This manuscript presents several applications for solving special kinds of ordinary and partial differential equations using iteration methods such as Adomian decomposition method (ADM), Variation iterative method (VIM) and Taylor series method. These methods can be applied as well as to solve nonperturbed problems and 3rd order parabolic PDEs with variable coefficient. Moreover, we compare the results using ADM, VIM and Taylor series method. These methods are a commination of the two initial conditions.

In the present article, we implement the new iterative method proposed by Daftardar-Gejji and Jafari (NIM) [V. Daftardar-Gejji, H. Jafari, An iterative method for solving nonlinear functional equations, J. Math. Anal. Appl. 316 (2006) 753-763] to solve two problems; the first one is the problem of spread of a non-fatal disease in a population which is assumed to have constant size over the period of the epidemic, and the other one is the problem of the prey and predator. The results demonstrate that the method has many merits such as being derivative-free, overcome the difficulty arising in calculating Adomian polynomials to handle the nonlinear terms in Adomian Decomposition Method (ADM), does not require to calculate Lagrange multiplier a

The flexible joint robot manipulators provide various benefits, but also present many control challenges such as nonlinearities, strong coupling, vibration, etc. This paper proposes optimal second order integral sliding mode control (OSOISMC) for a single link flexible joint manipulator to achieve robust and smooth performance. Firstly, the integral sliding mode control is designed, which consists of a linear quadratic regulator (LQR) as a nominal control, and switching control. This control guarantees the system robustness for the entire process. Then, a nonsingularterminal sliding surface is added to give a second order integral sliding mode control (SOISMC), which reduces chartering effect and gives the finite time convergence as well. S

In this work, an analytical approximation solution is presented, as well as a comparison of the Variational Iteration Adomian Decomposition Method (VIADM) and the Modified Sumudu Transform Adomian Decomposition Method (M STADM), both of which are capable of solving nonlinear partial differential equations (NPDEs) such as nonhomogeneous Kertewege-de Vries (kdv) problems and the nonlinear Klein-Gordon. The results demonstrate the solution’s dependability and excellent accuracy.

In this paper, the author established some new integral conditions for the oscillation of all solutions of nonlinear first order neutral delay differential equations. Examples are inserted to illustrate the results.

in this paper the second order neutral differential equations are incestigated are were we give some new suffucient conditions for all nonoscillatory

Markov chains are an application of stochastic models in operation research, helping the analysis and optimization of processes with random events and transitions. The method that will be deployed to obtain the transient solution to a Markov chain problem is an important part of this process. The present paper introduces a novel Ordinary Differential Equation (ODE) approach to solve the Markov chain problem. The probability distribution of a continuous-time Markov chain with an infinitesimal generator at a given time is considered, which is a resulting solution of the Chapman-Kolmogorov differential equation. This study presents a one-step second-derivative method with better accuracy in solving the first-order Initial Value Problem

This paper focuses on developing a self-starting numerical approach that can be used for direct integration of higher-order initial value problems of Ordinary Differential Equations. The method is derived from power series approximation with the resulting equations discretized at the selected grid and off-grid points. The method is applied in a block-by-block approach as a numerical integrator of higher-order initial value problems. The basic properties of the block method are investigated to authenticate its performance and then implemented with some tested experiments to validate the accuracy and convergence of the method.

This paper presents new modification of HPM to solve system of 3 rd order PDEs with initial condition, for finding suitable accurate solutions in a wider domain.