Journal of Mathematical Modeling
Volume:12 Issue: 1, Winter 2024

The main objective of this paper is to introduce the fourth and sixth-order compact finite difference methods for solving anti-periodic boundary value problems. Compact finite difference formulas can approximate the derivatives of a function more accurately than the standard finite difference formulas for the same number of grid points. The convergence analysis of the proposed method is also investigated. This analysis shows how the error between the approximate and exact solutions decreases as the grid space is reduced.  To validate the proposed method's accuracy and efficiency, some computational experiments are provided. Moreover, a comparison is performed between the standard and compact finite difference methods.  The experiments indicate that the compact finite difference method is more accurate and efficient than the standard one.

In this article, we propose an approximate technique for reconstructing a time-dependent reaction coefficient together with the surface heat flux histories and temperature distribution in a nonlinear inverse heat conduction problem (IHCP). We assume that the initial condition and the transient heat flux on the accessible boundary along with the temperature measured at specified interior locations in the domain of the problem are given as the input data. By applying the given measurements in a transformation, the main problem is reformulated as a certain parabolic problem and later a procedure based upon deploying the Ritz approximation along with the collocation method is applied which converts the problem to a nonlinear system of algebraic equations. Accurate numerical results in dealing with the exact initial and boundary data are obtained and regarding the perturbed boundary data, the regularization method based on cubic spline approximation is used, which results in obtaining stable numerical derivatives.

A class of two-parameter singularly perturbed nonlinear second order ordinary differential equations is considered in this article. A fitted mesh method which is a combination of finite difference scheme and a Shishkin mesh is developed to solve the problems. The method is proved to be essentially first order parameter independent convergent. Numerical experiments support the established theoretical results.

A mathematical analysis is performed for a system consisting of two coupled Cahn-Hilliard equations. These equations incorporate a diffusional mobility that depends on concentration. This modeling approach is often used to describe the process of phase separation in a thin layer of a binary liquid mixture covering a substrate, particularly when one of the components wets the substrate. The analysis establishes the existence of a weak formulation for this problem, which is supported by the use of a Lyapunov functional. Additionally, the analysis provides insights into the regularity properties of the weak formulation.

Due to the importance of the generalized nonlinear Klein-Gordon equation (NL-KGE) in describing the behavior of light waves and nonlinear optical materials, including phenomena such as optical switching by manipulating the dispersion and nonlinearity of optical fibers and stable solitons,  we explain the approximation of this model by evaluating different classical and fractional terms  in this paper. To estimate the fundamental function, we use a first-order finite difference approach in the temporal direction and a collocation method based on Gegenbauer polynomials (GP) in the spatial direction to solve the NL-KGE model. Moreover, the stability and convergence analysis is proved by examining the order of the new method in the time direction as $\mathcal{O}( \delta t )$. To demonstrate the efficiency of this design, we presented numerical examples and made comparisons with other methods in the literature.

We are interested in the numerical solution of the continuous-time Lyapunov equation. Generally, classical Krylov subspace methods for solving matrix equations use the Petrov-Galerkin condition to obtain projected equations from the original ones. The projected problems involves the restrictions of the coefficient matrices to a Krylov subspace. Alternatively, we propose a scheme based on the extended block Krylov subspace that leads to a smaller-scale equation, which also incorporates  the restriction of the inverse of the Lyapunov equation's square coefficient. The effectiveness of this approach is experimentally confirmed, particularly in terms of the required CPU time.

In this research work, a fifth-order weighted essentially non-oscillatory (WENO) scheme is created for traffic flow problems on networks. Street systems can be numerically demonstrated as a graph, whose edges are a limited number of streets that connect at intersections. A scalar hyperbolic conservation law can portray the advancement on each street, and traffic distribution matrices are considered to define coupling conditions at the network intersections. In this paper, numerical results for road networks with rich solution structures will be presented. These numerical results show that the new proposed scheme in this paper can generate essentially non-oscillatory and high resolution solutions.

This paper uses unit lower triangular matrices to solve the nonnegative inverse eigenvalue problem  for various sets of real  numbers. This problem  has remained unsolved for many years for $n \geq 5.$  The inverse of the unit lower triangular matrices can be easily calculated and the matrix similarities are also helpful to be able to solve this important problem to a considerable extent. It is assumed that in the given set of eigenvalues, the number of positive eigenvalues is less than or equal to the number of nonpositive  eigenvalues to find a nonnegative matrix such that the given set is its spectrum.

The use of preconditioning techniques has been shown to offer significant advantages in solving multi-linear systems involving nonsingular $\mathcal{M}$-tensors. In this paper, we introduce a new preconditioner that employs $(I+P)$-like preconditioning techniques, and give the proof of its convergence. We also present numerical examples and comparison results that demonstrate the superior efficiency of our preconditioner compared to both the original SOR method and the previously proposed preconditioned SOR method.

In this paper, we introduce the concept of Moore-Penrose inverse of a rectangular interval matrix based on a modified interval arithmetic. We determine the Moore-Penrose inverse in such a way that it satisfies all the four criteria similar to the real case. Also, we use the Moore-Penrose inverse for solving rectangular interval linear systems, algebraically.

This paper presents a numerical method for a class of singularly perturbed parabolic partial differential equations with integral boundary conditions (IBC). The solution to the considered problem exhibits pronounced boundary layers on both the left and right sides of the spatial domain. To address this challenging problem, we propose the use of the implicit Euler method for time discretization and a finite difference method on a well-designed piecewise uniform Shishkin mesh for spatial discretization. The integral boundary condition is approximated using Simpson's $\frac{1}{3}$ rule. The presented method demonstrates almost second-order uniform convergence in the discretization of the spatial derivative and first-order convergence in the discretization of the time derivative. To validate the applicability and accuracy of the proposed method, two illustrative examples are employed. The computational results not only accurately reflect the theoretical estimations but also highlight the method's effectiveness in capturing the intricate features of singularly perturbed parabolic partial differential equations with integral boundary conditions.

Considering multiple criteria and objectives simultaneously in a single real-world problem and the fuzzy nature of this type of problem is of particular importance and application. As a result,  multi-objective interval type-2 fuzzy linear programming problems have received much attention. However,  there are few and a limited number of methods available for solving multi-objective interval type-2 triangular fuzzy linear programming problems with ambiguity-type imprecision (interval type-2 triangular fuzzy numbers) in almost all of the problem parameters. This research first considers a multi-objective interval type-2 fuzzy linear programming problem with ambiguity in all coefficients,  in which,  all problem coefficients are interval triangular fuzzy numbers. In addition,  using the weighted sum method and the concept of nearest interval approximation,  the problem is solved,  and an example is provided.