Journal of Mathematical Modeling
Volume:10 Issue: 2, Spring 2022

In this paper, an efficient finite difference method is presented for solving singularly perturbed linear second order parabolic problems with large time lag. The comparable numerical model is related to automatically controlled system with spatial diffusion of reactants  in the processes. This study focuses on the formation of boundary layer behavior or oscillatory behaviors due to the presence of delay parameters and perturbation parameter. The numerical scheme comprising an exponentially fitted spline based difference scheme on a uniform mesh supported by Crank-Nicolson Method is constructed. It is found that the present method converges with second order accurate in both temporal and spatial variables. The convergence analysis and running time of the program with varied grid sizes are then used to do the efficiency analysis. The proposed scheme  accuracy and efficiency are also demonstrated through numerical experiments.

This paper is concerned with the existence of at least one   positive solution for a boundary value problem (BVP), with  $p$-Laplacian, of the form    begin{equation*}        begin{split}            (Phi_p(x^{'}))^{'} + g(t)f(t,x)  &= 0, quad t     in (0,1),\            x(0)-ax^{'}(0) = alpha[x], & quad            x(1)+bx^{'}(1) = beta[x],        end{split}    end{equation*}where $Phi_{p}(x) = |x|^{p-2}x$ is a one dimensional $p$-Laplacian operator with $p>1, a,b$ are real constants and $alpha,beta$ are  the Riemann-Stieltjes integrals    begin{equation*}        begin{split}            alpha[x] = int limits_{0}^{1} x(t)dA(t), quad  beta[x] = int limits_{0}^{1} x(t)dB(t),        end{split}    end{equation*}with $A$ and $B$ are functions of bounded variation. A Homotopy version of  Krasnosel'skii fixed point theorem is used to prove our results.

In the construction of efficient numerical methods for the stiff initial value problems, some second derivative multistep methods have been introduced equipping with super future point technique. In this paper, we are going to introduce a formula for the stability functions of a class of such methods. This group of methods encompasses SDBDF methods and their extensions with advanced step-point feature. This general formula, instead of obtaining the distinct stability functions for each of methods, will facilitate stability analysis of the methods.

The problem of constructing a matrix by its spectral information is called inverse eigenvalue problem (IEP) which arises in a variety of applications. In this paper, we study an IEP for arrowhead matrices in different cases. The problem involves constructing of the matrix by some eigenvalues of each of the leading principal submatrices and one eigenpair. We will also investigate this problem and its variants in the cases of matrix entries being real, nonnegative, positive definite, complex and equal diagonal entries. To solve the problems, a new method to establish a relationship between the IEP and properties of symmetric and general form of matrices is developed. The necessary and sufficient conditions of the solvability of the problems  are obtained. Finally, some numerical examples are presented.

Data Envelopment Analysis (DEA) is a non-parametric method for measuring the relative efficiency of peer decision-making units (DMUs), where the internal structures of DMUs are treated as a black box. Traditional DEA models do not pay attention to the internal structures and intermediate values. Network data envelopment analysis models addressed this shortcoming by considering intermediate measure. The results of two-stage DEA model not only provides an overall efficiency score for the entire process, but also yields an efficiency score for each of the individual stages. The centralized model has been widely used to evaluate the efficiency of two-stage systems, but the allocation problem of shared inputs and undesirable outputs has not been considered.  The aim of this paper is to develop a method based on bargaining for evaluation in network DEA considering shared inputs and undesirable outputs. The two stages are considered as players to bargain for a better payoff, which is offered by DEA ratio efficiency score of DMUs. The efficiency model is developed as a cooperative game model. Finally, a numerical example is given to evaluate the proposed model.

This paper provides a numerical solution for the fractional multi-compartmental models which are applied in pharmacokinetics. We implement the local discontinuous Galerkin method for these fractional models with the upwind numerical fluxes. To obtain high-order results with adequate accuracy, the third-order approximation polynomials are used. Finally, to validate the scheme, the results are compared with the solutions of a semi-analytical method.

In this paper, we tackle two important problems in low-rank learning, which are partial singular value decomposition  and numerical rank estimation of huge matrices. By using the concepts of Krylov subspaces such as  Golub-Kahan  bidiagonalization (GK-bidiagonalization) as well as Ritz vectors, we propose two methods for solving these problems in a fast and accurate way. Our experiments show the advantages of the proposed methods compared to the traditional and randomized singular value decomposition methods. The proposed methods are appropriate for applications involving huge matrices where the accuracy of the desired singular values and also all of their corresponding singular vectors are essential. As a real application, we evaluate the performance of our methods on the problem of Riemannian similarity learning between two different image datasets of MNIST and USPS.

This research describes a new fifth-order finite difference symmetrical WENO-Z scheme for solving Hamilton-Jacobi equations. This method employs the same six-point stencil as the original fifth-order WENO scheme (SIAM J. Sci. Comput. 21 (2000) 2126--2143) and a new WENO scheme recently proposed (Numer. Methods Partial Differential Eq. 33 (2017) 1095--1113), and could generate better results and create the same order of accuracy in smooth area without loss of accuracy at critical points simultaneously avoiding incorrect oscillations in the vicinity of the singularities. The new reconstruction is a convex combination of a fifth-order linear reconstruction and three third-order linear reconstructions. We prepare a detailed analysis of the approximation order of the designed WENO scheme. Some benchmark tests in 1D, 2D and 3D are performed to display the capability of the scheme.

In this paper, a wavelet-based numerical algorithm is described to obtain approximate numerical solution of a class of nonlinear Fredholm integral equations of second kind having smooth kernels. The algorithm involves  approximation of the unknown function in terms of Daubechies scale functions. The properties of Daubechies scale and wavelet functions together with one-point quadrature rule for the product of a smooth function and Daubechies scale functions are utilized to transform the integral equation to a system of nonlinear equations. The efficiency of the proposed method is demonstrated through three illustrative examples.

We study the combination of the Sinc and the Gaussian radial basis functions (GRBF) to develop the numerical methods for the time--space fractional diffusion equations with the Riesz fractional derivative. The GRBF is used to approximate the unknown function in spatial direction and the Sinc quadrature rule associated with double exponential transformation is applied to approximate the arising integrals.  Three practical examples are considered for testing the ability of the proposed method.

In this paper, the existence and the Ulam-Hyers stability of solutions for the implicit second-order  differential equations are investigated via fractional-orders integral boundary conditions by direct application of the Banach contraction principle.  Finally, we  present some particular cases and two examples to illustrate our results.

The main target of this paper is to solve a system of two-dimensional Volterra integral equations (2-DVIEs). Operational Matrices of two-dimensional hybrid of block-pulse functions and Legendre polynomials are applied to reduce these systems of integral equations to a system of algebraic equations. The main benefit of these basic functions is their efficiency in dealing with non-sufficiently smooth functions. An error bound is provided and some examples are prepared to verify the applicability of the offered numerical technique.