Approximate Controllability and Ulam Stability for Second-Order Impulsive Integrodifferential Evolution Equations with State-Dependent Delay
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Allen Cahn (AC) equation is highly nonlinear due to the presence of cubic term and also very stiff; therefore, it is not easy to find its exact analytical solution in the closed form. In the present work, an approximate analytical solution of the AC equation has been investigated. Here, we used the variational iteration method (VIM) to find approximate analytical solution for AC equation. The obtained results are compared with the hyperbolic function solution and traveling wave solution. Results are also compared with the numerical solution obtained by using the finite difference method (FDM). Absolute error analysis tables are used to validate the series solution. A convergent series solution obtained by VIM is found to be in a good agreement with the analytical and numerical solutions.
Conformable Fractional-Order Modeling and Analysis of HIV/AIDS Transmission Dynamics
The mathematical model of the dynamics of HIV/AIDS infection transmission is developed by adding the set of infected but noninfectious persons, using a conformable fractional derivative in the Liouville–Caputo sense. Some fixed point theorems are applied to this model to investigate the existence and uniqueness of the solutions. It is determined what the system’s fundamental reproduction number is. The disease-free equilibrium displays the model’s stability and the local stability around the equilibrium. The study also examined the effects of different biological features on the system through numerical simulations using the Adams–Moulton approach. Additionally, varied values of fractional orders are simulated numerically, demonstrating that the results generated by the conformable fractional derivative-based model are more physiologically plausible than integer-order derivatives.
Multiple Solutions for Singular Systems with Sign-Changing Weight, Nonlinear Singularities and Critical Exponent
This paper is an attempt to establish the existence and multiplicity results of nontrivial solutions to singular systems with sign-changing weight, nonlinear singularities, and critical exponent. By using variational methods, the Nehari manifold, and under sufficient conditions on the parameter which represent some physical meanings, we prove some existing results by researching the critical points as the minimizers of the energy functional associated with the proposed problem (2) on the constraint defined by the Nehari manifold, which are solutions of our system, under some sufficient conditions on the parameters , , , and . To the best of our knowledge, this paper is one of the first contributions to the study of singular systems with sign-changing weight, nonlinear singularities, and critical exponent.
Stability Results for Nonlinear Implicit -Caputo Fractional Differential Equations with Fractional Integral Boundary Conditions
This article examines the necessary conditions for the unique existence of solutions to nonlinear implicit -Caputo fractional differential equations accompanied by fractional order integral boundary conditions. The analysis draws upon Banach’s contraction principle and Krasnoselskii’s fixed point theorem. Furthermore, the circumstances leading to the attainment of Ulam–Hyers–Rassias forms of stability are established. An illustrative example is provided to demonstrate the derived findings.
Solving Nonlinear Fractional PDEs with Applications to Physics and Engineering Using the Laplace Residual Power Series Method
The Laplace residual power series (LRPS) method uses the Caputo fractional derivative definition to solve nonlinear fractional partial differential equations. This technique has been applied successfully to solve equations such as the fractional Kuramoto–Sivashinsky equation (FKSE) and the fractional generalized regularized long wave equation (GRLWE). By transforming the equation into the Laplace domain and replacing fractional derivatives with integer derivatives, the LRPS method can solve the resulting equation using a power series expansion. The resulting solution is accurate and convergent, as demonstrated in this paper by comparing it with other analytical methods. The LRPS approach offers both computational efficiency and solution accuracy, making it an effective technique for solving nonlinear fractional partial differential equations (NFPDEs). The results are presented in the form of graphs for various values of the order of the fractional derivative and time, and the essential objective is to reduce computation effort.
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