Iranian Journal of Numerical Analysis and Optimization
Volume:8 Issue: 2, Summer and Autumn 2018

In this paper, we study an extension of Riemann–Liouville fractional derivative for a class of Riemann integrable functions to Lebesgue measurable and integrable functions. Then we used this extension for the approximate solution of a particular fractional partial differential equation (FPDE) problems (linear space-time fractional order diffusion problems). To solve this problem, we reduce it approximately to a discrete optimization problem. Then, by using partition of measurable subsets of the domain of the original problem, we obtain some approximating solutions for it which are represented with acceptable accuracy. Indeed, by obtaining the suboptimal solutions of this optimization problem, we obtain the approximate solutions of the original problem. We show the efficiency of our approach by solving some numerical examples.

In this paper a novel Godunov-type finite volume technique is presented for the solution of one-dimensional Euler equations. The numerical scheme defined herein in is well-balanced and approximates the solution by propa gating a set of jump discontinuities from each Riemann cell interface. The corresponding source terms are then treated within the flux-differencing of the finite volume computational cells. First, the capability of the numerical solver under gravitational source term is examined and the results are val dated with reference solution and higher-order WENO scheme. Then, the well-balanced property of the scheme for the steady-state is tested and finally the proposed method is employed for the modeling small and large amplitude perturbation imposed to the polytropic atmosphere. It is found out that the defined well-balanced solver provides sensible prediction for all of the given test cases.

The minimum sum coloring problem (MSCP) is to find a legal vertex coloring for G using colors represented by natural numbers (1,2, . . .) such that the total sum of the colors assigned to the vertices is minimized. The aim of this paper is to present the skewed variable neighborhood search (SVNS) for this problem based on a new structure of neighborhoods. To increase the speed of the neighborhood search process, we present the new concepts of holder vertex and set. Tested on 23 commonly used benchmark instances, our algorithm shows acceptable competitive performance with respect to recently proposed heuristics.

One of the most critical issues for using data envelopment analysis models is the identification of technological returns to scale (TRTS). Recently, the angles method based on data mining is introduced for the identification of TRTS. This objective method uses the angles to measure the gap between the constant and variable TRTS. The gap is calculated in both the increasing and decreasing sections of the frontier. The larger the gap in the increasing and/or decreasing sections of the frontier, the closer TRTS is to the increasing and/or decreasing form of TRTS. In this paper, we propose a heuristic method for visualizing TRTS that would give a better understanding of identification of TRTS in the dataset. To this end, we introduce the maximum angles method for measuring the maximum possible deviation from constant TRTS assumption in the increasing and decreasing sections of the frontier. By the angles and the maximum angles , we can display the dataset’s TRTS in a two-dimensional space. To validate the proposed method, we consider six one input/one output cases. Also, we apply the angles method and the maximum angles method for the Maskan bank of Iran. Using the proposed method, we show that how TRTS of the bank dataset can be displayed in a two-dimensional space.

In this paper, the fractional–order form of a mouse population model is introduced. Some explicit and implicit schemes, which are Theta methods and nonstandard finite difference (NSFD) schemes, are implemented to give a numerical solution of nonlinear ordinary differential equation system named Hantavirus epidemic model. These methods are compared and discussed that the method preserves the positivity properties of the integer order system.
The numerical solutions are illustrated by means of some graphs. Numerical results of explicit and implicit methods denote that these methods are easy and accurate when applied to fractional–order Hantavirus model

An efficient discrete collocation method for solving Volterra type weakly singular integral equations with logarithmic kernels is investigated. One of features of these equations is that, in general the first erivative of solution behaves like as a logarithmic function, which is not continuous at the origin.
In this paper, to make a compatible approximate solution with the exact ones, we introduce a new collocation approach, which applies the M¨untz logarithmic polynomials(Muntz polynomials with logarithmic terms) as basis functions. Moreover, since implementation of this technique leads to integrals with logarithmic singularities that are often difficult to solve numerically, we apply a suitable quadrature method that allows the exact evaluation of integrals of polynomials with logarithmic weights. To this end, we first remind the well-known Jacobi–Gauss quadrature and then extend it to integrals with logarithmic weights. Convergence analysis of the proposed scheme are presented, and some numerical results are illustrated to demonstrate the efficiency and accuracy of the proposed method

In this paper a hybrid algorithm based on genetic algorithm (GA) and Nelder–Mead (NM) simplex search method is combined with least squares method for the determination of temperature in some nonlinear inverse parabolic problems (NIPP). The performance of hybrid algorithm is established with some examples of NIPP. Results show that hybrid algorithm is better than GA and NM separately. Numerical results are obtained by implementation expressed algorithms on 2.20GHz clock speed CPU.