IRANIAN JOURNAL OF NUMERICAL ANALYSIS AND OPTIMIZATION
Year:2015 | Volume:5 | Issue:2 |Papers Count:7

We describe the construction of second derivative general linear methods(SGLMs) of orders  ve and six. We will aim for methods which are A{stableand have Runge{Kutta stability property. Some numerical results are givento show the e ciency of the constructed methods in solving sti initial valueproblems.

LSMR (Least Squares Minimal Residual) is an iterative method for thesolution of the linear system of equations and least-squares problems. Thispaper presents a block version of the LSMR algorithm for solving linear sys-tems with multiple right-hand sides. The new algorithm is based on the blockbidiagonalization and derived by minimizing the Frobenius norm of the resid-ual matrix of normal equations. In addition, the convergence of the proposedalgorithm is discussed. In practice, it is also observed that the Frobeniusnorm of the residual matrix decreases monotonically. Finally, numerical ex-periments from real applications are employed to verify the e ectiveness ofthe presented method.

In this paper, a practical review of the Adomian decomposition method, to extend the procedure to handle the strongly nonlinear problems under themixed 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, pro-vides an explicit formula to calculate the Adomian polynomials of a nonlinearfunction. The e ciency of the approach will be shown by applying the pro-cedure on several interesting integro-di erential problems. The Mathematicaprograms generating the Adomian polynomials and Adomian solutions basedon the procedures in this paper are designed.

In this paper, an adaptive meshless method of line is applied to distributethe nodes in the spatial domain. In many cases in meshless methods, it isalso 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 theimplementation of an algorithm for selection of the nodes satisfying a givenconstraint, in the meshless method of line. This algorithm is applied to someillustrative examples to show the e ciency of the algorithm and its ability toincrease the accuracy.

In this paper, modi ed simple equation method has been applied to ob-tain generalized solutions of Burgers, Huxley equations and combined formsof these equations. The new exact solutions of these equations have beenobtained. It has been shown that the proposed method provides a very e ec-tive, and powerful mathematical tool for solving nonlinear partial di erentialequations.

In this paper, we investigate the application of the Homotopy Pertur-bation Method (HPM) for solving a one-dimensional nonlinear inverse heatconduction problem. In this problem the thermal conductivity term is a lin-ear function with respect to unknown heat temperature in bounded interval. Furthermore, the temperature histories are unknown at the end point of theinterval. This problem is ill-posed. So, using the  nite di erence scheme anddiscretizing the time interval, the partial di erential equation is reduced intoa System of Nonlinear Ordinary Di erential Equations (SNODE's). Then, using HPM, the approximated solution of the obtained Ordinary Di eren-tial 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 newadaptive nonmonotone trust region method is proposed for solving uncon-strained optimization problems. In our method, based on a slight modi ca-tion 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 isprovided. Our new proposed method is equipped with a new adaptive rulefor updating the radius and an appropriate nonmonotone technique. Undersome suitable and standard assumptions, the local and global convergenceproperties of the new algorithm as well as its convergence rate are investi-gated. Finally, the practical performance of the new proposed algorithm isveri ed on some test problems and compared with some existing algorithmsin the literature.