IRANIAN JOURNAL OF NUMERICAL ANALYSIS AND OPTIMIZATION
Year:2022 | Volume:12 | Issue:2|Papers Count:12

This paper presents a numerical solution for a time delay parabolic problem (reaction-diffusion) containing a small parameter. The numerical method combines the implicit Crank–Nicolson scheme for the time  derivative on the uniform mesh and the central difference scheme for the spatial derivative on the Shishkin type meshes. It is shown to be second-order uniformly convergent in time and space. Then Richardson extrapolation technique is applied to enhance the accuracy from second-order to fourth-order. The error analysis is carried out, and the method is proved to be uniformly convergent. These two methods are applied to two test examples in support of the theoretical results.

Sixth-order compact finite difference method is presented for solving the one-dimensional KdV-Burger equation. First, the given solution domain is discretized using a uniform discretization grid point in a spatial direction. Then, using the Taylor series expansion, we obtain a higher-order finite difference discretization of the KdV-Burger equation involving spatial variables and produce a system of nonlinear ordinary differential equa-tions. Then, the obtained system of a differential equation is solved by using the fourth-order Runge–Kutta method. To validate the applicability of proposed techniques, four model examples are considered. The stability and convergent analysis of the present method is worked by using von Neumann stability analysis techniques by supporting the theoretical and mathematical statements in order to verify the accuracy of the present solution. The quality of the attending method has been shown in the sense of root mean square error L2 and point-wise maximum absolute error L∞. This is used to show, how the present method approximates the exact solution very well and how it is quite efficient and practically well suited for solving the KdV-Burger equation. Numerical results of considered examples are presented in terms of L2 and L∞ in tables. The graph of obtained present numerical and its exact solution are also presented in this paper. The present approximate numeric solvent in the table and graph shows that the numerical solutions are in good agreement with the exact solution of the given model problem. Hence the technique is reliable and capable for solving the one-dimensional KdV-Burger equation.

In 1984, Abaffy, Broyden, and Spediacto (ABS) introduced a class of the so-called ABS algorithms to solve systems of real linear equations. Later, the scaled ABS, the extended ABS, the block ABS, and the integer ABS algorithms were introduced leading to various well-known matrix factorizations. Here, we present a generalization of ABS algorithms containing all matrix factorizations such as triangular, W Z, and ZW . We discuss the octant interlocking factorization and make use of the generalized ABS algorithm as a more general approach for producing the octant interlocking factorization.

Differential equations of fractional order have been the focus of many studies due to their frequent appearance in various applications in fluid mechanics, biology, physics, and engineering. In general, it is not easy to derive the analytical solutions to most of these equations. Therefore, it is vital to develop some reliable and efficient techniques to solve fractional differential equations. A numerical method for solving fractional differential equations is proposed in this paper. The method is based on a hybrid of Block-pulse and orthonormal Bernstein functions. Convergence analysis is given, and numerical examples are introduced to illustrate the effectiveness and simplicity of the method.

A deterministic mathematical model of terrorism with government inter-vention was constructed from five compartments and subdivided into two core and non-core groups. A non-core group is a general group G(t), while the core group is susceptible S(t), moderate I(t), terrorism T (t), and re-covered R(t). The Elzaki transform method with differential transform to handle the nonlinear terms is employed to solve the model. The results show that government intervention on susceptible groups proved to be 90% effective in reducing terrorist threats since the group of susceptible in the population appears to be at risk of adopting the ideology through different means of contact. Also, due to the government intervention, the moderate group reduces gradually in time.

An adaptive spline is used in this work to deal with singularly perturbed boundary value problems with layers in the interior region. To evaluate the layer behavior in the solution, a different technique on a uniform mesh is designed by replacing the first-order derivatives with nonstandard differences in the adaptive cubic spline. A tridiagonal solver is used to solve the tridiagonal system of the difference scheme. The fourth-order convergence of the approach is established. The validity of the suggested computational method is demonstrated through numerical experiments, which are compared to other methods in the literature. Layer profile is depicted in graphs.

