Partial Differential Equations in Applied Mathematics最新文献

This study introduces a novel analytical framework to explore the effects of Caputo spatial and temporal memory indices combined with a proportional time delay on (non)linear $(1+2)$-dimensional evolutionary models. The solution is expressed as a Cauchy product of an absolutely convergent series that effectively captures the dynamics of these parameters. By extending the differential transform method into higher-dimensional fractional space, we reformulate the evolution equation as a (non)linear higher-order recurrence relation, which enables the precise determination of fractional series coefficients. Our findings show that Caputo derivatives and time delay significantly impact the system’s behavior, with graphical analysis revealing a continuous transition from a stationary to an integer state solution. The study also identifies a quantitative analogy between the Caputo-time fractional derivative and proportional time delay that highlights the role of Caputo derivatives as memory indices. This method has proven highly effective in deriving solutions for fractional higher-dimensional extensions of evolutionary equations.

The stochastic SEIR model was employed to investigate the dynamics of influenza transmission. By incorporating transmission rates and prevalence ratios, this model provides the most comprehensive explanation of the virus’s unpredictable dissemination. To simulate the stochastic components of influenza transmission, we implemented conventional Brownian motions and stochastic differential equations. The investigation examines the uniqueness and presence of the solutions to demonstrate the conditions needed for eliminating the infection under random disturbances. The transmission rate coefficient ($\beta$) strongly impacts disease transmission speed. as demonstrated by the simulation results.Thus, the proper usage of safe transmission control methods is another decisive factor that determines the outcome of epidemics. Actual data of the Kingdom of Saudi Arabia was used. The results highlighted practicality of stochastic models and their usefulness to address and formulate and even execute the public health related policies. Regarding this, this study sets a high bar for other studies on modeling viral diseases on the grounds that stochastic and dynamic factors are also very important. These subsequent improvements in the model shall enable us to pinpoint the best strategies for the prevention and eradication of influenza and any other subsequent epidemic diseases, with reference to epidemic, epidemiology and public health.

The current research examines the combined effects of solar energy and thermal buoyancy around a laminar jet placed in a porous medium. The governing boundary layer equations are dimensionalized by using appropriate dimensionless variables. The numerical solution of the dimensionless boundary layer equations is obtained using the finite difference method. The impact of physical parameters, which are Darcy number, dimensionless porous medium inertia coefficient, Prandtl number, radiation parameter, and dimensionless fluid's absorption parameter, on velocity and temperature profile, is shown graphically, while the influence of the above parameters on the heat transfer rate is presented in tabular form. It is keenly observed that for Darcy number velocity profile decreases while reverse behavior is noted for temperature distribution. For the dimensionless radiation parameter, both the velocity and temperature profile decrease. The main novelty of the current work is to improve the thermal performance of natural convection heat transfer system in the presence of thermal radiation placed in porous medium.

Intraspecific mutualism is vital for ecosystems. Some interactions involve one species benefiting without reciprocating. Recent experiments suggest stable co-existence between mutualists and cheaters. This paper focuses on interactions between obligate mutualistic species and their cheaters, using a modified Lotka-Volterra model to analyze the negative impact of cheaters. We establish conditions for stability and verify a diffusive system's stability under spatial effects. Additionally, we study population dynamics fluctuations in the presence of Gaussian noise.

In this paper, an improvement for the ${\varphi }^6$-model expansion method is presented. In this approach, contrary to the classical ${\varphi }^6$-model expansion method, obtaining explicit solutions for nonlinear ordinary and partial differential equations is congenial and undemanding of any constraint conditions, where the method can be applied and used for obtaining solutions without having any conditions on them. Moreover, the new approach is used to obtain new solutions for the new $(3+1)$-dimensional integrable Kadomtsev–Petviashvili equation. We demonstrated that for the same equation, the classical ${\varphi }^6$-model expansion and the improved ${\varphi }^6$-model expansion approaches produce the same family of solutions. However, the improved ${\varphi }^6$-model expansion method is found to be more efficient and convenient.

