Numerical solution of Riccati equation using the cubic B-spline scaling functions and Chebyshev cardinal functions

Two numerical techniques are presented for solving the solution of Riccati differential equation. These methods use the cubic B-spline scaling functions and Chebyshev cardinal functions. The methods consist of expanding the required approximate solution as the elements of cubic B-spline scaling function or Chebyshev cardinal functions. Using the operational matrix of derivative, we reduce the problem to a set of algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the new techniques. The methods are easy to implement and produce very accurate results.     

An efficient method based on the second kind Chebyshev wavelets for solving variable-order fractional convection diffusion equations

In this paper, a class of variable-order fractional convection diffusion equations have been solved with assistance of the second kind Chebyshev wavelets operational matrix. The operational matrix of variable-order fractional derivative is derived for the second kind Chebyshev wavelets. By implementing the second kind Chebyshev wavelets functions and also the associated operational matrix, the considered equations will be reduced to the corresponding Sylvester equation, which can be solved by some appropriate iterative solvers. Also, the convergence analysis of the proposed numerical method to the exact solutions and error estimation are given. A variety of numerical examples are considered to show the efficiency and accuracy of the presented technique.     

Chebyshev expansions for wave functions

Chebyshev series expansion of solutions of linear differential equations which occur in atomic scattering problems is discussed. We apply this technique to obtain both the regular and the irregular radial Coulomb wave functions. The Chebyshev expansion technique is extended to evaluate linearly independent solutions for the modified Coulomb potential. It is further shown that relativistic Coulomb wave functions may also be evaluated using Chebyshev expansion techniques.An advantage of this technique is that wave functions and their derivatives can be represented to a very high accuracy in terms of only a small number of Chebyshev expansion coefficients over a wide range of values of the independent variable. Moreover, in certain cases it is possible to evaluate matrix elements involving functions so represented by using properties of Chebyshev polynomials and thus avoiding numerical integration altogether.     

Predictive control for a class of distributed delay systems using Chebyshev polynomials

In this paper, the model predictive control (MPC) is developed for linear time-varying systems with distributed time delay in state. The Chebyshev operational matrices of product, integration and delay are utilized to transform the solution of distributed delay differential equation to the solution of algebraic equations. The Chebyshev functions are also applied to derive approximate solution of finite horizon optimal control problem involved in MPC. The proposed method is simple and computationally advantageous. Illustrative example demonstrates the validity and applicability of the technique.     

Polynomial solutions of certain differential equations

In this paper, the Chebyshev matrix method is applied generalisations of the Hermite, Laguerre, Legendre and Chebyshev differential equations which have polynomial solution. The method is based on taking the truncated Chebyshev series expansions of the functions in equation, and then substituting their matrix forms into the result equation. Thereby the given equation reduces to a matrix equation, which corresponds to a system of linear algebraic equations with unknown Chebyshev coefficients.     

Chebyshev polynomial solution of the system of Cauchy-type singular integral equations of the first kind

This paper presents a Chebyshev series method for the numerical solutions of system of the first kind Cauchy type singular integral equation (SIE). The Chebyshev polynomials of the second kind with the corresponding weight function have been used to approximate the density functions. It is shown that the numerical solution of system of characteristic SIEs is identical to the exact solution when the force functions are cubic functions.     

Collocation method to solve inequality-constrained optimal control problems of arbitrary order

In this paper, the generalized fractional order of the Chebyshev functions (GFCFs) based on the classical Chebyshev polynomials of the first kind is used to obtain the solution of optimal control problems governed by inequality constraints. For this purpose positive slack functions are added to inequality conditions and then the operational matrix for the fractional derivative in the Caputo sense, reduces the problems to those of solving a system of algebraic equations. It is shown that the solutions converge as the number of approximating terms increases, and the solutions approach to classical solutions as the order of the fractional derivatives approach one. The applicability and validity of the method are shown by numerical results of some examples, moreover a comparison with the existing results shows the preference of this method.

     

A note on the numerical solution of non-linear parabolic equations in chebyshev series

A new method for the numerical solution of non linear parabolic equations is presented. The method is an extension of an existing algorithm for linear equations. Solutions are obtained in the form of a Chebyshev series, which is produced by approximating the partial differential equation by a set of ordinary differential equations over a small time interval. The method appears to be both accurate and economical.     

Nonlinear Constrained Optimal Control Problems and Cardinal Hermite Interpolant Multiscaling Functions

In this paper, a numerical method for solving nonlinear quadratic optimal control problems with inequality constraints is presented. The method is based upon cardinal Hermite interpolant multiscaling function approximation. The properties of these multiscaling functions are presented first. These properties are then utilized to reduce the solution of the nonlinear constrained optimal control to a nonlinear programming one, to which existing algorithms may be applied. Illustrative examples are included to demonstrate the efficiency and applicability of the technique.     

The Chebyshev multidomain approach to stiff problems in fluid mechanics

The Chebyshev multidomain technique for calculating solutions with steep gradients is first discussed regarding accuracy and stability in case of simple model equations. Then the method is described for the solution of the Navier-Stokes equations and applications to double-diffusive convection exhibiting thin inner layers are presented.     

