Iranian Journal of Numerical Analysis and Optimization
Volume:5 Issue: 2, Summer and Autumn 2015

We describe the construction of second derivative general linear methods (SGLMs) of orders five and six. We will aim for methods which are A--stable and have Runge--Kutta stability property. Some numerical results are given to show the efficiency of the constructed methods in solving stiff initial value problems.

LSMR (Least Squares Minimal Residual) is an iterative method for the solution of the linear system of equations and leastsquares problems. This paper presents a block version of the LSMR algorithm for solving linear systems with multiple right-hand sides. The new algorithm is based on the block bidiagonalization and derived by minimizing the Frobenius norm of the resid ual matrix of normal equations. In addition, the convergence of the proposed algorithm is discussed. In practice, it is also observed that the Frobenius norm of the residual matrix decreases monotonically. Finally, numerical experiments from real applications are employed to verify the effectiveness of the presented method.

In this paper, a practical review of the Adomian decomposition method, to extend the procedure to handle the strongly nonlinear problems under the mixed conditions, is given and the convergence of the algorithm is proved. For this respect, a new and simple way to generate the Adomian polynomials, for a general nonlinear function, is proposed. The proposed procedure, provides an explicit formula to calculate the Adomian polynomials of a nonlinear function. The efficiency of the approach will be shown by applying the procedure on several interesting integro-differential problems. The Mathematica programs generating the Adomian polynomials and Adomian solutions based on the procedures in this paper are designed.

In this paper, an adaptive meshless method of line is applied to distribute the nodes in the spatial domain. In many cases in meshless methods, it is also necessary for the chosen nodes to have certain smoothness properties. The set of nodes is also required to satisfy certain constraints. In this paper, one of these constraints is investigated. The aim of this manuscript is the implementation of an algorithm for selection of the nodes satisfying a given constraint, in the meshless method of line. This algorithm is applied to some illustrative examples to show the efficiency of the algorithm and its ability to increase the accuracy.

In this paper, modified simple equation method has been applied to ob-tain generalized solutions of Burgers, Huxley equations and combined forms of these equations. The new exact solutions of these equations have been obtained. It has been shown that the proposed method provides a very effective, and powerful mathematical tool for solving nonlinear partial differential equations.

In this paper, we investigate the application of the Homotopy Perturbation Method (HPM) for solving a one-dimensional nonlinear inverse heat conduction problem. In this problem the thermal conductivity term is a linear function with respect to unknown heat temperature in bounded interval. Furthermore, the temperature histories are unknown at the end point of the interval. This problem is ill-posed. So, using the finite difference scheme and discretizing the time interval, the partial differential equation is reduced into a System of Nonlinear Ordinary Differential Equations (SNODE''s). Then, using HPM, the approximated solution of the obtained Ordinary Differential Equation (ODE) system is determined. In the sequel, the stability andconvergence conditions of the proposed method are investigated. Finally, anupper bound of the error is provided.

Using a simple quadratic model in the trust region subproblem, a new adaptive nonmonotone trust region method is proposed for solving unconstrained optimization problems. In our method, based on a slight modification of the proposed approach in (J. Optim. Theory Appl. 158(2):626-635, 2013), a new scalar approximation of the Hessian at the current point is provided. Our new proposed method is equipped with a new adaptive rule for updating the radius and an appropriate nonmonotone technique. Under some suitable and standard assumptions, the local and global convergence properties of the new algorithm as well as its convergence rate are investigated. Finally, the practical performance of the new proposed algorithm is verified on some test problems and compared with some existing algorithms in the literature.