Modeling and Control of Uncertain Nonlinear Systems

A survey of the methodologies associated with the modeling and control of uncertain nonlinear systems has been given due importance in this paper. The basic criteria that highlights the work is relied on the various patterns of techniques incorporated for the solutions of fuzzy equations that corresponds to fuzzy controllability subject. The solutions which are generated by these equations are considered to be the controllers. Currently, numerical techniques have come out as superior techniques in order to solve these types of problems. The implementation of neural networks technique is contributed in the complex way of dealing the appropriate coefficients and solutions of the fuzzy systems

Solving Fuzzy Nonlinear Volterra-Fredholm Integral Equations by Using Homotopy Analysis and Adomian Decomposition Methods

In this paper, Adomian decomposition method (ADM) and homotopy analysis method (HAM) are proposed to solving the fuzzy nonlinear Volterra-Fredholm integral equation of the second kind$(FVFIE-2)$. we convert a fuzzy nonlinear Volterra-Fredholm integral equation to a nonlinear system of Volterra-Fredholm integral equation in crisp case. we use ADM , HAM and find the approximate solution of this system and hence obtain an approximation for fuzzy solution of the nonlinear fuzzy Volterra-Fredholm integral equation. Also, the existence and uniqueness of the solution and convergence of the proposed methods are proved. Examples is given and the results reveal that homotopy analysis method is very effective and simple compared with the Adomian decomposition method

Application of Adomian decomposition method to solve hybrid fuzzy differential equations

AbstractIn this paper, we study the numerical solution of hybrid fuzzy differential equations by using Adomian decomposition method (ADM). This is powerful method which consider the approximate solution of a nonlinear equation as an infinite series usually converging to the accurate solution. Several numerical examples are given and by comparing the numerical results obtained from ADM and predictor corrector method (PCM), we have studied their accuracy

Exact Solution of Coupled Nonlinear PDEs Via Sumudu Decomposition Method

In this paper, we apply the Sumudu Decomposition Method on system of coupled nonlinear partial differential equations to calculate the analytical solutions in closed form. The nonlinear term can easily be handled with the help of He’s polynomials. The proposed technique is tested on four problems. Calculated results show the potential of the technique. Keyword: Nonlinear PDEs, He’s polynomials, Sumudu transform, Adomian decomposition metho

Laplace Adomian Decomposition Method to study Chemical ion transport through soil

The paper deals with a theoretical study of chemical ion transport in soil under a uniform external force in the transverse direction, where the soil is taken as porous medium. The problem is formulated in terms of boundary value problem that consists of a set of partial differential equations, which is subsequently converted to a system of ordinary differential equations by applying similarity transformation along with boundary layer approximation. The equations hence obtained are solved by utilizing Laplace Adomian Decomposition Method (LADM). The merit of this method lies in the fact that much of simplifying assumptions need not be made to solve the non-linear problem. The decomposition parameter is used only for grouping the terms, therefore, the nonlinearities is handled easily in the operator equation and accurate approximate solution are obtained for the said physical problem. The computational outcomes are introduced graphically. By utilizing parametric variety, it has been demonstrated that the intensity of the external pressure extensively influences the flow behavior

APPLICATION OF DIFFERENTIAL TRANSFORMATION METHOD FOR SOLVING HIV MODEL WITH ANTI-VIRAL TREATMENT

Mathematical models have been widely used to understand complex phenomena. Generally, the model is in the form of system of differential equations. However, when the model becomes complex, analytical solutions are not easily used and hence a numerical approach has been used. A number of numerical schemes such as Euler, Runge-Kutta, and Finite Difference Scheme are generally used. There are also alternative numerical methods that can be used to solve system of differential equations such as the nonstandard finite difference scheme (NSFDS), the Adomian decomposition method (ADM), Variation iteration method (VIM), and the differential transformation method (DTM). In this paper, we apply the differential transformation method (DTM)  to solve system of differential equations. The DTM is semi-analytical numerical technique to solve the system of differential equations and provides an iterative procedure to obtain the power series of the solution in terms of initial value parameters. In this paper, we present a mathematical model of HIV with antiviral treatment and construct a numerical scheme based on the differential transformation method (DTM) for solving the model. The results are compared to that of Runge-Kutta method. We find a good agreement of the DTM and the Runge-Kutta method for smaller time step but it fails in the large time ste

On fuzzy and crisp solutions of a novel fractional pandemic model

Understanding disease dynamics is crucial for accurately predicting and effectively managing epidemic outbreaks. Mathematical modeling serves as an essential tool in such understanding. This study introduces an advanced susceptible, infected, recovered, and dead (SIRD) model that uniquely considers the evolution of the death parameter, alongside the susceptibility and infection states. This model accommodates the varying environmental factors influencing disease susceptibility. Moreover, our SIRD model introduces fractional changes in death cases, which adds a novel dimension to the traditional counts of susceptible and infected individuals. Given the model’s complexity, we employ the Laplace-Adomian decomposition method. The method allows us to explore various scenarios, including non-fuzzy non-fractional, non-fuzzy fractional, and fuzzy fractional cases. Our methodology enables us to determine the model’s equilibrium positions, compute the basic reproduction number, confirm stability, and provide computational simulations. Our study offers insightful understanding into the dynamics of pandemic diseases and underscores the critical role that mathematical modeling plays in devising effective public health strategies. The ultimate goal is to improve disease management through precise predictions of disease behavior and spread.ANCD -Agenția Națională pentru Cercetare și Dezvoltare(UIDP/00013/2020

(R1885) Analytical and Numerical Solutions of a Fractional-Order Mathematical Model of Tumor Growth for Variable Killing Rate

This work intends to analyze the dynamics of the most aggressive form of brain tumor, glioblastomas, by following a fractional calculus approach. In describing memory preserving models, the non-local fractional derivatives not only deliver enhanced results but also acknowledge new avenues to be further explored. We suggest a mathematical model of fractional-order Burgess equation for new research perspectives of gliomas, which shall be interesting for biomedical and mathematical researchers. We replace the classical derivative with a non-integer derivative and attempt to retrieve the classical solution as a particular case. The prime motive is to acquire both analytical and numerical solutions to the posed problem. At first, we employ the transform method, and then the Adomian decomposition method to obtain the solutions that shall be useful to provide information about the effect of medical care in the annihilation of gliomas. Finally, we discuss the applicability of this model with numerical simulations and graphical representations