Ferdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697711120210301Solving fuzzy multiobjective linear bilevel programming problems based on the extension principle1313946710.22067/ijnao.2021.11304.0ENA. Abbasi MolaiSchool of Mathematics and Computer Sciences, Damghan University, Damghan, P.O.Box 36715-364, Iran.Journal Article20200107Fuzzy 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’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.https://ijnao.um.ac.ir/article_39467_7c15ec370f1010dc0f6f7bf5a0733b0a.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697711120210301The time-dependent diffusion equation: An inverse diffusivity problem33543949810.22067/ijnao.2021.11309.0ENS.H. TabasiSchool of Mathematics and Computer Science, Damghan University, P.O.Box 36715-364,
Damghan, Iran.H.D. MazraehSchool of Mathematics and Computer Science, Damghan University, P.O.Box 36715-364,
Damghan, Iran.0000-0001-9619-4134A.A. IraniSchool of Mathematics and Computer Science, Damghan University, P.O.Box 36715-364,
Damghan, Iran.R. PourgholiSchool of Mathematics and Computer Science, Damghan University, P.O.Box 36715-364,
Damghan, Iran.0000-0003-4111-5130A. EsfahaniSchool of Mathematics and Computer Science, Damghan University, P.O.Box 36715-364,
Damghan, Iran.Journal Article20200205We find a solution of an unknown time-dependent diffusivity <em>a</em>(<em>t</em>) 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 <em>a</em>(<em>t</em>) in (<em>a</em>(<em>t</em>)<em>, T </em>(<em>x, t</em>)), which is a solution to the corresponding problem. Then, an optimal estimation for unknown <em>a</em>(<em>t</em>) 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.https://ijnao.um.ac.ir/article_39498_4e70303762f92c47958ef68e7d4ce560.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697711120210301Approximate solution for a system of fractional integro-differential equations by Müntz Legendre wavelets55723949710.22067/ijnao.2021.11327.0ENY. BarazandehDepartment of Mathematics, Lorestan University, Khorramabad, Iran.Journal Article20200728We use the Müntz 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.https://ijnao.um.ac.ir/article_39497_9fbb23ca8686f88aa7b256cdbe9e1ba9.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697711120210301Monotonicity-preserving splitting schemes for solving balance laws73943946610.22067/ijnao.2021.11328.0ENF. KhodadostiDepartment of Applied Mathematics, Faculty of Basic Sciences, Sahand University of Technology, Tabriz, Iran.0000-0003-4315-3907J. FarziDepartment of Applied Mathematics, Faculty of Basic Sciences, Sahand University of
Technology, Tabriz, Iran.0000-0001-9169-7920M.M. KhalsaraeiFaculty of Mathematical Science, University of Maragheh, Maragheh, Iran.Journal Article20200826<span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">In this paper, some monotonicity-preserving (MP) and positivity-preserving <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">(PP) splitting methods for solving the balance laws of the reaction and <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">diffusion source terms are investigated. To capture the solution with high <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">accuracy and resolution, the original equation with reaction source term<span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">is separated through the splitting method into two sub-problems including <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">the homogeneous conservation law and a simple ordinary differential equa<span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">tion (ODE). The resulting splitting methods preserve monotonicity and <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">positivity property for a normal CFL condition. A trenchant numerical <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">analysis made it clear that the computing time of the proposed methods <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">decreases when the so-called MP process for the homogeneous conserva<span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">tion law is imposed. Moreover, the proposed methods are successful in <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">recapturing the solution of the problem with high-resolution in the case <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">of both smooth and non-smooth initial profiles. To show the efficiency of <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">proposed methods and to verify the order of convergence and capability of <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">these methods, several numerical experiments are performed through some <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">prototype examples.</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><br style="font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;" /></span>https://ijnao.um.ac.ir/article_39466_dba8b413b2fc8a1621bf522c9463cc38.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697711120210301Toeplitz-like preconditioner for linear systems from spatial fractional diffusion equations951063953810.22067/ijnao.2021.11325.0ENN. AkhoundiSchool of mathematics and computer science, Damghan university, Damghan, Iran.Journal Article20200615<span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMSY8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI6; font-size: 5pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI6; font-size: 5pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMR8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: CMMI8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;"><span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">The article deals with constructing Toeplitz-like preconditioner for linear systems arising from finite difference discretization of the spatial fractional diffusion equations. The coefficient matrices of these linear systems have an $S+L$ structure, where $S$ is a symmetric positive definite (SPD) matrix and $L$ satisfies $\mbox{rank}(L)\leq 2$. We introduce an approximation for the SPD part $S$, which is called $P_S$, and then we show that the preconditioner $P=P_S+L$ has the Toeplitz-like structure and its displacement rank is 6. </span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span> <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">The analysis <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">shows that the eigenvalues of the corresponding preconditioned matrix are <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">clustered around 1. Numerical experiments exhibit that the Toeplitz-like <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">preconditioner can significantly improve the convergence properties of the <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">applied iteration method.