Solving Hammerstein Type Integral Equation by New Discrete Adomian Decomposition Methods

New discrete Adomian decomposition methods are presented by using some identified Clenshaw-Curtis quadrature rules. We investigate two mixed quadrature rules one of precision five and the other of precision seven. The first rule is formed by using the Fejér second rule of precision three and Simpson rule of precision three, while the second rule is formed by using the Fejér second rule of precision five and the Boole rule of precision five. Our methods were applied to a nonlinear integral equation of the Hammerstein type and some examples are given to illustrate the validity of our methods

A Comparative Study of Two Spatial Discretization Schemes for Advection Equation Keywords Advection Equation, Finite Difference Method, The Method of Lines, Von Neumann Method

Abstract In this paper, we describe a comparison of two spatial discretization schemes for the advection equation, namely the first finite difference method and the method of lines. The stability of the methods has been studied by Von Neumann method and with the matrix analysis. The methods are applied to a number of test problems to compare the accuracy and computational efficiency. We show that both discretization techniques approximate correctly solution of advection equation and compare their accuracy and performance

Exact Solution of Ambartsumian Delay Differential Equation and Comparison with Daftardar-Gejji and Jafari Approximate Method

The Ambartsumian equation, a linear differential equation involving a proportional delay term, is used in the theory of surface brightness in the Milky Way. In this paper, the Laplace-transform was first applied to this equation, and then the decomposition method was implemented to establish a closed-form solution. The present closed-form solution is reported for the first time for the Ambartsumian equation. Numerically, the calculations have demonstrated a rapid rate of convergence of the obtained approximate solutions, which are displayed in several graphs. It has also been shown that only a few terms of the new approximate solution were sufficient to achieve extremely accurate numerical results. Furthermore, comparisons of the present results with the existing methods in the literature were introduced

Modified Adomian Method through Efficient Inverse Integral Operators to Solve Nonlinear Initial-Value Problems for Ordinary Differential Equations

The present manuscript examines different forms of Initial-Value Problems (IVPs) featuring various types of Ordinary Differential Equations (ODEs) by proposing a proficient modification to the famous standard Adomian decomposition method (ADM). The present paper collected different forms of inverse integral operators and further successfully demonstrated their applicability on dissimilar nonlinear singular and nonsingular ODEs. Furthermore, we surveyed most cases in this very new method, and it was found to have a fast convergence rate and, on the other hand, have high precision whenever exact analytical solutions are reachable

A Method for the Solution of Coupled System of Emden–Fowler–Type Equations

A dependable semi-analytical method via the application of a modified Adomian Decomposition Method (ADM) to tackle the coupled system of Emden–Fowler-type equations has been proposed. More precisely, an effective differential operator together with its corresponding inverse is successfully constructed. Moreover, this operator is able to navigate to the closed-form solution easily without resorting to converting the coupled system to a system of Volterra integral equations; as in the case of a well-known reference in the literature. Lastly, the effectiveness of the method is demonstrated on some coupled systems of the governing model, and a speedier convergence rate was noted

A Method for the Solution of Coupled System of Emden–Fowler–Type Equations

A dependable semi-analytical method via the application of a modified Adomian Decomposition Method (ADM) to tackle the coupled system of Emden–Fowler-type equations has been proposed. More precisely, an effective differential operator together with its corresponding inverse is successfully constructed. Moreover, this operator is able to navigate to the closed-form solution easily without resorting to converting the coupled system to a system of Volterra integral equations; as in the case of a well-known reference in the literature. Lastly, the effectiveness of the method is demonstrated on some coupled systems of the governing model, and a speedier convergence rate was noted

Approximate analytical solution for 1-D problems of thermoelasticity with dirichlet condition

This paper presents the solution of the initial boundary-value problem for the system of 1-D thermoelasticity using a new modified decomposition method that takes into accounts both initial and boundary conditions. The obtained solution is based on the generalized form of the inverse operator and is given in the form of a finite series. Also, some numerical experiments were presented to the both the effectiveness and the accuracy of the presented method

Numerical Simulation of Cubic-Quartic Optical Solitons with Perturbed Fokas–Lenells Equation Using Improved Adomian Decomposition Algorithm

The current manuscript displays elegant numerical results for cubic-quartic optical solitons associated with the perturbed Fokas–Lenells equations. To do so, we devise a generalized iterative method for the model using the improved Adomian decomposition method (ADM) and further seek validation from certain well-known results in the literature. As proven, the proposed scheme is efficient and possess a high level of accuracy