Partial Differential Equations in Applied Mathematics最新文献

This paper aims at finding the numerical approximation of a classical Blasius flat plate problem using spectral collocation method. This technique is based on Chebyshev pseudospectral approach that involves the solution is approximated using Chebyshev polynomials, which are orthogonal polynomials defined on the interval [−1, 1]. The Chebyshev pseudospectral method employs Chebyshev- Gauss- Lobatto points, the extrema of the Chebyshev polynomials. The differential equation is approximated as a sum of Chebyshev polynomials. A differentiation matrix, based on these polynomials and their derivatives at the collocation points, transforms the differential equation into a system of algebraic equations. By evaluating the differential equation at these points and applying boundary conditions, the original boundary value problem reduced the solution to the solution of a system of algebraic equations. Solving for the coefficients of the polynomials yields the numerical approximation of the solution. The implementation of this method is carried out in Mathematica and its validity is ensured by comparing it with a built in MATLAB numerical routine called bvp4c.

The elasticity theory problem for a radially inhomogeneous cylinder of small thickness, whose elastic moduli are arbitrary continuous functions depending on the radius of the cylinder, is considered. It is assumed that the side surface of the cylinder is stress-free, and the boundary conditions that keep the cylinder in equilibrium are given at its seats. Asymptotic solutions are constructed by the asymptotic integration method. It is shown that the asymptotic solution consists of the sum of the penetrating solution, simple boundary effect and boundary layer solutions. The character of the stress-strain state corresponding to the penetrating solution, simple boundary effect and boundary layer solutions is determined. Asymptotic formulas are obtained for displacements and stresses, which allow to calculate the stress-strain state of a cylinder.

The problem of torsion of a radially inhomogeneous cylinder is studied, with its lateral surface free from stress and the boundary conditions keeping it in equilibrium at its seats. By applying the asymptotic integration method, it is determined that the asymptotic solution for the torsion problem consists of the sum of the penetrating solution and boundary layer solutions.

Numerical analysis is performed and the effect of material inhomogeneity on the stress-strain state of the cylinder is evaluated.

The modified Helmholtz equation $q_{xx}+q_{yy}-4{\beta }^{2}q=0$, is one of the basic equations of classical mathematical physics. In this paper we have obtained the solution of the boundary-value problems for the modified Helmholtz equation in an equilateral triangle. An additional mixed boundary condition related to the symmetry of the solution is taken into consideration. We have analysed the Global relation and only used the algebraic techniques to obtain the explicit solution of modified Helmholtz equation bypassing the Riemann Hilbert approach. This solution is applied to the problem of diffusion-limited coalescence, $A+A\rightleftharpoons A$.

The Nipah virus (NiV) is one of the most lethal viruses which can infect humans and lead to fatal encephalitis. The recent significant awareness of NiV is due to its elevated death rate and effective transmission capabilities among humans. With its recurrent outbreaks and exceptionally high mortality rate, the NiV infections have emerged as one of the most concerning hazards to public health. The exploration of NiV and its characteristics revealed that NiV has two distinct strains, namely, the Malaysia (NiVM) strain and the Bangladesh (NiVB) strain. In this paper, we propose a novel Caputo fractional order mathematical model to simulate the dynamics of the two strains NiV. The positivity and boundedness of the model’s solutions are investigated. The existence and asymptotic stability of the equilibrium points of the model are examined. The analysis of basic reproduction number is presented to determine whether the infection will die out or persist. The effects of key parameters of the model on its dynamical behaviors are also explored. Finally, efficient numerical technique is used to confirm the analytical results through detailed numerical simulations.

This study presents a predator-prey model within a food web, incorporating the impact of fear responses in prey and the effects of harvesting. The analysis begins by exploring equilibrium points and assessing the boundedness and uniqueness of the model's solutions. The local stability of each of the seven possible equilibrium points is then evaluated. To determine global stability, a suitable Lyapunov function is utilized. Finally, numerical simulations are carried out across different parameter values, with graphical representations provided to support the analytical findings.

In this study, we introduce a computational algorithm for solving Integro-Differential Equations (IDEs) using Bernstein polynomials as basis functions. The algorithm approximates the solution by expressing it in terms of Bernstein polynomials and substituting this assumed solution into the IDE. Collocating the resulting equation at evenly spaced points yields a system of linear algebraic equations, which is solved via matrix inversion to find the Bernstein coefficients. These coefficients are then used to construct the approximate solution. Numerical examples demonstrate the method's accuracy and efficiency, highlighting its advantages in reducing computational effort.

