Computational Mathematics and Computer Modeling with Applications (CMCMA)
https://cmcma.sbu.ac.ir/
Computational Mathematics and Computer Modeling with Applications (CMCMA)endaily1Thu, 01 Dec 2022 00:00:00 +0330Thu, 01 Dec 2022 00:00:00 +0330Cover for Volume 1, Issue 2, 2022
https://cmcma.sbu.ac.ir/article_103273.html
A hybrid method of successive linearization method (SLM) and collocation method to steady regime of the reaction-diffusion equation
https://cmcma.sbu.ac.ir/article_102617.html
This article presents a method based on combination of successive linearization method (SLM) and pseudo-spectral collocation method and then is applied on a nonlinear model of coupled diffusion and chemical reaction in a spherical catalyst pellet. It is obtained that this method can be used for nonlinear boundary value problems without difficulty because the nonlinear part of the equation becomes inactive by SLM and more, to treat the linear equation, even in the case of complicatedness, is straightforward by pseudo-spectral collocation method. Also, the results reveal the high efficiency with reliable accuracy of this hybrid method.An efficient iterative method for finding the Moore-Penrose and Drazin inverse of a matrix
https://cmcma.sbu.ac.ir/article_102767.html
In this paper, a third order convergent method for finding the Moore-Penrose inverse of a matrix is presented and analysed. Then, we develop the method to find Drazin inversion. This method is very robust to find the Moore-Penrose and Drazin inverse of a matrix. Finally, numerical examples show that the efficiency of the proposed method is superior over other proposed methods.A class of efficient derivative free iterative method with and without memory for solving nonlinear equations
https://cmcma.sbu.ac.ir/article_102771.html
In the present paper, at first, we propose a new two-step iterative method for solving nonlinear equations. This scheme is based on the Steffensen's method, in which the order of convergence is four. This iterative method requires only three functions evaluation in each iteration, therefore it is optimal in the sense of the Kung and Traub conjecture. Then we extend it to the method with memory, which the order of convergence is six. Finally, numerical examples indicate that theobtained methods in terms of accuracy and computational cost are superior to thefamous forth-order methods.Reproducing kernel method for Abel's second kind singular integral equations
https://cmcma.sbu.ac.ir/article_102919.html
Singular integral equations (SIEs) are often encountered in certain contact and fracture problems in solid mechanics. In this paper, we apply the reproducing kernel method (RKM) to give the approximate solution of Abel's second-kind singular integral equations. For solving this problem, difficulties lie in its singular term. In order to remove the singular term of the equation, an equivalent transformation is made. Solution representations are obtained in reproducing kernel Hilbert space. Numerical experiments show that our reproducing kernel method is efficient. To show the high accuracy of the method the results are compared to other numerical methods and satisfactory agreements are achieved.Newton-Krylov generalized minimal residual algorithm in solving the nonlinear two-dimensional integral equations of the second kind on non-rectangular domains with an error estimate
https://cmcma.sbu.ac.ir/article_103020.html
In this paper, an applicable numerical approximation has been proposed for solving nonlinear two-dimensional integral equations (2DIEs) of the second kind on non-rectangular domains. Because directly applying the collocation methods on non-rectangular domains is difficult, in this work, at first, the integral equation is converted to an equal integral equation on a rectangular domain, then the solution is approximated by applying 2D Jacobi collocation method, the implementation of these instructions reduces the integral equation to a system of nonlinear algebraic equations, therefore, solving this system has an important role to approximate the solution. In this paper, Newton-Krylov generalized minimal residual (NK-GMRes) algorithm is used for solving the system of nonlinear algebraic equations. Furthermore, an error estimate for the presented method is investigated and several examples confirm the accuracy and efficiency of the proposed instructions.Numerical investigation of differential biological models via Gaussian RBF collocation method with genetic strategy
https://cmcma.sbu.ac.ir/article_103078.html
In this paper, we use radial basis function collocation method for solving the system of differential equations in the area of biology. One of the challenges in RBF method is picking out an optimal value for shape parameter in Radial basis function to achieve the best result of the method because there are not any available analytical approaches for obtaining optimal shape parameter. For this reason, we design a genetic algorithm to detect a close optimal shape parameter. The population convergence figures, the residuals of the equations and the examination of the ASN2R and ARE measures all show the accurate selection of the shape parameter by the proposed genetic algorithm. Then, the experimental results show that this strategy is efficient in the systems of differential models in biology such as HIV and Influenza. Furthermore, we show that using our pseudo-combination formula for crossover in genetic strategy leads to convergence in the nearly best selection of shape parameter.A computational method to solve fractional-order Fokker-Planck equations based on Touchard polynomials
https://cmcma.sbu.ac.ir/article_103191.html
This manuscript presents a new approximation method for fractional-order Fokker-Planck equations based on Touchard polynomial approximation. We provide new Caputo and extra Caputo pseudo-operational matrices for these polynomials. Then, utilizing mentioned pseudo-operational matrices and an optimal method, the considered equation leads to a system of algebraic equations which can be solved by mathematical software. Finally, we illustrate the advantages of the suggested technique through several numerical examples.Transforming Ostrowski's method into a derivative-free method and its dynamics
https://cmcma.sbu.ac.ir/article_103763.html
The current research develops a derivative-free family without memory methods. The proposed method consisting of two steps and one parameter for solving nonlinear equations is brought forward.\,The basin of attraction of the proposed methods has investigated using different weight functions.\,Numerical examples are experimented with to check the performance of the proposed schemes. Furthermore, the theoretical order of convergence is confirmed on the experiment work.Analyzing the influence of treatment awareness rate on COVID-19 pandemic by fractional derivative-based modeling and simulation
