This article presents a method based on combination of successive linearization method (SLM) and pseudo-spectral collocation method and then is applied on a nonlinear model of coupled diffusion and chemical reaction in a spherical catalyst pellet. It is obtained that this method can be used for nonlinear boundary value problems without difficulty because the nonlinear part of the equation becomes inactive by SLM and more, to treat the linear equation, even in the case of complicatedness, is straightforward by pseudo-spectral collocation method. Also, the results reveal the high efficiency with reliable accuracy of this hybrid method.

In this paper, a third order convergent method for finding the Moore-Penrose inverse of a matrix is presented and analysed. Then, we develop the method to find Drazin inversion. This method is very robust to find the Moore-Penrose and Drazin inverse of a matrix. Finally, numerical examples show that the efficiency of the proposed method is superior over other proposed methods.

In the present paper, at first, we propose a new two-step iterative method for solving nonlinear equations. This scheme is based on the Steffensen's method, in which the order of convergence is four. This iterative method requires only three functions evaluation in each iteration, therefore it is optimal in the sense of the Kung and Traub conjecture. Then we extend it to the method with memory, which the order of convergence is six. Finally, numerical examples indicate that theobtained methods in terms of accuracy and computational cost are superior to thefamous forth-order methods.

Singular integral equations (SIEs) are often encountered in certain contact and fracture problems in solid mechanics. In this paper, we apply the reproducing kernel method (RKM) to give the approximate solution of Abel's second-kind singular integral equations. For solving this problem, difficulties lie in its singular term. In order to remove the singular term of the equation, an equivalent transformation is made. Solution representations are obtained in reproducing kernel Hilbert space. Numerical experiments show that our reproducing kernel method is efficient. To show the high accuracy of the method the results are compared to other numerical methods and satisfactory agreements are achieved.

In this paper, an applicable numerical approximation has been proposed for solving nonlinear two-dimensional integral equations (2DIEs) of the second kind on non-rectangular domains. Because directly applying the collocation methods on non-rectangular domains is difficult, in this work, at first, the integral equation is converted to an equal integral equation on a rectangular domain, then the solution is approximated by applying 2D Jacobi collocation method, the implementation of these instructions reduces the integral equation to a system of nonlinear algebraic equations, therefore, solving this system has an important role to approximate the solution. In this paper, Newton-Krylov generalized minimal residual (NK-GMRes) algorithm is used for solving the system of nonlinear algebraic equations. Furthermore, an error estimate for the presented method is investigated and several examples confirm the accuracy and efficiency of the proposed instructions.

In this paper, we use radial basis function collocation method for solving the system of differential equations in the area of biology. One of the challenges in RBF method is picking out an optimal value for shape parameter in Radial basis function to achieve the best result of the method because there are not any available analytical approaches for obtaining optimal shape parameter. For this reason, we design a genetic algorithm to detect a close optimal shape parameter. The population convergence figures, the residuals of the equations and the examination of the ASN2R and ARE measures all show the accurate selection of the shape parameter by the proposed genetic algorithm. Then, the experimental results show that this strategy is efficient in the systems of differential models in biology such as HIV and Influenza. Furthermore, we show that using our pseudo-combination formula for crossover in genetic strategy leads to convergence in the nearly best selection of shape parameter.

This manuscript presents a new approximation method for fractional-order Fokker-Planck equations based on Touchard polynomial approximation. We provide new Caputo and extra Caputo pseudo-operational matrices for these polynomials. Then, utilizing mentioned pseudo-operational matrices and an optimal method, the considered equation leads to a system of algebraic equations which can be solved by mathematical software. Finally, we illustrate the advantages of the suggested technique through several numerical examples.