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.     

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.     

A wavelet approach for the multi-term time fractional diffusion-wave equation

In this paper, a Galerkin method based on the second kind Chebyshev wavelets (SKCWs) is established for solving the multi-term time fractional diffusion-wave equation. To do this, a new operational matrix of fractional integration for the SKCWs must be derived and in order to improve the computational efficiency, the hat functions are proposed to create a general procedure for constructing this matrix. Implementation of these wavelet basis functions and their operational matrix of fractional integration simplifies the problem under consideration to a system of linear algebraic equations, which greatly decreases the computational cost for finding an approximate solution. The main privilege of the proposed method is adjusting the initial and boundary conditions in the final system automatically. Theoretical error and convergence analysis of the SKCWs expansion approve the reliability of the approach. Also, numerical investigation reveals the applicability and accuracy of the presented method.     

A statistical approach for economization of the polynomial functions

Due to having the minimax property, Chebyshev polynomials are used today to economize the arbitrary polynomial functions. In this work, we present a statistical approach to show that, contrary to current thought, the Chebyshev polynomials of the first kind are not appropriate for economizing these polynomials if one uses this statistical approach. In this way, a numerical results section is also given to clearly prove our claim.     

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.     

Modified method for determining an approximate solution of the Fredholm–Volterra integral equations by Taylor’s expansion

A modified method for determining an approximate solution of the Fredholm–Volterra integral equations of the second kind is developed. Via Taylor’s expansion of the unknown function, the integral equation to be solved is approximately transformed into a system of linear equations for the unknown and its derivatives, which can be dealt with in an easy way. The obtained nth-order approximate solution is of high accuracy, and is exact for polynomials of degree n. In particular, an approximate solution with satisfactory accuracy of the weakly singular Volterra integral equation is also given. The efficiency of the method is illustrated by some numerical examples.     

A matrix formulation to the wave equation with non-local boundary condition

Recently, various sequential numerical schemes have been proposed for the solution of non-classical hyperbolic initial value problems which involve non-local integral terms over the spatial domain. In this paper, we focus on the wave equation with the non-local boundary condition. Two matrix formulation techniques based on the shifted standard and shifted Chebyshev bases are proposed for the numerical solution. Several numerical examples and also some comparisons with another methods will be investigated to confirm the efficiency of this procedure.     

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.     

Wavelets Galerkin method for solving stochastic heat equation

In this paper, a new computational method based on the second kind Chebyshev wavelets (SKCWs) together with the Galerkin method is proposed for solving a class of stochastic heat equation. For this purpose, a new stochastic operational matrix for the SKCWs is derived. A collocation method based on block pulse functions is employed to derive a general procedure for forming this matrix. The SKCWs and their operational matrices of integration and stochastic Itô-integration are used to transform the under consideration problem into the corresponding linear system of algebraic equations which can be simply solved to achieve the solution of the problem. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account automatically. Moreover, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.     

A separation of the boundary geometry from the boundary functions in PIES for 3D problems modeled by the Navier–Lamé equation

In this paper, we present a modification of the Somigliana identity for the 3D Navier–Lamé equation in order to analytically include in its mathematical formalism the boundary represented by Coons and Bézier parametric surface patches. As a result, the equations called the parametric integral equation system (PIES) with integrated boundary shape are obtained. The PIES formulation is independent from the boundary shape representation and it is always, for any shape, defined in the parametric domain and not on the physical boundary as in the traditional boundary integral equations (BIE). This feature is also helpful during numerical solving of PIES, as from a formal point of view, a separation between the approximation of the boundary and the boundary functions is obtained. In this paper, the generalized Chebyshev series are used to approximate the boundary functions. Numerical examples demonstrate the effectiveness of the presented strategy for boundary representation and indicate the high accuracy of the obtained results.     

A set of new Chebyshev kernel functions for support vector machine pattern classification

In this study, we introduce a set of new kernel functions derived from the generalized Chebyshev polynomials. The proposed generalized Chebyshev polynomials allow us to derive different kernel functions. By using these polynomial functions, we generalize recently introduced Chebyshev kernel function for vector inputs and, as a result, we obtain a robust set of kernel functions for Support Vector Machine (SVM) classification. Thus in this study, besides clarifying how to apply the Chebyshev kernel functions on vector inputs, we also increase the generalization capability of the previously proposed Chebyshev kernels and show how to derive new kernel functions by using the generalized Chebyshev polynomials. The proposed set of kernel functions provides competitive performance when compared to all other common kernel functions on average for the simulation datasets. The results indicate that they can be used as a good alternative to other common kernel functions for SVM classification in order to obtain better accuracy. Moreover, test results show that the generalized Chebyshev kernel approaches to the minimum support vector number for classification in general.     

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.     

A method for the numerical solution of singular integral equations with logarithmic singularities

A method of numerical solution of singular integral equations of the first kind with logarithmic singularities in their kernels along the integration interval is proposed. This method is based on the reduction of these equations to equivalent singular integral equations with Cauchy-type singularities in their kernels and the application to the latter of the methods of numerical solution, based on the use of an appropriate numerical integration rule for the reduction to a system of linear algebraic equations. The aforementioned method is presented in two forms giving slightly different numerical results. Furthermore, numerical applications of the proposed methods are made. Some further possibilities are finally investigated     

A Chebyshev Finite Difference Method For Solving A Class Of Optimal Control Problems

A new Chebyshev finite difference method for solving class of optimal control problem is proposed. The algorithm is based on Chebyshev approximations of the derivatives arising in system dynamics. In the performance index, we use Chebyshev approximations for integration. The numerical examples illustrate the robustness, accuracy and efficiency of the proposed technique.     

Perturbed Chebyshev rational approximation

In this work, we give a perturbed Chebyshev rational approximation for a function f (x) which has a Chebyshev expansion. This approximation contains a perturbation parameter ~ which is calculated so that the perturbed Chebyshev rational approximation agrees with the Chebyshev expansion to a certain number of terms. Also, we introduce a perturbed Chebyshev rational approximation for the definite integral of a function f (x) having Chebyshev expansion and show that this method can be used iteratively to approximate the multiple integral of the considered function. The method has been applied to approximate some functions and their definite integrals.     

Pseudospectral integration matrix and boundary value problems

This paper presents a new spectral successive integration matrix. This matrix is used to construct a Chebyshev expansion method for the solution of boundary value problems. The method employs the pseudospectral approximation of the highest-order derivative to generate an approximation to the lower-order derivatives. Application to the linear stability problem for plane Poiseuille flow is presented. The present numerical results are in satisfactory agreement with the exact 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 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.     

Second Chebyshev wavelet operational matrix of integration and its application in the calculus of variations

In this paper, we extend the application of the second Chebyshev wavelet (SCW) method to solve variational problems and establish a clear solving procedure for this kind of problems. An operational matrix of integration based on the SCW is presented and a general procedure for forming this matrix is given. The main characteristic of SCW operational matrix method is that it can transform a variational problem to a system of algebraic equations. Thus, it can simplify the variational problems solving. The proposed method is illustrated by some applications, which can show the validity of the present approach.     

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.