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The purpose of this paper is to propose an iterative Legendre technique to deal with a continuous optimal control problem (OCP).

For the system in the considered problem, the control variable is a function of the state variables and their derivatives. State variables in the problem are approximated by Legendre expansions as functions of time t. A constant matrix is given to express the derivatives of state variables. Therefore, control variables can be described as functions of time t. After that, the OCP is converted to an unconstrained optimization problem whose decision variables are the unknown coefficients in the Legendre expansions.

The convergence of the proposed algorithm is proved. Experimental results, which contain the controlled Duffing oscillator problem demonstrate that the proposed technique is faster than existing methods.

Experimental results, which contained the controlled Duffing oscillator problem demonstrate that the proposed technique can be faster while securing exactness.

Fokker–Planck equation appears in various areas in natural science, it is used to describe solute transport and Brownian motion of particles. This paper aims to present an efficient and convenient numerical algorithm for space-time fractional differential equations of the Fokker–Planck type.

The main idea of the presented algorithm is to combine polynomials function approximation and fractional differential operator matrices to reduce the studied complex equations to easily solved algebraic equations.

Based on Taylor basis, simple and useful fractional differential operator matrices of alternative Legendre polynomials can be quickly obtained, by which the studied space-time fractional partial differential equations can be transformed into easily solved algebraic equations. Numerical examples and error date are presented to illustrate the accuracy and efficiency of this technique.

Various numerical methods are proposed in complex steps and are computationally expensive. However, the advantage of this paper is its convenient technique, i.e. using the simple fractional differential operator matrices of polynomials, numerical solutions can be quickly obtained in high precision. Presented numerical examples can also indicate that the technique is feasible for this kind of fractional partial differential equations.

The purpose of the paper is to extend the differential quadrature method (DQM) for solving time and space fractional non-linear partial differential equations on a semi-infinite domain.

The proposed method is the combination of the Legendre polynomials and differential quadrature method. The authors derived and constructed the new operational matrices for the fractional derivatives, which are used for the solutions of non-linear time and space fractional partial differential equations.

The fractional derivative of Lagrange polynomial is a big hurdle in classical DQM. To overcome this problem, the authors represent the Lagrange polynomial in terms of shifted Legendre polynomial. They construct a transformation matrix which transforms the Lagrange polynomial into shifted Legendre polynomial of arbitrary order. Then, they obtain the new weighting coefficients matrices for space fractional derivatives by shifted Legendre polynomials and use these in conversion of a non-linear fractional partial differential equation into a system of fractional ordinary differential equations. Convergence analysis for the proposed method is also discussed.

Many engineers can use the presented method for solving their time and space fractional non-linear partial differential equation models. To the best of the authors’ knowledge, the differential quadrature method has never been extended or implemented for non-linear time and space fractional partial differential equations.

A novel technique based on Bernoulli wavelets has been proposed to solve two-dimensional Fredholm integral equation of second kind. Bernoulli wavelets have been created by dilation and translation of Bernoulli polynomials. This paper aims to introduce properties of Bernoulli wavelets and Bernoulli polynomials.

To solve the two-dimensional Fredholm integral equation of second kind, the proposed method has been used to transform the integral equation into a system of algebraic equations.

Numerical experiments shows that the proposed two-dimensional wavelets technique can give high-accurate solutions and good convergence rate.

The efficiency of newly developed two-dimensional wavelets technique has been validated by different illustrative numerical examples to solve two-dimensional Fredholm integral equations.

This paper aims to address the issue of model discontinuity typically encountered in traditional Denavit-Hartenberg (DH) models. To achieve this, we propose the use of a local Product of Exponentials (POE) approach. Additionally, a modified calibration model is presented which takes into account both kinematic errors and high-order joint-dependent kinematic errors. Both kinematic errors and high-order joint-dependent kinematic errors are analyzed to modify the model.

Robot positioning accuracy is critically important in high-speed and heavy-load manufacturing applications. One essential problem encountered in calibration of series robot is that the traditional methods only consider fitting kinematic errors, while ignoring joint-dependent kinematic errors.

Laguerre polynomials are chosen to fitting kinematic errors and high-order joint-dependent kinematic errors which can avoid the Runge phenomenon of curve fitting to a great extent. Levenberg–Marquard algorithm, which is insensitive to overparameterization and can effectively deal with redundant parameters, is used to quickly calibrate the modified model. Experiments on an EFFORT ER50 robot are implemented to validate the efficiency of the proposed method; compared with the Chebyshev polynomial calibration methods, the positioning accuracy is improved from 0.2301 to 0.2224 mm.

The results demonstrate the substantial improvement in the absolute positioning accuracy achieved by the proposed calibration methods on an industrial serial robot.

Adomian's method is completed to obtain the analytic solution of any partial differential equation with boundary conditions defined on the four sides of a rectangle. Adomian's decomposition method is first used to obtain the N‐order approximant to the one direction partial solution that satisfies the boundary conditions on that direction. Then the functions obtained are variationally modified on the four sides to make them compatible with a given experimental error. The product of these transformations is an analytic approximation to the solution which is compatible with both the weak norm imposed on the boundaries and the accuracy imposed in the whole domain.

This paper aims to propose a new numerical method for the solution of the Blasius and magnetohydrodynamic (MHD) Falkner-Skan boundary-layer equations. The Blasius and MHD Falkner-Skan equations are third-order nonlinear boundary value problems on the semi-infinite domain.

The approach is based upon modified rational Bernoulli functions. The operational matrices of derivative and product of modified rational Bernoulli functions are presented. These matrices together with the collocation method are then utilized to reduce the solution of the Blasius and MHD Falkner-Skan boundary-layer equations to the solution of a system of algebraic equations.

The method is computationally very attractive and gives very accurate results.

Many problems in science and engineering are set in unbounded domains. One approach to solve these problems is based on rational functions. In this work, a new rational function is used to find solutions of the Blasius and MHD Falkner-Skan boundary-layer equations.

This paper aims to introduce a novel algorithm to solve initial/boundary value problems of high-order ordinary differential equations (ODEs) and high-order system of ordinary differential equations (SODEs).

The proposed method is based on Hermite polynomials and extreme learning machine (ELM) algorithm. The Hermite polynomials are chosen as basis function of hidden neurons. The approximate solution and its derivatives are expressed by utilizing Hermite network. The model function is designed to automatically meet the initial or boundary conditions. The network parameters are obtained by solving a system of linear equations using the ELM algorithm.

To demonstrate the effectiveness of the proposed method, a variety of differential equations are selected and their numerical solutions are obtained by utilizing the Hermite extreme learning machine (H-ELM) algorithm. Experiments on the common and random data sets indicate that the H-ELM model achieves much higher accuracy, lower complexity but stronger generalization ability than existed methods. The proposed H-ELM algorithm could be a good tool to solve higher order linear ODEs and higher order linear SODEs.

The H-ELM algorithm is developed for solving higher order linear ODEs and higher order linear SODEs; this method has higher numerical accuracy and stronger superiority compared with other existing methods.

This paper is motivated by the need for a generic approach to evaluate the volumetric accuracy of rapid prototyping (RP) machines. The approach presented in this paper is inspired in large part by the techniques developed over the years for the parametric evaluation of coordinate measuring machine (CMM) errors. In CMM metrology, the parametric error functions for the machine are determined by actual measurement of a master reference artifact with known characteristics. In our approach, the RP machine is used to produce a generic artifact, which is then measured by a master CMM, and measurement results are used to infer the RP machine's parametric error functions. The results presented demonstrate the feasibility of such an approach on a two‐dimensional model.