ANALYSIS OF TIME-FRACTIONAL KAWAHARA EQUATION UNDER MITTAG-LEFFLER POWER LAW

In this paper, we study a newly updated nonlinear fractional Kawahara equation (KE) using Atangana–Baleanu fractional operator in the sense of Caputo (ABC). To find the approximate solution, one of the famous techniques of the Laplace Adomian decomposition method (LADM) is used along with a time-fractional derivative. For evaluation, the required quantity is decomposing into small particles along with the application of Adomian polynomial to the nonlinear term. By the addition of the first few evaluating terms, the required convergent quantity is obtained. To explain the authenticity and the manageability of the procedure, few examples are present at different fractional orders both in three and two dimensions. Further, to compare the obtained results between fractional derivative and integer derivative, some graphical presentations are given. So, the newly updated version of the KE equation is analyzed in fraction operator providing the whole density of the total dynamics at any fractional value between two different integers.

ON ANALYSIS OF FRACTIONAL ORDER MATHEMATICAL MODEL OF HEPATITIS B USING ATANGANA–BALEANU CAPUTO (ABC) DERIVATIVE

The scaling exponent of a hierarchy of cities used to be regarded as a fractional. This paper investigates a newly constructed system of equation for Hepatitis B disease in sense of Atanganaa–Baleanu Caputo (ABC) fractional order derivative. The proposed approach has five distinctive quantities, namely, susceptible, acute infections, chronic infection, immunized and vaccinated populace. By applying some well-known results of fixed point theory, we find the Ulam–Hyers type stability and qualitative analysis of the candidate solution. The deterministic stability for the proposed system is also computed. We apply well-known transform due to Laplace and decomposition techniques (LADM) and Adomian polynomial for nonlinear terms for computing the series solution for the proposed model. Graphical results show that LADM is an efficient and robust method for solving nonlinear problems.

CAPUTO TYPE FRACTIONAL OPERATOR APPLIED TO HEPATITIS B SYSTEM

In our research work, we develop the analysis of a noninteger-order model for hepatitis B (HBV) under singular type Caputo fractional-order derivative. We investigated our proposed system for an approximate or semi-analytical solution using Laplace transform along with decomposition techniques by Adomian polynomial of nonlinear terms and some perturbation techniques of Homotopy (HPM). The obtained solutions have been compared with each other against some real data by simulation via MATLAB. The graphical simulation in fractional form shows a better general result as compared to integer-order simulation.

Efficient Iterative Methods for Solving the SIR Epidemic Model

In this article, the numerical and approximate solutions for the nonlinear differential equation systems, represented by the epidemic SIR model, are determined. The effective iterative methods, namely the Daftardar-Jafari method (DJM), Temimi-Ansari method (TAM), and the Banach contraction method (BCM), are used to obtain the approximate solutions. The results showed many advantages over other iterative methods, such as Adomian decomposition method (ADM) and the variation iteration method (VIM) which were applied to the non-linear terms of the Adomian polynomial and the Lagrange multiplier, respectively. Furthermore, numerical solutions were obtained by using the fourth-orde Runge-Kutta (RK4), where the maximum remaining errors showed that the methods are reliable. In addition, the fixed point theorem was used to show the convergence of the proposed methods. Our calculation was carried out with MATHEMATICA®10 to evaluate the terms of the approximate solutions.

Homotopy Analysis Decomposition Method for the Solution of Viscous Boundary Layer Flow Due to a Moving Sheet

This paper investigates a new approach called Homotopy Analysis Decomposition Method (HADM) for solving nonlinear differential equations, the method was developed by incorporating Adomian polynomial into Homotopy Analysis Method. The Adomian polynomial was used to decompose the nonlinear term in the equation then apply the scheme of homotopy analysis method. The accuracy and efficiency of the proposed method was validated by considering algebraically decaying viscous boundary layer flow due to a moving sheet. Diagonal Pade approximation was used to get the skin friction. The obtained results were presented along with other methods in the literature in tabular form to show the computational efficiency of the new approach. The results were found to agree with those in literature. Owing to its small size of computation, the method is not aected by discretization error as the results are presented in form of polynomials.

Modification of Laplace Adomian Decomposition Method for Solving Nonlinear Volterra Integral and Integro-differential Equations based on Newton Raphson Formula

In this paper, we establish a modified Laplace decomposition method for nonlinear volterra integral and integro-differential equations. This technique differs from the general Laplace decomposition method because of the terms involved in Adomian polynomial.We have used Newton Raphson formula in place of the $u_{i}$ in Adomian polynomial. The proposed scheme is investigated with some illustrative examples and has given reliable results.

A modified Adomian decomposition method for singular initial value Emden-Fowler type equations

<p>Traditional Adomian decomposition method (ADM) usually fails to solve singular initial value problems of Emden-Fowler type. To overcome this shortcoming, a new and effective modification of ADM that only requires calculation of the first Adomian polynomial is formally proposed in the present paper. Three singular initial value problems of Emden-Fowler type with alpha=1, 2, and &gt;2, have been selected to demonstrate the efficiency of the method. </p>