The Numerical Solutions of Nonlinear Time-Fractional Differential Equations by LMADM

This paper presents a numerical scheme for solving nonlinear time-fractional differential equations in the sense of Caputo. This method relies on the Laplace transform together with the modified Adomian method (LMADM), compared with the Laplace transform combined with the standard Adomian Method (LADM). Furthermore, for the comparison purpose, we applied LMADM and LADM for solving nonlinear time-fractional differential equations to identify the differences and similarities. Finally, we provided two examples regarding the nonlinear time-fractional differential equations, which showed that the convergence of the current scheme results in high accuracy and small frequency to solve this type of equations.

Analyzing a New Third Order Iterative Method for Solving Nonlinear Problems

In this paper, we propose and analyse a new iterative method for solving nonlinear equations. The method is constructed by applying Adomian method to Taylor’s series expansion. Using one-way analysis of variance (ANOVA), the method is being compared with other existing methods in terms of the number of iterations and solution to convergence between the individual methods used. Numerical examples are used in the comparison to justify the efficiency of the new iterative method.

An analytic study of the fractional order model of HIV-1 virus and CD4+ T-cells using adomian method

In this article, we study the fractional mathematical model of HIV-1 infection of CD4+ T-cells, by studying a system of fractional differential equations of first order with some initial conditions, we study the changing effect of many parameters. The fractional derivative is described in the caputo sense. The adomian decomposition method (Shortly, ADM) method was used to calculate an approximate solution for the system under study. The nonlinear term is dealt with the help of adomian polynomials. Numerical results are presented with graphical justifications to show the accuracy of the proposed methods.

Approximate Solutions for Systems of Volterra Integro-differential Equations Using Laplace –Adomian Method

Some modified techniques are used in this article in order to have approximate solutions for systems of Volterra integro-differential equations. The suggested techniques are the so called Laplace-Adomian decomposition method and Laplace iterative method. The proposed methods are robust and accurate as can be seen from the given illustrative examples and from the comparison that are made with the exact solution.

On Semi-Analytical Solutions for Linearized Dispersive KdV Equations

The most well-known equations both in the theory of nonlinearity and dispersion, KdV equations, have received tremendous attention over the years and have been used as model equations for the advancement of the theory of solitons. In this paper, some semi-analytic methods are applied to solve linearized dispersive KdV equations with homogeneous and inhomogeneous source terms. These methods are the Laplace-Adomian decomposition method (LADM), Homotopy perturbation method (HPM), Bernstein-Laplace-Adomian Method (BALDM), and Reduced Differential Transform Method (RDTM). Three numerical experiments are considered. As the main contribution, we proposed a new scheme, known as BALDM, which involves Bernstein polynomials, Laplace transform and Adomian decomposition method to solve inhomogeneous linearized dispersive KdV equations. Besides, some modifications of HPM are also considered to solve certain inhomogeneous KdV equations by first constructing a newly modified homotopy on the source term and secondly by modifying Laplace’s transform with HPM to build HPTM. Both modifications of HPM numerically confirm the efficiency and validity of the methods for some test problems of dispersive KdV-like equations. We also applied LADM and RDTM to both homogeneous as well as inhomogeneous KdV equations to compare the obtained results and extended to higher dimensions. As a result, RDTM is applied to a 3D-dispersive KdV equation. The proposed iterative schemes determined the approximate solution without any discretization, linearization, or restrictive assumptions. The performance of the four methods is gauged over short and long propagation times and we compute absolute and relative errors at a given time for some spatial nodes.

Existence theory and approximate solution to prey–predator coupled system involving nonsingular kernel type derivative

This manuscript considers a nonlinear coupled system under nonsingular kernel type derivative. The considered problem is investigated from two aspects including existence theory and approximate analytical solution. For the concerned qualitative theory, some fixed point results are used. While for approximate solution, the Laplace transform coupled with Adomian method is applied. Finally, by a pertinent example of prey–predator system, we support our results. Some graphical presentations are given using Matlab.