Iranian Journal of Numerical Analysis and Optimization
Volume:9 Issue: 1, Winter and Spring 2019

A class of optimal shape design problems is studied in this paper. The boundary shape of a domain is determined such that the solution of the underlying partial differential equation matches, as well as possible, a given desired state. In the original optimal shape design problem, the variable domain is parameterized by a class of functions in such a way that the optimal design problem is changed to an optimal control problem on a fixed domain. Then, the resulting distributed control problem is embedded in a measure theoretical form, in fact, an infinite-dimensional linear programming problem. The optimal measure representing the optimal shape is approximated by a solution of a finite-dimensional linear programming problem. The method is evaluated via a numerical example.

In this paper, an explicit exact finite-difference scheme for the Huxley equation is presented based on the nonstandard finite-difference (NSFD) scheme. Afterwards, an NSFD scheme is proposed for the numerical solution of the Huxley equation. The positivity and boundedness of the scheme is discussed. It is shown through analysis that the proposed scheme is consistent, stable, and convergence. The numerical results obtained by the NSFD scheme is compared with the exact solution and some available methods, to verify the accuracy and efficiency of the NSFD scheme.

The aim of the current paper is to construct the shifted fractional-order Jacobi functions (SFJFs) based on the Jacobi polynomials to numerically solve the fractional-order pantograph differential equations. To achieve this purpose, first the operational matrices of integration, product, and pantograph, related to the fractional-order basis, are derived (operational matrix of integration is derived in Riemann–Liouville fractional sense). Then, these matrices are utilized to reduce the main problem to a set of algebraic equations. Finally, the reliability and efficiency of the proposed scheme are demonstrated by some numerical experiments. Also, some theorems are presented on existence of solution of the problem under study and convergence of our method.

In this paper, the 1-median location problem on an undirected network with discrete random demand weights and traveling times is investigated. The objective function is to maximize the probability that the expected sum of weighted distances from the existing nodes to the selected median does not exceed a prespecified given threshold. An analytical algorithm is proposed to get the optimal solution in small-sized networks. Then, by using the centrallimit theorem, the problem is studied in large-sized networks and reduced to a nonlinear problem. The numerical examples are given to illustrate the efficiency of the proposed methods.

Let T = (V, E) be a tree with | V |= n. A 2-(k, l)-core of T is two subtrees with at most k leaves and with a diameter of at most l, which the sum of the distances from all vertices to these subtrees is minimized. In this paper, we first investigate the problem of finding 2-(k, l)-core on an unweighted tree and show that there exists a solution that none of (k, l)-cores is a vertex. Also in the case that the sum of the weights of vertices is negative, we show that one of (k, l)-cores is a single vertex. Then an algorithm for finding the 2-(k, l)-core of a tree with the pos/neg weight is presented.

The class of strong stochastic Runge–Kutta (SRK) methods for stochas tic differential equations with a commutative noise proposed by R¨ oßler (2010) is considered. Motivated by Komori and Burrage (2013), we design a class of explicit stochastic orthogonal Runge–Kutta Chebyshev (SROCKC2) meth ods of strong order one for the approximation of the solution of Itˆo SDEs with an m-dimensional commutative noise.The mean-square and asymptotic stability analysis of the newly proposed methods applied to a scalar linear test equation with a multiplicative noise is presented. Finally, some numer ical experiments for stochastic models arising in applications are given that confirm the theoretical discussion.

‎‎ It is important to note that mixed systems of first and second-kind Volterra–Fredholm integral equations are ill-posed problems, so that solving discretized system of such problems has a lot of difficulties. We will apply the regularization method to convert this mixed system (ill-posed problem) to system of the second kind Volterra–Fredholm integral equations (well-posed problem). A numerical method based on Chebyshev wavelets is suggested for solving the obtained well-posed problem, and convergence analysis of the method is discussed. For showing efficiency of the method, some test problems, for which the exact solution is known, are considered.