</span></span></span></span><br style="font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;" /></span>https://ijnao.um.ac.ir/article_39538_c609307c0a3e168ca95fd4e025727180.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697711120210301A high-order algorithm for solving nonlinear algebraic equations1071153953710.22067/ijnao.2021.11329.0ENA. GhorbaniDepartment of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.M. GachpazanDepartment of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.0000-0001-5662-0207Journal Article20200828<span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">A fourth-order and rapid numerical algorithm, utilizing a procedure as <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">Runge–Kutta methods, is derived for solving nonlinear equations. The <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">method proposed in this article has the advantage that it, requiring no <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">calculation of higher derivatives, is faster than the other methods with the <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">same order of convergence. The numerical results obtained using the devel<span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">oped approach are compared to those obtained using some existing iterative <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">methods, and they demonstrate the efficiency of the present approach.</span></span></span></span></span></span></span>https://ijnao.um.ac.ir/article_39537_c4286c78e9af13cd4003d34ca620d548.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697711120210301Computation of eigenvalues of fractional Sturm–Liouville problems1171333962210.22067/ijnao.2020.11305.0ENE.M. MaralaniDepartment of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran.F.D. SaeiDepartment of Mathematics, Tabriz Branch, Islamic Azad University, Tabriz, Iran.0000-0001-9829-2141A.A.J. AkbarfamUniversity of Tabriz, Tabriz, Iran.0000-0003-0339-225XK. GhanbariSahand university of Technology, Tabriz, Iran.Journal Article20200112We consider the eigenvalues of the fractional-order <span style="text-decoration: underline;">Sturm</span>--<span style="text-decoration: underline;">Liouville</span> equation of the form<br />\begin{equation*}<br />-{}^{c}D_{0^+}^{\alpha}\circ D_{0^+}^{\alpha} y(t)+q(t)y(t)=\lambda y(t),\quad 0<\alpha\leq 1,\quad t\in[0,1], <br />\end{equation*}<br />with Dirichlet boundary conditions<br />$$I_{0^+}^{1-\alpha}y(t)\vert_{t=0}=0\quad\mbox{and}\quad I_{0^+}^{1-\alpha}y(t)\vert_{t=1}=0,$$<br />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.https://ijnao.um.ac.ir/article_39622_986a1d19d921e33a18083f766e24c2b7.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697711120210301Using approximate endpoint property on existing solutions for two inclusion problems of the fractional q-differential1351573956010.22067/ijnao.2021.11300.0ENG.K. RanjbarDepartment of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan,
Iran.M.E. SameiDepartment of Mathematics, Faculty of Science, Bu-Ali Sina University, Hamedan,
Iran.0000-0002-5450-3127Journal Article20200101<span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">Using the approximate endpoint property, we describe a technique for exist<span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">ing solutions of the fractional <span style="font-family: CMMI8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>q</em><span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">-differential inclusion with boundary value <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">conditions on multifunctions. For this, we use an approximate endpoint <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">result on multifunctions. Also, we give an example to elaborate on our <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">results and to present the obtained results by fractional calculus.</span></span></span></span><br style="font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;" /></span></span></span>https://ijnao.um.ac.ir/article_39560_8535e30023161a5722024c665642705e.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697711120210301Hopf bifurcation analysis in a delayed model of tumor therapy with oncolytic viruses1591943959010.22067/ijnao.2021.11319.0ENN. AkbariDepartment of Mathematical Sciences, Isfahan University of Techonology, Isfahan, Iran.R. AsheghiDepartment of Mathematical Sciences, Isfahan University of Techonology, Isfahan, Iran.Journal Article20200421<span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">The stability and Hopf bifurcation of a nonlinear mathematical model are <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">described by the delay differential equation proposed by Wodarz for inter<span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">action between uninfected tumor cells and infected tumor cells with the <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">virus. By choosing <span style="font-family: CMMI8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>τ </em><span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">as a bifurcation parameter, we show that the Hopf <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">bifurcation can occur for a critical value <span style="font-family: CMMI8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>τ</em><span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">. Using the normal form theory <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">and the center manifold theory, formulas are given to determine the sta<span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">bility and the direction of bifurcation and other properties of bifurcating <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">periodic solutions. Then, by changing the infection rate to two nonlinear <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">infection rates, we investigate the stability and existence of a limit cycle for <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">the appropriate value of <span style="font-family: CMMI8; font-size: 7pt; color: #000000; font-style: normal; font-variant: normal;"><em>τ</em><span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">, numerically. Lastly, we present some numerical <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">simulations to justify our theoretical results.