Journal of Functional Programming

In this paper, an algebraic decay rate for the radius of spatial analyticity of solutions to the Kawahara equation is investigated. With given analytic initial data having a fixed radius of analyticity σ0, we derive an algebraic decay rate σ(t) ~ |t|−1/2 for the uniform radius of spatial analyticity of solutions to the Kawahara equation. This improves a recent result due to Ahn et al.’s study, where they demonstrated a decay rate of order |t|−1. Our strategy mainly relies on an approximate conservation law in a modified Gevrey space and bilinear estimate in Bourgain space.

In this paper, we shall establish sufficient conditions for the existence, approximate controllability, and Ulam–Hyers–Rassias stability of solutions for impulsive integrodifferential equations of second order with state-dependent delay using the resolvent operator theory, the approximating technique, Picard operators, and the theory of fixed point with measures of noncompactness. An example is presented to illustrate the efficiency of the result obtained.

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.

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 R0 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.

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 η 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 α, β, μ, and η. 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.

This article examines the necessary conditions for the unique existence of solutions to nonlinear implicit ϑ-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.

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.

In the realm of ecology, species naturally strive to enhance their own survival odds. This study introduces and investigates a predator-prey model incorporating reaction-diffusion through a system of differential equations. We scrutinize how diffusion impacts the model’s stability. By analysing the stability of the model’s uniform equilibrium state, we identify a condition leading to Turing instability. The study delves into how diffusion influences pattern formation within a predator-prey system. Our findings reveal that various spatiotemporal patterns, such as patches, spots, and even chaos, emerge based on species diffusion rates. We derive the amplitude equation by employing the weak nonlinear multiple scales analysis technique and the Taylor series expansion. A novel sinc interpolation approach is introduced. Numerical simulations elucidate the interplay between diffusion and Turing parameters. In a two-dimensional domain, spatial pattern analysis illustrates population density dynamics resulting in isolated groups, spots, stripes, or labyrinthine patterns. Simulation results underscore the method’s effectiveness. The article concludes by discussing the biological implications of these outcomes.

Heat transfer in fluid mechanisms has a stronghold in everyday activities. To this end, nanofluids take a leading position in the advent of the betterment of thermal conductivity. The present study examines numerical investigations of incompressible magnetohydrodynamic (MHD) flow of Carreau nanofluid by considering nonlinear thermal radiation, Joule heating, temperature-dependent heat source/sink, and chemical reactions with attached Brownian movement and thermophoresis above a stretching sheet that saturates the porous medium. Pertaining similarity assumptions are used to change the flow equations into tractable forms of higher order nonlinear ordinary differential equations (ODEs). The continuation technique is adopted in the MATLAB bvp4c package for the numerical outcomes. The velocity, temperature, and nanoparticle concentration distributions in contrast to the leading parameters are availed in graphical and tabular descriptions. Among the many outcomes, increasing the radiation parameter from 0.2 to 0.8 surged the heat transfer rate by 47.78 % at n = 1.5 and lifted it only by 8.5 % at n = 0.5. By boosting the magnetic parameter from 0 to 1.5, respective 37.64 % and 20.17 % rises in local drag forces are achieved in shear-thickening and thinning regions. On top of that, chemical reactions and Brownian motion parameters decay the concentration field. The distinctiveness of this method is that a solution is secured for the problem, which is highly sensitive to initial and boundary conditions. It will be worth mentioning that these fluid flow models will be applicable in various fields, such as engineering, petroleum, nuclear safety processes, and medical science.

