Global dynamics of solutions for a sixth-order parabolic equation describing continuum evolution of film-free surface

This paper is concerned with a sixth-order diffusion equation, which describes continuum evolution of film-free surface. By using the regularity estimates for the semigroups, iteration technique and the classical existence theorem of global attractors we verified the existence of global attractor for this surface diffusion equation in the spaces H3(Ω) and fractional-order spaces Hk(Ω), where 0 ≤ k < ∞.

Analysis of the Fractional-Order Kaup–Kupershmidt Equation via Novel Transforms

In this article, we develop a technique to determine the analytical result of some Kaup–Kupershmidt equations with the aid of a modified technique called the new iteration transform method. This technique is a mixture of the novel integral transformation Elzaki transformation and the new iteration technique. The nonlinear term can be handled easily by a new iteration technique. The results show that the combination of the Elzaki transformation and the new iteration technique is quite capable and basically well suited for applying in such problems and that it can be implemented to other nonlinear models. This technique is viewed as an effective alternative approach to certain existing approaches for such accurate models.

Numerical Methods for Fractional-Order Fornberg-Whitham Equations in the Sense of Atangana-Baleanu Derivative

In this paper, we investigate the numerical solution of the Fornberg-Whitham equations with the help of two powerful techniques: the modified decomposition technique and the modified variational iteration technique involving fractional-order derivatives with Mittag-Leffler kernel. To confirm and illustrate the accuracy of the proposed approach, we evaluated in terms of fractional order the projected models. Furthermore, the physical attitude of the results obtained has been acquired for the fractional-order different value graphs. The results demonstrated that the future method is easy to implement, highly methodical, and very effective in analyzing the behavior of complicated fractional-order linear and nonlinear differential equations existing in the related areas of applied science.

Semianalytical Solutions of Some Nonlinear-Time Fractional Models Using Variational Iteration Laplace Transform Method

In this work, we combined two techniques, the variational iteration technique and the Laplace transform method, in order to solve some nonlinear-time fractional partial differential equations. Although the exact solutions may exist, we introduced the technique VITM that approximates the solutions that are difficult to find. Even a single iteration best approximates the exact solutions. The fractional derivatives being used are in the Caputo-Fabrizio sense. The reliability and efficiency of this newly introduced method is discussed in details from its numerical results and their graphical approximations. Moreover, possible consequences of these results as an application of fixed-point theorem are placed before the experts as an open problem.

Stress Analysis of Functionally Graded Disk with Exponentially Varying Thickness using Iterative Method

In this paper, variable thickness disk made up of functionally graded material (FGM) under internal and external pressure is analyzed using a simple iteration technique. Thickness of FGM disk and the material property, namely, Young’s modulus are varying exponentially in radial direction. Poisson’s ratio is considered invariant for the material. Navier equation is used to formulate the problem in the differential equation form under plane stress condition. Displacement, stresses, and strains are obtained under the influence of material gradation and variable thickness. Three different material combinations are considered for the FGM disk. The mechanical response of disk obtained for different functionally graded material combinations are compared with the homogenous disk, and results are plotted graphically

Iterative Self-Tuning Minimum Variance Control of a Nonlinear Autonomous Underwater Vehicle Maneuvering Model

This paper addresses the problem of control design for a nonlinear maneuvering model of an autonomous underwater vehicle. The control algorithm is based on an iteration technique that approximates the original nonlinear model by a sequence of linear time-varying equations equivalent to the original nonlinear problem and a self-tuning control method so that the controller is designed at each time point on the interval for trajectory tracking and heading angle control. This work makes use of self-tuning minimum variance principles. The benefit of this approach is that the nonlinearities and couplings of the system are preserved, unlike in the cases of control design based on linearized systems, reducing in this manner the uncertainty in the model and increasing the robustness of the controller. The simulations here presented use a torpedo-shaped underwater vehicle model and show the good performance of the controller and accurate tracking for certain maneuvering cases.

Application of modified Mickens iteration procedure to a pendulum and the motion of a mass attached to a stretched elastic wire

The paper deals with the application of a strong method called the modified Mickens iteration technique which is used for solving a strongly nonlinear system. The system describes the motion of a simple mathematical pendulum with a particle attached to it through a stretched wire. This model has great applications especially in the area of nonlinear vibrations and oscillation systems. The proposed method depends on determining the frequency and amplitude of the system through the modified Mickens iterative approach which is a modification of the regular Mickens approach. The preliminaries of the proposed technique are present and the application to the model is discussed. The method depends on the Mickens iteration approach which transforms the considered equation into a linear form and then is solving this equation result in the approximate solution. Some examples are given to validate and illustrate the effectiveness and convenience of the method. These results are compared with other relative techniques from the literature in terms of finding the frequency of the two examined models. The method produces more accurate results when compared to these methods and is considered a strong candidate for solving other nonlinear problems with applications in science and engineering.

Stability and Hopf Bifurcation of a Delayed Viral Infection Dynamics Model with Immune Impairment

In this paper, we consider a viral infection dynamics model with immune impairment and time delay in immune expansion. By calculation, it is shown that the model has three equilibria: infection-free equilibrium, immunity-inactivated infection equilibrium, and immunity-activated infection equilibrium. By analyzing the distributions of roots of corresponding characteristic equations, the local stability of the infection-free equilibrium and the immunity-inactivated infection equilibrium is established. Furthermore, we discuss the existence of Hopf bifurcation at the immunity-activated infection equilibrium. Sufficient conditions are obtained for the global asymptotic stability of each feasible equilibria of the model by using LaSalle’s invariance principle and iteration technique, respectively. Numerical simulations are carried out to illustrate the main theoretical results.

Numerical Analysis of Time-Fractional Diffusion Equations via a Novel Approach

The aim of this paper is a new semianalytical technique called the variational iteration transform method for solving fractional-order diffusion equations. In the variational iteration technique, identifying of the Lagrange multiplier is an essential rule, and variational theory is commonly used for this purpose. The current technique has the edge over other methods as it does not need extra parameters and polynomials. The validity of the proposed method is verified by considering some numerical problems. The solution achieved has shown that the better accuracy of the proposed technique. This paper proposes a simpler method to calculate the multiplier using the Shehu transformation, making a valuable technique to researchers dealing with various linear and nonlinear problems.

Autonomous path planning through application of rotated two-parameter overrelaxation 9-point Laplacian iteration technique

Autonomous path navigation is one of the important studies in robotics since a robot’s ability to navigate through an environment brings about many advancements with it. This paper suggests the iteration technique called half-sweep two parameter overrelaxation 9-point laplacian (HSTOR-9P) to be applied on autonomous path navigation and aims to investigate its effectiveness in performing computation for path planning in an indoor static environment. This iteration technique is a harmonic function that solves the Laplace’s equation where the modelling of the environment is based on. The harmonic functions are an appropriate method to be used on autonomous path planning because it satisfies the min-max principle, therefore avoiding the occurrence of local minima which traps robot’s movements, and that it offers complete path planning algorithm. Its performance is tested against its predecessor iteration technique. Results shown that HSTOR-9P iteration technique enables path construction in a lower number of iterations, thus, performs better than its predecessors.