</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>https://ijnao.um.ac.ir/article_39590_d59280a45084e7d4d270d87ea04b5559.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697711120210301ADI method of credit spread option pricing based on jump-diffusion model1952103960510.22067/ijnao.2021.11333.0ENR. MohamadinejadDepartment of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran.A. NeisyDepartment of Mathematics, Faculty of Mathematics Science and Computer, Allameh Tabataba’i University, Tehran, IranJ. BiazarDepartment of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran.0000-0001-8026-2999Journal Article20201021<span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">As the main contribution of this article, we establish an option on a credit <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">spread under a stochastic interest rate. The intense volatilities in financial <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">markets cause interest rates to change greatly; thus, we consider a jump <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">term in addition to a diffusion term in our interest rate model. However, <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">this decision leads us to a partial integral differential equation. Since the <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">integral part might bring some difficulties, we put forward a fairly new nu<span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">merical scheme based on the alternating direction implicit method. In the <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">remainder of the article, we discuss consistency, stability, and convergence <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">of the proposed approach. As the final step, with the help of the MATLAB <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">program, we provide numerical results of implementing our method on the <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">governing equation.</span></span></span></span></span></span></span></span></span></span><br style="font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;" /></span>https://ijnao.um.ac.ir/article_39605_9802545b7bf4bb1604c0ab0c9dccf299.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697711120210301An adaptive descent extension of the Polak–Rebière–Polyak conjugate gradient method based on the concept of maximum magnification2112193961110.22067/ijnao.2021.67048.0ENZ. AminifardDepartment of Mathematics, Semnan University, P.O. Box: 35195–363, Semnan, Iran.0000-0001-8828-0227S. Babaie-KafakiDepartment of Mathematics, Semnan University, P.O. Box: 35195–363, Semnan, Iran.0000-0003-0122-8384Journal Article20201119<span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">Recently, a one-parameter extension of the Polak–Rebière–Polyak method <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">has been suggested, having acceptable theoretical features and promising <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">numerical behavior. Here, based on an eigenvalue analysis on the method <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">with the aim of avoiding a search direction in the direction of the maximum <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">magnification by a symmetric version of the search direction matrix, an <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">adaptive formula for computing parameter of the method is proposed. Un<span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">der standard assumptions, the given formula ensures the sufficient descent <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">property and guarantees the global convergence of the method. Numerical <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">experiments are done on a collection of CUTEr test problems. They show <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">practical effectiveness of the suggested formula for the parameter of the <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">method.</span></span></span></span></span></span></span></span></span></span></span>https://ijnao.um.ac.ir/article_39611_8ea68f6640f8fea945e5c90509a7630e.pdfFerdowsi University of MashhadIranian Journal of Numerical Analysis and Optimization2423-697711120210301Some applications of Sigmoid functions2212333976110.22067/ijnao.2021.68901.1015ENM. A. JafariDepartment of Mathematical Finance, Faculty of Financial Sciences, Kharazmi University, Tehran, Iran.A. AminataeiDepartment of Mathematical Finance, Faculty of Financial Sciences, Kharazmi University, Tehran, Iran.0000-0001-5247-4492Journal Article20210213<span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">In numerical analysis, the process of fitting a function via given data is <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">called interpolation. Interpolation has many applications in engineering <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">and science. There are several formal kinds of interpolation, including <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">linear interpolation, polynomial interpolation, piecewise constant interpo<span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">lation, trigonometric interpolation, and so on. In this article, by using <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">Sigmoid functions, a new type of interpolation formula is presented. To il<span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">lustrate the efficiency of the proposed new interpolation formulas, some ap <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">plications in quadrature formulas (in both open and closed types), numer<span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">ical integration for double integral, and numerical solution of an ordinary <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">differential equation are included. The advantage of this new approach is <span style="font-family: LMRoman9-Regular; font-size: 8pt; color: #000000; font-style: normal; font-variant: normal;">shown in the numerical applications section. </span></span></span></span></span></span></span></span></span></span><br style="font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;" /></span>https://ijnao.um.ac.ir/article_39761_c12a127108811ae188dce79e3ca1794f.pdf