https://cmcma.sbu.ac.ir/article_103764.html
Covid-19 disease is a respiratory illness caused by SARS-Cov-2 and poses a serious public health risk. It usually spread from person-to-person. The fractional- order of covid-19 was determined and basic reproduction number using the next generation matrix was calculated. The stability of disease-free equilibrium and endemic equilibrium of the model were investigated. Also, sensitivity analysis of the reproduction number with respect to the model parameters were carried out. It was observed that in the absence of infected persons, disease free equilibrium is achievable and is asymptotically stable.Numerical simulations were presented graphically. The results of the model analysis indicated that $R_{0}$ $\mathrm{&lt;}$ 1 is adequate enough to reducing the spread of disease and disease persevere in the population when $R_{0}$ $\mathrm{&gt;}$ 1 The numerical results showed that effective vaccination of the population helps in curtailing the spread of the viral disease.In order to know whether the disease may die out or persist, basic reproduction number, $R_{0}$ was obtained using Next Generation Matrix Method. It was observed that the value of $R_{0}$ is high when the depletion of awareness programme is high while the value of $R_{o}$ is very low when the rate of implementation of awareness programme is high. So, neglecting the implementation of awareness program can have serious effect on the population. The model shows the implementation of awareness program is the key eradication to the pandemic.A new approach for solving nonlinear equations
https://cmcma.sbu.ac.ir/article_103591.html
In this paper, we describe and analyze an efficient method to find the roots of a general one variable function $f:\mathbb{R}\rightarrow \mathbb{R}$. The proposed method is based on partitioning an interval (that probably contains root(s) of $f$) into subintervals. From this point of view, we name this method a finite element approach for root finding. Also the convergence analysis of the presented method is presented. The new approach can be generalized to estimate the roots of the multivariable functions in higher dimensions. Also it is capable to find all of the roots of the function on a determined interval. Finally, numerical examples are given to illustrate the effectiveness of the new method.On two convex variational models and their iterative solutions for selective segmentation of images with intensity inhomogeneity
https://cmcma.sbu.ac.ir/article_103592.html
Treating images as functions and using variational calculus,mathematical imaging offers to design novel and continuous methods, outperforming traditional methods based on matrices, for modelling real life tasks in image processing.Image segmentation is one of such fundamental tasks &nbsp;as &nbsp;many application areas demand a reliable segmentation method. Developing reliable selective segmentation algorithms isparticularly important in relation to training data preparation in modern machine learning as accurately isolating a specific object in an image with minimal user input is a valuable tool. When an image's intensity is consisted of mainly piecewise constants, convex models are available.Different from previous works, this paperproposes two convex models that are capable of segmenting local features defined by geometric constraints for images having intensity inhomogeneity.Our new, local, selective and convex variants are extended from the non-convex Mumford-Shah model intended for global segmentation.They have fundamentally improved on previous selective models that assume intensity &nbsp;of piecewise constants. Comparisons with related models are conducted to illustrate the advantages of &nbsp;our new models.Numerical solution for solving magnetohydrodynamic (MHD) flow of nanofluid by least squares support vector regression
https://cmcma.sbu.ac.ir/article_103630.html
This paper introduces a new numerical solution based on the least squares support vector machine (LS-SVR) for solving nonlinear ordinary differential equations of high dimensionality. We apply the quasilinearization method to linearize the magnetohydrodynamic (MHD) flow of nanofluid around a stretching cylinder, thereby transforming it into a linear problem. We then utilize LS-SVR with fractional Hermite functions as basis functions to solve this problem over a semi-infinite interval. Our numerical results confirm the effectiveness of this approach.On modified decomposition of interval matrices and its application
https://cmcma.sbu.ac.ir/article_103631.html
In this paper we present the LU decomposition of a generalized interval matrix ${\bf{A}}$ under a modified interval arithmetic. This modified interval arithmetic is defined on generalized intervals and possesses group properties with respect to the addition and multiplication operations. These properties cause that the two computed generalized interval matrices ${\bf{L}}$ and ${\bf{U}}$ from the LU decomposition satisfy ${\bf{A}}={\bf{L}}{\bf{U}}$, with equality in modified interval arithmetic instead of the weaker inclusion in the classical interval arithmetic. Some applications of the new technique for solving interval linear systems are given and effectiveness of the new approach is investigated along some numerical tests.A hybrid numerical method based on the generalized pseudospectral method for solving nonlinear differential equations
https://cmcma.sbu.ac.ir/article_103741.html
In this paper, a hybrid numerical method using generalized pseudospectral and Newton-Kantorovich quasilinearization methods is presented to solve nonlinear differential equations. Initially, generalized Lagrange functions as basic functions are introduced and then derivative operational matrices for these functions are presented. Then using these new functions, the generalized pseudospectral method is constructed as a numerical method. Finally, this method and the Newton-Kantorovich quasilinearization method are combined to produce an efficient method. Because of the use of derivative operating matrices and the conversion of any nonlinear differential equation into sequences of linear differential equations, the implementation of this method does not require mathematically to calculate the derivative and the computational costs are also reduced. To illustrate the efficiency, accuracy, and convergence of the method, the proposed method is implemented on two famous equations and the results are compared with other methods.