A note on the convergence of the Zakharov-Kuznetsov equation by homotopy analysis method

Abstract. In this paper, the convergence of Zakharov-Kuznetsov (ZK) equation by homotopy analysis method (HAM) is investigated. A theorem is proved to guarantee the convergence of HAM and to find the series solution of this equation via a reliable algorithm

Application of Reproducing Kernel Hilbert Space Method for Solving a Class of Nonlinear Integral Equations

A new approach based on the Reproducing Kernel Hilbert Space Method is proposed to approximate the solution of the second-kind nonlinear integral equations. In this case, the Gram-Schmidt process is substituted by another process so that a satisfactory result is obtained. In this method, the solution is expressed in the form of a series. Furthermore, the convergence of the proposed technique is proved. In order to illustrate the effectiveness and efficiency of the method, four sample integral equations arising in electromagnetics are solved via the given algorithm

Dynamical Control of Accuracy Using the Stochastic Arithmetic to Estimate Double and Improper Integrals

The CESTAC (Control et Estimation STochastique des Arrondis de Calculs) method is based on a probabilistic approach of the round-off error propagation which replaces the floating-point arithmetic by the stochastic arithmetic. This is an efficient method to estimate the accuracy of the results. In this paper, we present the reliable schemes using the CESTAC method to estimate the definite double integral I = ({int_a^b}{int_c^d})f(x,y)dydx and the improper integral I = (int_a^infty)f(x)dx , where a, b, c, d ∈ R, by applying the trapezoidal or Simpson\u27s rule. For each kind of integrals, we prove a theorem to show the accuracy of the results. According to these theorems, one can find an optimal value number of the points which we can find the best approximation of I from the computer point of view. Also, we observe that by using the stochastic arithmetic, we are able to validate the results

Inherited Fuzzy Interpolation Based on the Inherited Lower-upper Factorization

In this paper, a new scheme is proposed to find the fuzzy interpolation polynomial. In this case, the nodes are crisp data and the values are fuzzy numbers. In order to obtain the interpolation polynomial, a linear system is solved with crisp coefficients matrix and fuzzy right hand side. Then, the inherited lower-upper (LU) triangular factorization and inherited interpolation are applied to solve this system. The examples illustrate the applicability, simplicity and efficiency of the proposed method

A Novel Approach to Find Optimal Parameter in the Homotopy-Regularization Method for Solving Integral Equations

The regularization method is one of the important schemes to solve the ill-posed problems. In this work, by combining the Wazwaz’s regularization method and the homotopy analysis method, a new and robust approach is presented to solve integral equations which is called the homotopy-regularization method. The solution which is produced by the homotopy-regularization method depends on the regularization parameter. In order to find the optimal value of this parameter, the Controle et Estimation Stochastique des Arrondis de Calculs method is applied which is based on the stochastic arithmetic. A theorem is presented to show the accuracy of the proposed approach. Also, in order to implement the algorithm, the Control of Accuracy and Debugging for Numerical Applications library is applied to perform the Controle et Estimation Stochastique des Arrondis de Calculs method in the stochastic arithmetic automatically. Some examples of the singular and ill-posed integral equations are illustrated. The numerical results show the abilities of the Controle et Estimation Stochastique des Arrondis de Calculs method to find the optimal regularization parameter and the optimal approximation of the homotopy-regularization method

Solving generalized quintic complex Ginzburg–Landau equation by homotopy analysis method

In this paper, the generalized quintic complex Ginzburg–Landau equation is considered to be solved, by means of the homotopy analysis method (HAM). Two examples are solved to illustrate the efficiency of the proposed method. By plotting the h-curve of the examples, the region of convergence is determined. Keywords: Generalized quintic complex Ginzburg–Landau (GCGL) equation, Homotopy analysis method (HAM), Nonlinearity, h-curve, Variational iteration method (VIM

THE USE OF SEMI INHERITED LU FACTORIZATION OF MATRICES IN INTERPOLATION OF DATA

The polynomial interpolation in one dimensional space R is an important method to approximate the functions. The Lagrange and Newton methods are two well known types of interpolations. In this work, we describe the semi inherited interpolation for approximating the values of a function. In this case, the interpolation matrix has the semi inherited LU factorization

A method to obtain the best uniform polynomial approximation for the family of rational function

In this article, by using Chebyshev’s polynomials and Chebyshev’s expansion, we obtain the best uniform polynomial approximation out of P2n to a class of rational functions of the form (ax2+c)-1 on any non symmetric interval [d,e]. Using the obtained approximation, we provide the best uniform polynomial approximation to a class of rational functions of the form (ax2+bx+c)-1 for both cases b2-4ac L 0 and b2-4ac G 0