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.Exponentially fitted tension spline method for singularly perturbed differential difference equations
https://ijnao.um.ac.ir/article_39980.html
In this paper, singularly perturbed differential difference equations having delay and advance in the reaction terms are considered. The highest order derivative term of the equation is multipliedby a perturbation parameter&nbsp; &epsilon; taking arbitrary values in the interval $(0, 1]$. For the small value of &epsilon;, the solution of the equation exhibits a boundary layer on the left or right side of the domain depending on the sign of the convective term. The terms with the shifts are approximated by using the Taylor series approximation. The resulting singularly perturbed boundary value problem is solved using the exponentially fitted tension spline method. The stability and the uniform convergence of the scheme are discussed and proved. Numerical examples are considered for validating the theoretical analysis of the scheme. The developed scheme gives accurate results with linear order uniform convergence.New class of hybrid explicit methods for numerical solution of optimal control problems
https://ijnao.um.ac.ir/article_40089.html
&lrm;Forward-backward sweep method (FBSM) is an indirect numerical method used for solving optimal control problems&lrm;, &lrm;in which differential equation arising from this method is solved by the &lrm;Pontryagin'&lrm;s&lrm; &lrm;&lrm;&lrm;maximum principle&lrm;. &lrm;I&lrm;n this paper&lrm;, &lrm;&lrm;&lrm;a set of hybrid methods based on explicit 6th order Runge&lrm;- &lrm;Kutta method &lrm;presented&lrm; &lrm;for &lrm;the &lrm;FBSM &lrm;solution&lrm; of optimal control problems&lrm;. &lrm;O&lrm;rder &lrm;of truncation error&lrm;, stability region and numerical results of the &lrm;new hybrid &lrm;methods were compared with &lrm;those&lrm; of the 6th order Runge-Kutta method&lrm;. &lrm;Numerical results showed that &lrm;new &lrm;hybrid&lrm; methods &lrm;&lrm;&lrm; &lrm;are more accurate than the 6th order Runge-Kutta method and their stability regions are &lrm;also&lrm; wider than &lrm;that&lrm; of the 6th order Runge&lrm;- &lrm;Kutta method.&lrm;The strict complementarity in linear fractional optimization
https://ijnao.um.ac.ir/article_40153.html
As an important duality result in linear optimization, the Goldman--Tucker theorem establishes strict complementarity between a pair of primal and dual linear programs. Our study extends this result into the framework of linear fractional optimization. Associated with a linear fractional program, a dual program can be defined as the dual of the equivalent linear program obtained from applying the Charnes--Cooper transformation to the given program. Based on this definition, we propose new criteria for primal and dual optimality by showing that the primal and dual optimal sets can be equivalently modeled as the optimal sets of a pair of primal and dual linear programs. Then, we define the concept of strict complementarity and establish the existence of at least one, called strict complementary, pair of primal and dual optimal solutions such that in every pair of complementary variables exactly one variable is positive and the other is zero. We geometrically interpret the strict complementarity in terms of the relative interiors of two sets that represent the primal and dual optimal sets in higher dimensions. Finally, using this interpretation, we develop two approaches for finding a strict complementary solution in linear fractional optimization. We illustrate our results with two numerical examples.Solving quantum optimal control problems by wavelets method
https://ijnao.um.ac.ir/article_40206.html
In this paper, we present the quantum equation and synthesize an optimal control procedure for this equation. We develop a theoretical method for the analysis of quantum optimal control system given by time depending Schrodinger equation. The Legendre wavelet method is proposed for solving this problem. This can be used as an efficient and accurate computational method in obtaining numerical solutions for different quantum optimal control problems. The distinguishing feature of this paper is that it makes the method previously used to solve non-quantum control equations based on Legendre wavelets usable by using a change of variables for quantum control equations.Singularly perturbed robin type boundary value problems with discontinuous source term in geophysical fluid dynamics.