The present manuscript relates to the movement of three immersed viscous magnetic liquids in porous media that are positioned between three concentric cylindrical interfaces. Every cylindrical layer has an axial extension that extends in both vertical directions to infinity. Moreover, the pressure gradients are the driving force behind the uniform motion of all fluids in a similar upward motion at varying velocities. Permanent magnetic fields are tangentially oriented exert stress on the system. The computations are rendered simpler by applying the viscous potential theory. The Maxwell equations are utilized for the magnetic field, while the Brinkman-Darcy equations define the fluid mobility. Moreover, the rise of the disturbance and the subsequent extraction of fuel from the tiny cracks of the reservoir stone can be managed through the implementation of engineering techniques such as electromagnetic field impacts and oil extraction engineering. Typically, the nonlinear strategy involves incorporating the applicable nonlinear boundary conditions and analyzing the linearized equations of motion. This research offers a framework for verifying theoretical models and simulations based on experimental observations. The inclusion of a homogeneous magnetic field introduces complexity to the system, rendering it a suitable candidate for validating and improving models in the field Magneto hydrodynamics. The originality of the problem lies in the dual nonlinear stability of cylindrical interfaces when subjected to uniform magnetic field. Accordingly, two nonlinear characteristic differential equations controlling the surface displacements are produced. The nonlinear stability prerequisite is met by applying the matrix concept and the multiple scale technique in conjunction with a theoretical analysis of stability. Additionally, the Routh-Hrutwitz criterion is encompassed to judge the stability cofiguration. A detailed examination of the associated nonlinear stability requirements is showed. Meanwhile, the estimated limited solutions of perturbed surfaces are achieved. For the cylindrical middle layer, it is found that the outer cylindrical interface has a more stabilizing effect than the inner one. The approximate solutions for displacements at the interface are calculated. The influence of Weber numeral of the problem on the stability profile is investigated.

This paper presents a mathematical model aimed at comprehensively understanding the thermal processes associated with arc erosion of closed electrical contacts triggered by the instantaneous explosion of micro-asperities including Thomson effect and Joule heat source. The phenomenon involves vaporization and liquid zones, and the temperature distribution is governed by a generalized heat equation, accounting for the heating effect due to current flow in temperature gradients and effect of heat source. The proposed model allows for a nuanced analysis of the thermal dynamics, shedding light on the complex phenomena involved in arc erosion. The proposed method in this paper employs similarity transformation techniques to effectively reduce the complexity of the problem, transforming it into a set of manageable ordinary differential equations. The study establishes the existence and uniqueness of the solution through rigorous analysis. The behavior of the solution of the problem is successfully considered for special cases of Thomson and thermal coefficients. By leveraging similarity transformations, the paper offers a powerful approach for unraveling the intricacies of the thermal processes in arc erosion of closed electrical contacts, providing valuable insights into the phenomena associated with current-carrying heating in the presence of temperature gradients.

A rotating system that includes an initially stressed piezo-electric substrate was the subject of an analytical study of the SH waves. The substrate and vacuum bonding are in good contact with the corrugation, and the top border is treated as stress-free corrugation. About the upper barrier that is both electrically open as well as electrically short scenarios are taken into consideration. Dispersion equations for circumstances with a corrugated interface that are both electrically short as well as electrically open have been found. The SH wave characteristics through the framework suggested and circumstances dependent on different geometrical and physical factors have been investigated based on the numerical findings. The study looks at the simultaneous simulated outcomes of a number of physical factors that were produced in Mathematica 7 and include rotation, inhomogeneity, phase velocity, initial stress, and corrugated interface of SH wave distribution in a structure under discussion. The model under examination has potential applications in the creation of surface acoustic wave devices.

In this work, we are concerned with a nonlinear wave equation with variable exponents. In the presence of the logarithmic nonlinear source, we established a global nonexistence result with negative initial data and without imposing the Sobolev Logarithmic Inequality. The blow-up time is established with upper bound and lower bound. In addition, under some conditions on the initial data and for a specific class of relaxation functions, we established an infinite time blow-up result.