Journal of Mathematical Modeling
Volume:8 Issue: 3, Spring 2020

In this paper, a $2n$-by-$2n$ circulant preconditioner  is introduced for a system of linear equations arising from discretization of the spatial fractional diffusion equations (FDEs). We show that the eigenvalues of our preconditioned system  are clustered around 1, even if the diffusion coefficients of FDEs are not constants. Numerical experiments are presented to demonstrate that the preconditioning technique is very efficient.

The present work sheds new light on the computation of the heat distribution on the boundary of the human eye. Due to different values of the thermal conductivity on each region of the human eye, the domain decomposition technique is introduced and an optimization formulation is analysed and studied to derive a proposed algorithm. All obtained partial differential equations are approached by discontinuous dual reciprocity boundary element method. The validity of the proposed approaches is confirmed by comparing to results reported with previous experimental and numerical studies.

The relationship among vehicles on the road is  modeled using fundamental traffic equations. In traffic modeling, a   particular speed-density equation  usually fits a peculiar dataset. The study seeks to parameterize some existing fundamental models so that a given equation could match different  dataset. The new  equations are surmisal offshoots from existing equations.    The parameterized equations are  used in the LWR model and  solved  using the Lax-Friedrichs differencing scheme. The simulation results illustrate different scenarios of acceleration and deceleration traffic  wave profiles.  The proposed  models appropriately explain the varying transitions of  different traffic regimes.

In this article, we study some existence, uniqueness and Ulam type stability results for a class of boundary value problem for nonlinear fractional integro--differential equations with positive constant coefficient involving the Caputo fractional derivative. The main tools used in our analysis is based on Banach contraction principle, Schaefer's fixed point theorem and Pachpatte's integral inequality. Finally, results are illustrated with suitable example.

The considered stochastic travelling salesman problem is defined where the costs are distributed exponentially. The costs are symmetric and they satisfy the triangular inequality. A discrete time Markov chain is established in some periods of time. A stochastic tour is created in a dynamic recursive way and the best node is detected to traverse in each period. Then, a simulated annealing based heuristic method is applied to select the best state. All the nodes should be traversed exactly once. An initial $rho$-approximate solution is applied for some benchmark problems and the obtained solutions are improved by a simulated annealing heuristic method.

This paper introduces an efficient numerical scheme  for solving a significant class of  nonlinear parabolic integro-differential equations (PIDEs). The major contributions made in this paper are applying a direct approach based on a combination of   group preserving scheme (GPS) and  spectral meshless radial point interpolation (SMRPI) method   to transcribe the partial differential problem under study into a system of ordinary differential equations (ODEs). The resulting problem is then solved by employing the numerical method of lines, which is also a well-developed numerical method.   Two numerical experiments are carried out to evaluate the performance and effectiveness of the suggested framework.

Recovering sparse signals from linear measurements has demonstrated outstanding utility in a vast variety of real-world applications.  Compressive sensing is the topic that studies the associated raised questions for the possibility of a successful recovery. This topic is well-nourished and numerous results are available in the literature. However, their dispersity makes it  time-consuming for  practitioners to quickly grasp its main ideas and classical algorithms, and further touch upon the recent advancements. In this survey, we overview vital classical  tools and algorithms in compressive sensing and describe its significant recent advancements. We conclude  by a numerical comparison of the performance of described approaches.