A new fractional Chebyshev FDM: an application for solving the fractional differential equations generated by optimisation problem

In this paper, we introduce a new numerical technique which we call fractional Chebyshev finite difference method. The algorithm is based on a combination of the useful properties of Chebyshev polynomial approximation and finite difference method. We implement this technique to solve numerically the non-linear programming problem which are governed by fractional differential equations (FDEs). The proposed technique is based on using matrix operator expressions which applies to the differential terms. The operational matrix method is derived in our approach in order to approximate the Caputo fractional derivatives. This operational matrix method can be regarded as a non-uniform finite difference scheme. The error bound for the fractional derivatives is introduced. The application of the method to the generated FDEs leads to algebraic systems which can be solved by an appropriate method. Two numerical examples are provided to confirm the accuracy and the effectiveness of the proposed method. A comparison with the fourth-order Runge–Kutta method is given.     

A modified Chebyshev pseudospectral DD algorithm for the GBH equation

In this paper, a Chebyshev spectral collocation domain decomposition (DD) semi-discretization by using a grid mapping, derived by Kosloff and Tal-Ezer in space is applied to the numerical solution of the generalized Burger’s-Huxley (GBH) equation. To reduce roundoff error in computing derivatives we use the above mentioned grid mapping. In this work, we compose the Chebyshev spectral collocation domain decomposition and Kosloff and Tal-Ezer grid mapping, elaborately. Firstly, the theory of application of the Chebyshev spectral collocation method with grid mapping and DD on the GBH equation is presented. This method yields a system of ordinary differential algebraic equations (DAEs). Secondly, we use a fourth order Runge-Kutta formula for the numerical integration of the system of DAEs. Application of this modified method to the GBH equation show that this method (M-DD) is faster and more accurate than the standard Chebyshev spectral collocation DD (S-DD) method.     

Pseudospectral method of solution of the Fitzhugh–Nagumo equation

We present a study of the convergence of different numerical schemes in the solution of the Fitzhugh–Nagumo equations in the form of two coupled reaction diffusion equations for activator and inhibitor variables. The diffusion coefficient for the inhibitor is taken to be zero. The Fitzhugh–Nagumo equations, have spatial and temporal dynamics in two different scales and the solutions exhibit shock-like waves. The numerical schemes employed are a Chebyshev multidomain method, a finite difference method and the method developed by Barkley [D. Barkley, A model for fast computer simulation of excitable media, Physica D, 49 (1991) 61–70]. We consider two different models for the local dynamics. We present results for plane wave propagation in one dimension and spiral waves for two dimensions. We use an operator splitting method with the Chebyshev multidomain approach in order to reduce the computational time. Zero flux boundary conditions are imposed on the solutions.     

Integration of Dynamic Equation of Spring-Mass-Damper Systems via Chebyshev Polynomials

A numerical scheme based on Chebyshev polynomials for the determination of the response of spring-mass-damper systems is presented. The state vector of the differential equation of the spring-mass-damper system is expanded in terms of Chebyshev polynomials. This expansion reduces the original differential equations to a set of linear algebraic equations where the unknowns are the coefficient of Chebyshev polynomials. A formal procedure to generate the coefficient matrix and the right-hand side vector of this system of algebraic equations is discussed. The numerical efficiency of the proposed method is compared with that of Runge-Kutta method. It is shown that this scheme is accurate and is computationally efficient.     

A Chebyshev technique for solving nonlinear optimal controlproblems

A numerical technique for solving nonlinear optimal control problems is introduced. The state and control variables are expanded in the Chebyshev series, and an algorithm is provided for approximating the system dynamics, boundary conditions, and performance index. Application of this method results in the transformation of differential and integral expressions into systems of algebraic or transcendental expressions in the Chebyshev coefficients. The optimum condition is obtained by applying the method of constrained extremum. For linear-quadratic optimal control problems, the state and control variables are determined by solving a set of linear equations in the Chebyshev coefficients. Applicability is illustrated with the minimum-time and maximum-radius orbit transfer problems     

An Optimized Iterative Method for Numerical Solution of Large Systems of Equations Based on the Extremal Property of Zeroes of Chebyshev Polynomials

An iterative method suitable for numerical solution of large systems of equations is presented. An extremal property of the Chebyshev polynomials is established, providing a logical foundation for the proposed procedure. A modification of the method is applicable for evaluation of the maximal eigenvalue of a matrix with real eigenvalues and of the associated eigenvector.     

Analysis and identification of linear distributed systems via Chebyshev series

A double Chebyshev series is introduced to approximate functions of two independent variables and then applied to analyse and identify linear distributed systems. The solution for the coefficient matrices can be obtained directly from a Kronecker product formula. In addition, the algorithm for formulating the algebraic equations to estimate unknown parameters is derived, with the method used being algebraic and computer-oriented. Two illustrative examples are given and excellent results are obtained owing to the rapid convergence property of the Chebyshev series     

An incomplete factorization iterative technique for elliptic equations

This paper describes efficient iterative techniques for solving the large sparse symmetric linear systems that arise from application of finite difference approximations to self-adjoint elliptic equations. We use an incomplete factorization technique with the method of D'Yakonov type, generalized conjugate gradient and Chebyshev semi-iterative methods. We compare these methods with numerical examples. Bounds for the 4-norm of the error vector of the Chebyshev semi-iterative method in terms of the spectral radius of the iteration matrix are derived.     

Chebyshev series approximation for solving differential–algebraic equations (DAEs)

The numerical solution of differential–algebraic equations (DAEs) using the Chebyshev series approximation is considered in this article. Two different problems are solved using the Chebyshev series approximation and the solutions are compared with the exact solutions. First, we calculate the power series of a given equation system and then transform it into Chebyshev series form, which gives an arbitrary order for solving the DAE numerically.