The behavior of nonlinear waves within a modified Zakharov–Kuznetsov equation and their interactions with discrete electric lattice structures are examined in this study. The ${\varphi }^6-$model expansion method is utilized to acquire substantial knowledge into the complex dynamics of the system under consideration, particularly with regard to the discrete electric lattice and analytical electrical solitons. By incorporating higher-order effects and improving accuracy in representing specific physical conditions, the study achieves a more realistic portrayal of nonlinear wave dynamics. The investigation also sheds light on the relationship between non-linearity, discreteness, and equation dynamics by exploring the conditions that lead to the formation of solitons and other nonlinear structures. In addition, a unique set of electrical solitons is defined to explore dynamic behaviors such as chaotic, quasi-periodic, and periodic motions under various parameterized conditions, including an external damping force. Phase plane analysis is visualized by using dynamic structure 3D and 2D phase plots, is used for bifurcation and sensitivity inspections. Finally, time series graphs are offered as mathematical depictions of solitary waves, and Lyapunov exponents with real and complex eigenvalues are used to study the stability and chaotic behaviors of the system.

In this article, we explore several significant nonlinear physical models, including the Benjamin–Bona–Mahony–Peregrine–Burgers (BBMPB) equation, the Burgers–Korteweg–De Vries (BK) equation, the one-dimensional Oskolkov (OSK) equation, the Klein–Gordon (KG) equation with quadratic non-linearity, and the improved Boussinesq (IB) equation. Utilizing the $\left(\frac{G^{\prime }}{G}\right)$-expansion method ansatz, we derive new exact traveling wave solutions for these models. These solutions, expressed in the forms of rational, hyperbolic, and trigonometric functions, present a novel contribution distinct from existing literature. The physical dynamics of these solutions are elucidated through Mathematica simulations.

In this paper, we proposed an accurate $ϵ$-uniformly convergent numerical method to solve singularly perturbed time-fractional convection–diffusion equations via exponential fitted operator scheme. The time-fractional derivative is defined in the sense of Caputo with order $\eta \in (0,1)$. The time-fractional derivative is discretized by employing the Crank–Nicolson method on a uniform mesh, and an exponential fitted operator scheme along with the standard upwind method is used to mesh-grid the space domain. The truncation error and uniform stability of the discretized problems are examined in order to prove the parameter uniform convergence of the proposed scheme. It is demonstrated that the scheme is $ϵ$-uniformly convergent of order $O({\left(\Delta t\right)}^{2-\eta }+\Delta x),$ where $\Delta t$ and $\Delta x$ represent the step sizes of the time and space domains, respectively. Two numerical examples are provided in order to assess the accuracy of the suggested scheme and validate the theoretical concepts discussed. To demonstrate the efficiency of the numerical scheme presented, comparisons have been made with the numerical solution obtained by the finite difference method that exists in the literature. Consequently, it is observed that the results obtained by the present scheme are more accurate and have a better convergence rate.

In this study, the Perturbation Iteration transform method, namely PITM, is in short presented and implemented for solving a class of fractional integro-differential equations. The fractional derivative will be in the Atangana–Baleanu Caputo fractional derivative sense (ABC). The (PITM) is consists of merging Laplace transform method and the perturbation iteration algorithm (PIM). The proposed method furnish the solution in the form of a fastly convergent series. Some illustrative examples are presented to illustrate that the PITM is a powerful, efficient and accurate method and it can be enforced to other nonlinear problems.

This research paper focuses on the analysis of a discrete FitzHugh–Nagumo reaction–diffusion system. We begin by discretizing the FitzHugh–Nagumo reaction–diffusion model using the second-order and L1-difference approximations. Our study examines the local stability of the equilibrium points of the system. To identify conditions that ensure the global asymptotic stability of the steady-state solution, we employ various techniques, with a primary focus on the direct Lyapunov method. Theoretical results are supported by numerical simulations that demonstrate the practical validity of the asymptotic stability conclusions. Our findings provide new insights into the stability characteristics of discrete FitzHugh–Nagumo reaction–diffusion systems and contribute to the broader understanding of such systems in mathematical biology.