https://ijnao.um.ac.ir/article_40208.html
In this paper, singularly perturbed robin type boundary value problems with discontinuous source term applicable in geophysical fluid is considered. Due to the discontinuity, interior layers appears in the solution. To fit the interior and boundary layers fitted non-standard numerical method is constructed. To treat the robin boundary condition we used finite difference formula. The stability and parameter uniform convergence of the proposed method are proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter, $varepsilon$ and mesh size, $h$. The numerical result are tabulated and it is observed that the present method is more accurate and uniformly convergent with order of convergence of $O(h)$.Two new approximations to Caputo-Fabrizio fractional equation on non-uniform meshes and its applications
https://ijnao.um.ac.ir/article_40297.html
In this paper, we present two numerical approximations with non-uniform meshes to the Caputo-Fabrizio derivative of order &alpha; (0 &lt; &alpha; &lt; 1). First, the L1 formula is obtained using the linearinterpolation approximation for constructing the second-order approximation. Next, the quadratic interpolation approximation is used for improving the accuracy in the &lrm;temporal&lrm; &lrm;direction. Besides, we discretize the spatial derivative using the compact finite difference (CFD) scheme. The accuracy of the suggested schemes is not dependent on the fractional &alpha;.The coefficients and the truncation errors are carefully investigated for two schemes, separately. Three examples are carried out to support the convergence orders and show the efficiency of the suggested scheme.Application of Newton-Cotes quadrature rule for nonlinear Hammerstein integral equations
https://ijnao.um.ac.ir/article_40298.html
In this paper, a numerical method for solving Fredholm and Volterra integral equations of the second kind is presented. The method is based on the use of Newton-Cotes quadrature rule and Lagrange interpolation polynomials. By the proposed method, the main problem is reduced to solve some nonlinear algebraic equations that can be solved by Newton&rsquo;s method. Also, we prove some statements about the convergence of the method. It is shown that the approximated solution is uniformly convergent to the exact solution. In addition, to demonstrate efficiencyand applicability of the proposed method, several numerical examples are included which confirms the convergenceresults.Investigating a claim about resource complexity measure
https://ijnao.um.ac.ir/article_40312.html
&lrm;The utilization factor (UF) measures the ratio of the total resources&rsquo; amount required to the availability of resources&rsquo; amount during the life cycle of a project&lrm;. &lrm;In 1982 in the journal of Management Science&lrm;, &lrm;Kurtulus and Davis claimed that &ldquo;If two resource-constrained problems for each type of resource have the same UF&rsquo;s value in each period of time&lrm;, &lrm;then each problem is subjected to the same amount of delay provided that the same sequencing rule is used (If different tie-breaking rules are used&lrm;, &lrm;a different schedule may be obtained)&rdquo;&lrm;. &lrm;In this paper&lrm;, &lrm;with a counterexample&lrm;, &lrm;we show that the claim of authors cannot be justified&lrm;.A new algorithm for solving linear programming problems with bipolar fuzzy relation equation constraints
https://ijnao.um.ac.ir/article_40356.html
This paper studies the linear optimization problem subject to a system of bipolar fuzzy relation equations with the max-product composition operator. Its feasible domain is briefly characterized by its lower and upper bound and its consistency is considered. Also, some sufficient conditions are proposed to reduce the size of the search domain of the optimal solution of the problem.Under these conditions, some equations can be deleted to compute the minimum objective value. Some sufficient conditions are then proposed which under them, one of the optimal solutions of the problem is explicitly determined and the unique conditions of the optimal solution are expressed. Moreover, a modified branch-and-bound method based on a value matrix is proposed to solve the reduced problem. A new algorithm is finally designed to solve the problem based on the conditions and modified branch-and-bound method. The algorithm is compared to the methods in other papers to show its efficiency.Modified ADMM algorithm for solving proximal bound formulation of multi-delay optimal control problem with bounded control
https://ijnao.um.ac.ir/article_40357.html
This study presents an algorithm for solving Optimal Control Problems(OCPs) with objective function of the Lagrange - type and multiple delayson both the state and control variables of the constraints; with bounds on thecontrol variable. The full discretization of the objective functional and the multipledelay constraints was carried out using the Simpson numerical scheme.The discrete recurrence relations generated from the discretization of both theobjective functional and constraints were used to develop the matrix operatorswhich satisfy the basic spectral properties The primal-dual residuals of the algorithmwere derived in order to ascertain the rate of convergence of the algorithm;which performs faster when relaxed with an accelerator variant in the sense ofNesterov. The direct numerical approach for handling the multi-delay controlproblem was observed to obtain an accurate result at a faster rate of convergencewhen over-relaxed with an accelerator variant. This research problemis limited to linear constraints and objective functional of the Lagrange-typeand can address real-life models with multiple delays as applicable to quadraticoptimization of Intensity Modulated Radiation Theory (IMRT) planning. Thenovelty of this research paper lies in the method of discretization and its adaptationto handle linearly and proximal bound constrained program formulatedfrom the multiple delay optimal control problems.Review of the strain-based formulation for analysis of plane structures Part I: Formulation of basics and the existing elements
https://ijnao.um.ac.ir/article_40376.html
Since the introduction of the finite element approach, as a numerical solution scheme for structural and solid mechanics applications, various formulation methodologies have been proposed. These ways offer different advantages and shortcomings. Among these techniques, the standard displacement-based approach has attracted more interests due to its straightforward scheme and generality. Investigators have proven that the other strategies, such as, the force-based, hybrid, assumed stress and assumed strain provide special advantages in comparison with the classic finite elements. For instance, the mentioned techniques are able to solve difficulties, like shear locking, shear parasitic error, mesh sensitivity, poor convergence and rotational dependency. The main goal of this two-part study is to present a brief yet clear portrait of the basics and advantages of the direct strain-based method for development of high-performance plane finite elements. In this paper, which is the first part of this study, assumptions and the basics of this method are introduced. Then, a detailed review of all the existing strain-based membrane elements is presented. Although the strain formulation is applicable for different types of structures, most of the existing elements pertain to the plane structures. The second part of this study deals with application and performance of the reviewed elements in the analysis of plane stress/strain problems.Review of the strain-based formulation for analysis of plane structures Part II: Evaluation of the numerical performance
https://ijnao.um.ac.ir/article_40377.html
&nbsp;In this part of the study, several benchmark problems are solved to evaluate the performance of the existing strain-based membrane elements which were reviewed in the first part. This numerical evaluation provides a basis for comparison between these elements. Detailed discussions are offered after each benchmark problem. Based on the attained results, it is concluded that the inclusion of drilling degrees of freedom and also utilization of higher-order assumed strain field results in higher accuracy of the elements. Moreover, it is evident that imposing the optimal criteria such as equilibrium and compatibility on the assumed strain field, in addition to reducing the number of degrees of freedom of the element, increases the convergence speed of the resulting strain-based finite elements.