Iranian Journal of Numerical Analysis and Optimization
Volume:10 Issue: 2, Summer and Autumn 2020

We present a numerical method for solving linear and nonlinear fractional partial differential equations (FPDEs) with variable coefficients. The main aim of the proposed method is to introduce an orthogonal basis of twodimensional fractional Muntz–Legendre polynomials. By using these polynomials, we approximate the unknown functions. Furthermore, an operational matrix of fractional derivative in the Caputo sense is provided for computing the fractional derivatives. The proposed approximation together with the Tau method reduces the solution of the FPDEs to the solution of a system of algebraic equations. Finally, to show the validity and accuracy of the presented method, we give some numerical examples.

We apply a primal-dual simplex algorithm for solving the biobjective min imum cost-time network flow problem such that the total shipping cost and the total shipping fixed time are considered as the first and second objective functions, respectively. To convert the proposed model into a single-objective parametric one, the weighted sum scalarization technique is commonly used. This problem is a mixed-integer programming, which the decision variables are directly dependent together. Generally, the previous works have consid ered the linear biobjective problem with the traditional network flow con straints, while in this paper, corresponding to each flow variable, a binary variable is defined. These zero-one variables are utilized to describe a fixed shipping time for positive flows. The proposed method is successful in finding all supported efficient solutions of a real numerical example.

A nonoverlapping domain decomposition technique applied to a finite difference method is presented for the numerical solution of the forward backward heat equation in the case of one-dimension. While the previous at tempts in dealing with this problem have been based on an iterative domain decomposition scheme, the current work avoids iterations. Also a physical matching condition is suggested to avoid difficulties caused by the interface boundary nodes. Furthermore, we obtain a square system of equations. In addition, the convergence and stability of the proposed method are investi gated. Some numerical experiments are given to show the effectiveness of the proposed method.

We study two numerical techniques based on the homotopy perturba tion transform method (HPTM) and the fractional Adams–Bashforth method (FABM) for solving a class of nonlinear time-fractional differential equations involving the Caputo–Prabhakar fractional derivatives. In this manuscript, the convergence for numerical solutions obtained using HPTM and the con vergence and stability for numerical solutions obtained using FABM are inves tigated. We compare the solutions obtained by the HPTM and the FABM for some nonlinear time-fractional differential equations. Moreover, some numer ical examples are demonstrated in order to show the validity and reliability of the suggested methods.

In this paper, based on a discrete total variation model, a modified discretization of total variation (TV) is introduced for image processing problems. Two optimization problems corresponding to compressed sensing magnetic resonance imaging (MRI) data reconstruction problem and image denoising are proposed. In the proposed method, instead of applying isotropic TV whose gradient field is a two directions vector, a four directions discretization with some modification is applied for the inverse problems. A dual formulation for the proposed TV is explained and an efficient primal dual algorithm is employed to solve the problem. Some important image test problems in MRI and image denoising problems are considered in the numerical experiments. We compare our model with the state of the art methods.

We propose a maximum probability model to estimate the origin-destination trip matrix in the networks, where the observed traffic counts of links and the target origin-destination trip demands are independent discrete random variables with known probabilities. The problem is formulated by using the least squares approach in which the objective is to maximize the probability that the sum of squared errors between the estimated values and the observed (target) ones does not exceed a pre-specified threshold. An enumeration so lution approach is proposed to solve the problem in small-sized networks, while a normal approximation based on the central limit theorem is applied in large-sized networks to transform the problem into a deterministic nonlin ear fractional model. Some numerical examples are provided to illustrate the efficiency of the proposed method.

We introduce a new family of multivalue and multistage methods based on Hermite–Birkhoff interpolation for solving nonlinear Volterra integro differential equations. The proposed methods that have high order and ex tensive stability region, use the approximated values of the first derivative of the solution in the m collocation points and the approximated values of the solution as well as its first derivative in the r previous steps. Convergence order of the new methods is determined and their linear stability is analyzed. Efficiency of the methods is shown by some numerical experiments.

The most important purpose in location problems is usually to locate some facilities and allocate the demands of nodes so that the total transportation cost of the network is minimized. However, in real networks, there are some other influencing factors, aside from the transportation costs, for determin ing the allocation mode. In this paper, a minimum information approach is applied to the capacitated p-median problem to estimate the most likely allo cation solution based on some prior probabilities. Indeed, the most probable solution is achieved through minimizing a log-based objective function, while the total transportation cost should be less than or equal to a predetermined budget. The problem is solved by using a decomposition method combined with the Karush–Kuhn–Tucker optimality conditions, and some numerical examples are provided to verify the added value of the proposed model and solution approach.

We consider a fully-discrete approximation of the Allen-Cahn equation, such that the forward Euler/Crank–Nicolson scheme (in time) combined with the RBF collocation method based on “shifted” surface spline (in space). Numerical solvability and stability of the method, by using second order finite difference matrices are discussed. We show that, in the proposed scheme, the nonlinear term can be treated explicitly and the resultant numerical scheme is linear and easy to implement. Numerical results that show the effciency and reliability of the proposed method are presented, and two types of collocation nodes for solving this equation are compared.

We present a new numerical approach to solve the optimal control problems (OCPs) with a quadratic performance index. Our method is based on the Bell polynomials basis. The properties of Bell polynomials are explained. We also introduce the operational matrix of derivative for Bell polynomials. The chief feature of this matrix is reducing the OCPs to an optimization problem. Finally, we discuss the convergence of the new technique and present some illustrative examples to show the effectiveness and applicability of the proposed scheme. Comparison of the proposed method with other previous methods shows that this method is accurate.

We apply a new method to solve fractional partial differential equations (FPDEs) with proportional delays. The method is based on expanding the unknown solution of FPDEs with proportional delays by the basis of Bernstein polynomials with unknown control points and uses operational matrices with the least-squares method to convert the FPDEs with proportional de lays to an algebraic system in terms of Bernstein coefficients (control points) approximating the solution of FPDEs. We use the Caputo derivatives of de gree 0 < α ≤ 1 as the fractional derivatives in our work. The main advantage of using this technique is that the method can easily be employed to a variety of FPDEs with or without proportional delays, and also the method offers a very simple and flexible framework for direct approximating of the solution of FPDEs with proportional delays. The convergence analysis of the present method is discussed. We show the effectiveness and superiority of the method by comparing the results obtained by our method with the results of some available methods in two numerical examples.

Functionally graded materials (FGMs) are materials that show different properties in different areas due to the gradual change of chemical composition, distribution, and orientation, or the size of the reinforcing phase in one or more dimensions. In this paper, the free vibrations of a thin cylindrical shell made of FGM is investigated. In order to investigate this problem, the first-order shear theory is used, by using relations related to the propagation of waves and fluid-structure interaction. Also, due to the rotational iner tia of first-order shear deformation and the fluid velocity potential, dynamic equation of functionally graded cylinder shell, containing current is obtained. Convergence of the solutions obtained from this method in different modes of boundary conditions as well as different geometric characteristics for the submerged cylinder and results of other studies and articles is showed. Also the effects of different parameters on the FGM cylindrical shell frequencies for the classical boundary conditions (compositions of simple, clamped, and free boundary conditions) are investigated against the ratio of length to the radius and the ratio of thickness to radius for different values of exponential power (exponential order) of FGM material. The results show that if the more density of the fluid in which the cylinder is submerged is lower, then the frequency values will be higher. Also, by examining the different fluid velocities, it can be seen that the effect of thickness change so that increas ing thickness causes the increase of effect of speed on the natural frequency reduction, especially in higher modes.