Despite the recent progress of global control efforts, tuberculosis (TB) remains a significant public health threat worldwide, especially in developing countries, including Ethiopia. Furthermore, the emergence of multidrug-resistant tuberculosis (MDR-TB) has further complicated the situation. This study aims at identifying the most effective strategies for combating MDR-TB in Ethiopia. We first present a compartmental model of MDR-TB transmission dynamics in Ethiopia. The model is shown to have positive solutions, and the stability of the equilibrium points is analyzed. Then, we extend the model by incorporating time-dependent control variables. These control variables are vaccination, distancing, and treatment for DS-TB and MDR-TB. Finally, the optimality system is numerically simulated by considering different combinations of the strategies, and their cost effectiveness is analysed. Our finding shows that, among single control strategies, the successful treatment of drug-susceptible tuberculosis (DS-TB) is the most effective control factor for eliminating MDR-TB transmission in Ethiopia. Furthermore, within the six dual control strategies, the combination of distancing and successful treatment of DS-TB is less costly and more effective than other strategies. Finally, out of the triple control strategies, the combination of distancing, successful treatment for DS-TB, and treatment for MDR-TB is the most efficient strategy for curbing the MDR-TB disease in Ethiopia. Thus, to reduce MDR-TB efficiently, it is recommended that authorities focus on treating MDR-TB, effective treatment of DS-TB, and promoting social distancing through public health education and awareness programs.

In this study, the classical Lie symmetry method is successfully applied to investigate the symmetries of the time-fractional generalized foam drainage equation with the Riemann–Liouville derivative. With the help of the obtained Lie point symmetries, the equation is reduced to nonlinear fractional ordinary differential equations (NLFODEs) which contain the Erdélyi–Kober fractional differential operator. The equation is also studied by applying the power series method, which enables us to obtain extra solutions. The obtained power series solution is further examined for convergence. Conservation laws for this equation are obtained with the aid of the new conservation theorem and the fractional generalization of the Noether operators.

One way in which nonlinear descriptor systems of (index-k) naturally arise is through semiexplicit differential-algebraic equations. The study considers the nonbilinear dynamical systems which are described by the class of higher-index differential-algebraic equations (DAEs). Their nature is analysed both quantitatively and qualitatively, and stability characteristics are presented for their solution. Higher-index differential-algebraic systems seem to show inherent shaky around their solution manifolds. The often use of logarithmic norms is for the estimation of stability and perturbation bounds in linear ordinary differential equations (ODEs). The question of how to apply the notation of logarithmic norms to nonlinear DAEs has long been an open question. Other problem extensions including nonlinear dynamics and nonbilinear DAEs need subtle modification of the logarithmic norms. The logarithmic norm is combined by conceptual focus with the finite-time stability criterion in order to treat nonbilinear DAEs with the aim of covering some unbounded operators. This means we obtain the perturbation bounds from differential inequalities for a norm by the use of the relationship between Dini derivatives and semi-inner products. A numerical result obtained when tested on the nonbilinear mechanical system with a larger scale showed that the method was highly efficient and accurate and particularly suitable for nonbilinear DAEs.

In this paper, oscillation and asymptotic behavior of three-dimensional third-order delay systems are discussed. Some sufficient conditions are obtained to ensure that every solution of the system is either oscillatory or nonoscillatory and converges to zero or diverges as t goes to infinity. A special technique is adopted to include all possible cases for all nonoscillatory solutions (NOSs). The obtained results included illustrative examples.

This manuscript aims to highlight the existence and uniqueness results for the following Schrödinger problem in the extended Colombeau algebra of generalized functions. 1 / ı ∂ / ∂ t u t , x − △ u t , x + v x u t , x = 0 , t ∈ R + , x ∈ R n , v x = δ x , u 0 , x = δ x , where δ is the Dirac distribution. The proofs of our main results are based on the Gronwall inequality and regularization method. We conclude our article by establishing the association concept of solutions.

In this article we study the existence and stability of bounded solutions for semilinear abstract dynamic equations on time scales in Banach spaces. In order to do so, we use the definition of the Riemann delta-integral to prove a result about closed operator in Banach spaces and then we just use the representation of bounded solutions as an improper delta-integral from minus infinite to t . We prove the existence, uniqueness, and exponential stability of such bounded solutions. As particular cases, we study the existence of periodic and almost periodic solutions as well. Finally, we present some equations on time scales where our results can be applied.