Iranian Journal of Numerical Analysis and Optimization
https://ijnao.um.ac.ir/
Iranian Journal of Numerical Analysis and Optimizationendaily1Mon, 01 Mar 2021 00:00:00 +0330Mon, 01 Mar 2021 00:00:00 +0330Solving fuzzy multiobjective linear bilevel programming problems based on the extension principle
https://ijnao.um.ac.ir/article_39467.html
Fuzzy multiobjective linear bilevel programming (FMOLBP) problems are studied in this paper. The existing methods replace one or some deterministic model(s) instead of the problem and solve the model(s). Doing this work, we lose much information about the compromise decision, and it does not make sense for the uncertain conditions. To overcome the difficulties, Zadeh&rsquo;s extension principle is applied to solve the FMOLBP problems. Two crisp multiobjective linear three-level programming problems are proposed to find the lower and upper bound of its objective values in different levels. The problems are reduced to some linear optimization problems using one of the scalarization approaches, called the weighting method, the dual theory, and the vertex enumeration method. The lower and upper bounds are estimated by the resolution of the corresponding linear optimization problems. Hence, the membership functions of compromise objective values are produced, which is the main contribution of this paper. This technique is applied for the problem for the first time. This method applies all information of a fuzzy number and does not estimate it by a crisp number. Hence, the compromise decision resulted from the proposed method is consistent with reality. This point can minimize the gap between theory and practice. The results are compared with the results of existing approaches. It shows the efficiency of the proposed approach.The time-dependent diffusion equation: An inverse diffusivity problem
https://ijnao.um.ac.ir/article_39498.html
We find a solution of an unknown time-dependent diffusivity a(t) in a linear inverse parabolic problem by a modified genetic algorithm. At first, it is shown that under certain conditions of data, there exists at least one solution for unknown a(t) in (a(t), T (x, t)), which is a solution to the corresponding problem. Then, an optimal estimation for unknown a(t) is found by applying the least-squares method and a modified genetic algo rithm. Results show that an excellent estimation can be obtained by the implementation of a modified real-valued genetic algorithm within an Intel Pentium (R) dual-core CPU with a clock speed of 2.4 GHz.Approximate solution for a system of fractional integro-differential equations by Müntz Legendre wavelets
https://ijnao.um.ac.ir/article_39497.html
We use the&nbsp;M&uuml;ntz Legendre wavelets and operational matrix to solve a system of fractional integro-differential equations. In this method, the system of integro-differential equations shifts into the systems of the algebraic equation, which can be solved easily. Finally, some examples confirming the applicability, accuracy, and efficiency of the proposed method are given.Monotonicity-preserving splitting schemes for solving balance laws
https://ijnao.um.ac.ir/article_39466.html
In this paper, some monotonicity-preserving (MP) and positivity-preserving (PP) splitting methods for solving the balance laws of the reaction and diffusion source terms are investigated. To capture the solution with high accuracy and resolution, the original equation with reaction source termis separated through the splitting method into two sub-problems including the homogeneous conservation law and a simple ordinary differential equation (ODE). The resulting splitting methods preserve monotonicity and positivity property for a normal CFL condition. A trenchant numerical analysis made it clear that the computing time of the proposed methods decreases when the so-called MP process for the homogeneous conservation law is imposed. Moreover, the proposed methods are successful in recapturing the solution of the problem with high-resolution in the case of both smooth and non-smooth initial profiles. To show the efficiency of proposed methods and to verify the order of convergence and capability of these methods, several numerical experiments are performed through some prototype examples.Toeplitz-like preconditioner for linear systems from spatial fractional diffusion equations
https://ijnao.um.ac.ir/article_39538.html
&lrm;The article deals with constructing Toeplitz-like preconditioner for linear systems arising from finite difference discretization of the spatial fractional diffusion equations&lrm;. &lrm;The coefficient matrices of these linear systems have an $S+L$ structure&lrm;, &lrm;where $S$ is a symmetric positive definite (SPD) matrix and $L$ satisfies $\mbox{rank}(L)\leq 2$&lrm;. &lrm;We introduce an approximation for the SPD part $S$&lrm;, &lrm;which is called $P_S$&lrm;, &lrm;and then we show that the preconditioner $P=P_S+L$ has the Toeplitz-like structure and its displacement rank is 6&lrm;.&nbsp; &lrm;The analysis shows that the eigenvalues of the corresponding preconditioned matrix are clustered around 1. Numerical experiments exhibit that the Toeplitz-like preconditioner can significantly improve the convergence properties of the applied iteration method.A high-order algorithm for solving nonlinear algebraic equations
https://ijnao.um.ac.ir/article_39537.html
A fourth-order and rapid numerical algorithm, utilizing a procedure as Runge&ndash;Kutta methods, is derived for solving nonlinear equations. The method proposed in this article has the advantage that it, requiring no calculation of higher derivatives, is faster than the other methods with the same order of convergence. The numerical results obtained using the developed approach are compared to those obtained using some existing iterative methods, and they demonstrate the efficiency of the present approach.Computation of eigenvalues of fractional Sturm–Liouville problems
