Iranian Journal of Numerical Analysis and Optimization
Volume:14 Issue: 2, Spring 2024

In the present paper, we precisely conduct a quantum calculus method for the numerical solutions of PDEs. A nonlinear Schrödinger equation is considered. Instead of the known classical discretization methods based on the finite difference scheme, Adomian method, and third modified ver-sions, we consider a discretization scheme leading to subdomains according to q-calculus and provide an approximate solution due to a specific value of the parameter q. Error estimates show that q-calculus may produce effi-cient numerical solutions for PDEs. The q-discretization leads effectively to higher orders of convergence provided with faster algorithms. The numer-ical tests are applied to both propagation and interaction of soliton-type solutions.

In this article, we explore the discontinuous Galerkin finite element method for two-parametric singularly perturbed convection-diffusion problems with a discontinuous source term. Due to the discontinuity in the source term, the problem typically shows a weak interior layer. Also, the presence of multiple perturbation parameters in the problem causes boundary layers on both sides of the boundary. In this work, we develop the nonsymmetric discontinuous Galerkin finite element method with interior penalties to handle the layer phenomenon. With the use of a typical Shishkin mesh, the domain is discretized, and a uniform error estimate is obtained. Numerical experiments are conducted to validate the theoretical conclusions.

In this paper, we introduce second derivative multistep collocation meth-ods for the numerical integration of ordinary differential equations (ODEs). These methods combine the concepts of both multistep methods and col-location methods, using second derivative of the solution in the collocation points, to achieve an accurate and efficient solution with strong stability properties, that is, A-stability for ODEs. Using the second-order deriva-tives leads to high order of convergency in the proposed methods. These methods approximate the ODE solution by using the numerical solution in some points in the r previous steps and by matching the function values and its derivatives at a set of collocation methods. Also, these methods utilize information from the second derivative of the solution in the colloca-tion methods. We present the construction of the technique and discuss the analysis of the order of accuracy and linear stability properties. Finally, some numerical results are provided to confirm the theoretical expecta-tions. A stiff system of ODEs, the Robertson chemical kinetics problem, and the two-body Pleiades problem are the case studies for comparing the efficiency of the proposed methods with existing methods.

This manuscript puts forward two new generalized families of Jarratt’s iterative schemes for deciding the solution of scalar and systems of non-linear equations. The schemes involve weight functions that are based on bi-variate rational approximation polynomial of degree two in both its numerator and denominator. The convergence study conducted on the schemes, indicated that they have convergence order (CO) four in scalar space and retain the same number of CO in vector space. The numerical experiments conducted on the schemes when used to decide the solutions of some real-life nonlinear models show that they are good challengers of some well-known and robust existing iterative schemes.

Software testing is a crucial step in the development of software that guar-antees the dependability and quality of software products. A crucial step in software testing is test case minimization, which seeks to minimize the number of test cases while ensuring maximum coverage of the system being tested. It is observed that the existing algorithms for test case minimization still suffer in efficiency and precision. This paper proposes a new optimiza-tion algorithm for efficient test case minimization in software testing. The proposed algorithm is designed on the base parameters of the metaheuristic algorithms, inspired by scientific principles. We evaluate the performance of the proposed algorithm on a benchmark suite of test cases from the literature. Our experimental results show that the proposed algorithm is highly effective in reducing the number of test cases while maintaining high coverage of the system under test. The algorithm outperforms the existing optimization algorithms in terms of efficiency and accuracy. We also con-duct a sensitivity analysis to investigate the effect of different parameters on the performance of the proposed algorithm. The sensitivity analysis results show that the performance of the algorithm is robust to changes in the parameter values. The proposed algorithm can help software testers reduce the time and effort required for testing while ensuring maximum coverage of the system under test.

We derive a deterministic mathematical model that scrutinizes the dy-namics of cholera pathogen carriers and the hygiene consciousness of in-dividuals, before the illness, during its prevalence, and after the disease’s outbreaks. The dynamics can effectively help in curtailing the disease, but its effects had less coverage in the literature. Boundedness of the solu-tion of the model, its existence, and uniqueness are ascertained. Effects of cholera pathogen carriers and hygiene consciousness of individuals in controlling the disease or allowing its further spread are analyzed. The differential transformation method is used to obtain series solutions of the differential equations that make the system of the model. Simulations of the series solutions of the model are carried out and displayed in graphs. The dynamics of the concerned state variables and parameters in the model are interpreted via the obtained graphs. It is observed that higher hygiene consciousness of individuals can drastically reduce catching cholera disease at onset and further spread of its infections in the population, this in turn, shortens the period of cholera epidemic.

Atherosclerosis is one of the most common diseases in the world. Med-ication with metal stents plays an important role in treating this disease. There are many models for delivering drugs from stents to the arterial wall. This paper presents a model that describes drug delivery from the stent coating layers to the arterial wall tissue. This model complements the previous models by considering the mec hanical properties of the arte-rial wall tissue, which changes due to atherosclerosis and improves results for designing stents. The stability behavior of the model is analyzed, and a number of numerical results are provided with explanations. A compar-ison between numerical and experimental results, which examine a more accurate match between the in vivo and in vitro, is shown.