# Journal of Computational Mathematics and Computer Modeling with Applications Volume:1 Issue: 2, Summer and Autumn 2022

• تاریخ انتشار: 1401/09/10
• تعداد عناوین: 12
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• Elyas Shivanian *, Eghbal Mohammadi Pages 1-7
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.
Keywords: successive linearization method, diffusion reaction equation, pseudo-spectral collocation method
• Raziyeh Erfanifar * Pages 8-19
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.
Keywords: Moore-Penrose inverse, Iterative method, Third-order convergence
• Raziyeh Erfanifar * Pages 20-26
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.
Keywords: Nonlinear equations, Two-step methods, Efficiency index, Order of convergence, Simple root, Iterative method with memory
• Nazi Abdollahi, Saeid Abbasbandy * Pages 27-34
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.
Keywords: Abel integral equation, Reproducing Hilbert kernel space, Approximate solution
• Hafez Yari, Mehdi Delkhosh * Pages 35-45
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.
Keywords: Non-rectangular domains, 2D integral equations, Jacobi polynomials, Collocation method, Newton-Krylov GMRes algorithm
• Fardin Salehi, Soleiman Hashemi Shahraki, MohammadKazem Fallah, Mohammad Hemami * Pages 46-64

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.

Keywords: Radial Basis Function, Genetic algorithm, HIV, Influenza, Shape parameter
• Sedigheh Sabermahani, Yadollah Ordokhani * Pages 65-73
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.
Keywords: Fractional-order Fokker-Planck equations, Touchard polynomials, Pseudo-operational matrix
• Marzieh Dehghani-Madiseh * Pages 74-85
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.
Keywords: Nonlinear function, finite element approach, piecewise Lagrange interpolation
• Liam Burrowsa, Ke Chen *, Francesco Torella Pages 86-103
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  as  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  of piecewise constants. Comparisons with related models are conducted to illustrate the advantages of  our new models.
Keywords: Variational calculus, Inverse problems, image segmentation, Mumford-Shah, Intensity inhomogeneity, Geometric constraints, iterative methods
• Aida Pakniyat * Pages 104-121
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.
Keywords: Cylinder, Spectral methods, Least squares support vector machine, Magnetohydrodynamic (MHD), Nonlinear ODE, fractional Hermite functions, Semi-infinite
• Marzieh Dehghani-Madiseh * Pages 122-128
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.
Keywords: Decomposition of matrices, Interval linear systems
• Mehdi Delkhosh *, Reza Arefi Shirvan Pages 129-138
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.
Keywords: Generalized pseudospectral method, Newton-Kantorovich quasilinearization method, Generalized Lagrange functions, Derivative operational matrix