (channel densities, non-uniform dendritic geometries), as well as mixed linear boundary conditions can easily be implemented, and can be chosen from the large class of piecewise smooth functions.

Numerical integration of functions with poles near the interval of integration

An automatic quadrature scheme is presented for approximating integrals of functions that are analytic in the interval of integration but contain pole (or poles) of order 2, i.e., a double pole on the real axis or a complex conjugate pair of double poles, near the interval of integration. The present scheme is based on product integration rules of interpolatory type, using function values of the abscissae only in the interval of integration. The integral is approximated and evaluated by using recurrence relations and some extrapolation method after the smooth part of the integrand is expanded in terms of the Chebyshev polynomials. The fast Fourier transform (FFT) technique is used to generate efficiently the sequence of the finite Chebyshev series expansions until an approximation of the integral satisfying the required tolerance is obtained with an adequate estimate of the error. Numerical examples are included to illustrate the performance of the method.

Compressed planetary and lunar ephemerides

A package of FORTRAN software has been developed which provides planetary and lunar positions, with respect to the solar system barycenter, for all times in the interval 18012049; positions agree to 1 milliarcsecond with those generated by Jet Propulsion Laboratory Development Ephemeris 200 (DE200). The system consists of approximately 800 kilobytes of ephemeris files and 40 kilobytes of programs, totalling 5% of the storage required by DE200. After removal of reference orbits, segments of DE200 positions were fitted by finite Chebyshev series of degree 40. The Chebyshev coefficients were rounded to integer multiples of a suitable unit and packed to form the ephemeris files.

Differential eigenvalue problems in which the parameter appears nonlinearly

Several methods are examined for determining the eigenvalues of a system of equations in which the parameter appears nonlinearly. The equations are the result of the discretization of differential eigenvalue problems using a finite Chebyshev series. Two global methods are considered which determine the spectrum of eigenvalues without an initial estimate. A local iteration scheme with cubic convergence is presented. Calculations are performed for a model second order differential problem and the Orr-Sommerfeld problem for plane Poiseuille flow.

Approximate optimal control of distributed parameter systems.

An approximation technique for determining the optimal control of distributed parameter systems is developed. Treated explicitly, to illustrate the basic algorithm, are systems with a single state variable and control variables which appear linearly. The system partial differential equation may be nonlinear. Extension can be made to systems with more than one state variable and/or nonlinear control variables. The basis of the approach involves the intro- duction of a finite Chebyshev series approximation for the state variable. State and control variable constraints are treated using penalty functions. Numerical results are obtained for a representative example involving the optimal heating of a metal slab. The computational results obtained compare favorably with results presented by other authors. vI approximation technique for determining the optimal control of distributed parameter systems is given in this paper. The technique represents an extension of one used by Johnson1 for a finite-thrust trajectory problem involving the solution of ordinary differential equations. The basis of the approach developed here involves the introduction of a finite Chebyshev series approximation for the state variable. The coefficients of the approximation are determined by insisting that the state variable approximation be equal to specified values of the true state variable at the finite set of nonuniformly distributed Chebyshev points. The values of the state variable at the Chebyshev points become the free parameters of the problem. Since a closed form analytic expression for the state variable is available it is possible to eliminate all numerical differentiation and integration. Consequently a problem with a functional performance index is transformed into one involving minimization of an ordinary function of many variables. Optimization of the performance index takes place when the values of the state variable are manipulated using one of the well known mathematical programming algorithms. Thus for the sake of computation, the roles of the state and control variable are reversed, from that prevalent in most common optimization techniques. To illustrate the basic algorithm to be developed, a one- dimensional heat conduction problem is considered. 2-3 It should be noted that this problem was chosen for illustrative purposes only and does not represent a limitation on the type of distributed parameter system for which the technique is applicable.

Chebyshev Methods for Ordinary Differential Equations

The solution of linear ordinary differential systems, with polynomial coefficients, can be approximated by a finite polynomial or a finite Chebyshev series. The computation can be performed so that the solution satisfies exactly a perturbed differential system, the perturbations being computed multiples of one or more Chebyshev polynomials. An upper bound to the errors in the solution can often be estimated by approximate solution of the differential system satisfied by the error. Various devices are used to make the errors smaller, including a priori integration of the given differential various aspects of these topics, for both initial-value and boundary-value problems, and also suggests a method for automatic computation.