A new hybrid variational model is presented for image denoising, which in-corporates the merits of Shannon interpolation, total generalized variation (TGV) regularization, and a symmetrized derivative regularization term based on l1-norm. In this model, the regularization term is a combination of a TGV functional and the symmetrized derivative regularization term, while the data fidelity term is characterized by the l2-norm. Unlike most variational models that are discretized using a finite-difference scheme, our approach in structure is based on Shannon interpolation. Quantitative and qualitative assessments of the new model indicate its effectiveness in restoration accuracy and staircase effect suppression. Numerical experi-ments are carried out using the primal-dual algorithm. Numerous real- world examples are conducted to confirm that the newly proposed method outperforms several current state-of-the-art numerical methods in terms of the peak signal to noise ratio and the structural similarity (SSIM) index.

The crocodiles have a good strategy for hunting the fishes in nature. These creatures are divided into two groups of chasers and ambushers when hunt-ing. The chasers direct prey toward shallow water with a powerful splash of its tail without catching them, and the ambushers wait in the shallow and try to snatch the fishes. Such behavior inspires the development of a new population-based optimization algorithm called the crocodile hunting strategy (CHS). In order to verify the performance of the CHS, several classical benchmark functions and four constrained engineering design op-timization problems are used. In the classical benchmark function, the comparisons are performed using ant colony optimization, differential evo-lution, genetic algorithm, and particle swarm optimization. Constrained engineering design problems are compared with firefly algorithm, harmony search, shuffled frog-leaping algorithm, and teaching-learning-based opti-mization. The results of the comparison show that different operators de-signed in the CHS algorithm lead to fast algorithm convergence and show better results compared to other algorithms.

This study aims to present an accelerated derivative-free method for solving systems of nonlinear equations using a double direction approach. The approach approximates the Jacobian using a suitably formed diag-onal matrix by applying the acceleration parameter. Moreover, a norm descent line search is employed in the scheme to compute the optimal step length. Under the primary conditions, the proposed method’s global con-vergence is proved. Numerical results are recorded in this paper using a set of large-scale test problems. Moreover, the new method is successfully used to address the problem of Chandrasekhar’s integral equation problem appearing in radiative heat transfer. This method outperforms the existing Newton and inexact double step length methods.

A class of Bernstein-like basis functions, equipped with a shape param-eter, is presented. Employing the introduced basis functions, the corre-sponding curve and surface in rectangular patches are defined based on some control points. It is verified that the new curve and surface have most properties of the classical Bézier curves and surfaces. The shape parameter helps to adjust the shape of the curve and surface while the control points are fixed. We prove that the proposed Bézier-like curves can preserve monotonicity and that Bézier-like surfaces can preserve axial monotonicity. Moreover, the presented curves and surfaces preserve bound constraints implied by the original data.

The Tau method based on the Bernoulli polynomials is implemented efficiently to approximate the Nash equilibrium of open-loop kind in non-linear differential games over a finite time horizon. By this treatment, the system of two-point boundary value problems of differential game ex-tracted from Pontryagin’s maximum principle is transferred to a system of algebraic equations that Newton’s iteration method can be used for solving it. Also, for the mentioned approximation by the Bernoulli polynomials, the convergence analysis and the error upper bound are discussed. To demonstrate the applicably and accuracy of the proposed approach, some illustrated examples are presented at the final.

In this paper, a class of second-order singularly perturbed interior layer problems is examined. A nonpolynomial mixed spline is used to develop the tridiagonal scheme. The developed method is second as well as fourth-order accurate based on the parameters. Error analysis is also carried out. The method is shown to converge point-wise to the true solution with higher accuracy. Linear and nonlinear second-order singularly perturbed boundary value problems have been solved by the presented method. Five numerical illustrations are given to demonstrate the applicability of the proposed method. Absolute errors are given in tables, which show that our method is more efficient than previously existing methods.