https://ijnao.um.ac.ir/article_39622.html
We consider the eigenvalues of the fractional-order Sturm--Liouville equation of the form \begin{equation*} -{}^{c}D_{0^+}^{\alpha}\circ D_{0^+}^{\alpha} y(t)+q(t)y(t)=\lambda y(t),\quad 0&lt;\alpha\leq 1,\quad t\in[0,1], \end{equation*} with Dirichlet boundary conditions $$I_{0^+}^{1-\alpha}y(t)\vert_{t=0}=0\quad\mbox{and}\quad I_{0^+}^{1-\alpha}y(t)\vert_{t=1}=0,$$ where $q\in L^2(0,1)$ is a real-valued potential function. The method is used based on a Picard's iterative procedure. We show that the eigenvalues are obtained from the zeros of the Mittag-Leffler function and its derivatives.Using approximate endpoint property on existing solutions for two inclusion problems of the fractional q-differential
https://ijnao.um.ac.ir/article_39560.html
Using the approximate endpoint property, we describe a technique for existing solutions of the fractional q-differential inclusion with boundary value conditions on multifunctions. For this, we use an approximate endpoint result on multifunctions. Also, we give an example to elaborate on our results and to present the obtained results by fractional calculus.Hopf bifurcation analysis in a delayed model of tumor therapy with oncolytic viruses
https://ijnao.um.ac.ir/article_39590.html
The stability and Hopf bifurcation of a nonlinear mathematical model are described by the delay differential equation proposed by Wodarz for interaction between uninfected tumor cells and infected tumor cells with the virus. By choosing &tau; as a bifurcation parameter, we show that the Hopf bifurcation can occur for a critical value &tau;. Using the normal form theory and the center manifold theory, formulas are given to determine the stability and the direction of bifurcation and other properties of bifurcating periodic solutions. Then, by changing the infection rate to two nonlinear infection rates, we investigate the stability and existence of a limit cycle for the appropriate value of &tau;, numerically. Lastly, we present some numerical simulations to justify our theoretical results.ADI method of credit spread option pricing based on jump-diffusion model
https://ijnao.um.ac.ir/article_39605.html
As the main contribution of this article, we establish an option on a credit spread under a stochastic interest rate. The intense volatilities in financial markets cause interest rates to change greatly; thus, we consider a jump term in addition to a diffusion term in our interest rate model. However, this decision leads us to a partial integral differential equation. Since the integral part might bring some difficulties, we put forward a fairly new numerical scheme based on the alternating direction implicit method. In the remainder of the article, we discuss consistency, stability, and convergence of the proposed approach. As the final step, with the help of the MATLAB program, we provide numerical results of implementing our method on the governing equation.An adaptive descent extension of the Polak–Rebière–Polyak conjugate gradient method based on the concept of maximum magnification
https://ijnao.um.ac.ir/article_39611.html
Recently, a one-parameter extension of the Polak&ndash;Rebi&egrave;re&ndash;Polyak method has been suggested, having acceptable theoretical features and promising numerical behavior. Here, based on an eigenvalue analysis on the method with the aim of avoiding a search direction in the direction of the maximum magnification by a symmetric version of the search direction matrix, an adaptive formula for computing parameter of the method is proposed. Under standard assumptions, the given formula ensures the sufficient descent property and guarantees the global convergence of the method. Numerical experiments are done on a collection of CUTEr test problems. They show practical effectiveness of the suggested formula for the parameter of the method.Some applications of Sigmoid functions
https://ijnao.um.ac.ir/article_39761.html
In numerical analysis, the process of fitting a function via given data is called interpolation. Interpolation has many applications in engineering and science. There are several formal kinds of interpolation, including linear interpolation, polynomial interpolation, piecewise constant interpolation, trigonometric interpolation, and so on. In this article, by using Sigmoid functions, a new type of interpolation formula is presented. To illustrate the efficiency of the proposed new interpolation formulas, some ap plications in quadrature formulas (in both open and closed types), numerical integration for double integral, and numerical solution of an ordinary differential equation are included. The advantage of this new approach is shown in the numerical applications section. Trainable fourth-order partial differential equations for image noise removal
https://ijnao.um.ac.ir/article_39916.html
Image processing by partial differential equations (PDEs) has been an active topic in the area of image de-noising, which is an important task in computer vision. In PDE-based methods for image processing the unprocessed, original image is considered as the initial value for the PDE and the solution of the equation is the outcome of the model. Despite the advantages of using PDEs in image processing, designing and modeling different equations for various types of applications has always been a challenging and interesting problem. In this paper, we aim to tackle this problem by introducing a fourth-order equation with flexible, trainable coefficients, and with the help of an optimal control problem, the coefficients are determined therefore the proposed model adapts itself to each particular application. At the final stage, the image enhancement is performed on the noisy test image and the performance of our proposed method is compared to other PDE-based models.