A Note on He’s Parameter-Expansion Method of Coupled Van der Pol–Duffing Oscillators

This paper presents the analytical and approximate solutions of the coupled chaotic Van der Pol-Duffing systems, by using the He\u27s parameter-expansion method (PEM). One iteration is sufficient to obtain a highly accurate solution, which is valid for the whole solution domain. From the obtained results, we can conclude that the suggest method, is of utter simplicity, and can be easily extended to all kinds of non-linear equations

Numerical Simulations of Some Real-Life Problems Governed by ODEs

In this chapter, some real-life model problems that can be formulated as ordinary differential equations (ODEs) are introduced and numerically studied. These models are the variable-order fractional Hodgkin–Huxley model of neuronal excitation (VOFHHM) and other models with the variable-order fractional (VOF) time delay, such as the 4-year life cycle of a population of lemmings model, the enzyme kinetics with an inhibitor molecule model, and the Chen system model. A class of numerical methods is used to study the above-mentioned models such as non-standard finite difference (NSFD) and Adams-Bashforth-Moulton (ABM) methods. Numerical test examples are presented

Numerical Simulation for Solving Fractional Riccati and Logistic Differential Equations as a Difference Equation

In this paper, we introduce a numerical treatment using the generalized Euler method (GEM) for the fractional (Caputo sense) Riccati and Logistic differential equations. In the proposed method, we invert the given model as a difference equation. We compare our numerical solutions with the exact solution and with those numerical solutions using the fourth-order Runge-Kutta method (RK4). The obtained numerical results of the two proposed problem models show the simplicity and efficiency of the proposed method

Numerical Studies for Solving Fractional Riccati Differential Equation

In this paper, finite difference method (FDM) and Pade\u27-variational iteration method (Pade\u27- VIM) are successfully implemented for solving the nonlinear fractional Riccati differential equation. The fractional derivative is described in the Caputo sense. The existence and the uniqueness of the proposed problem are given. The resulting nonlinear system of algebraic equations from FDM is solved by using Newton iteration method; moreover the condition of convergence is verified. The convergence\u27s domain of the solution is improved and enlarged by Pade\u27-VIM technique. The results obtained by using FDM is compared with Pade\u27-VIM. It should be noted that the Pade\u27-VIM is preferable because it always converges to the solution even for large domain

Semi-Analytical Solution for the Multicell Spheroid model for Vascular Tumor Growth

Abstract: The homotopy analysis method is used to obtain semi-analytic solutions for the mathematical model describing a solid tumor growth in the initial a vascular stage of growth. During a vascular tumor growth, the balance between cell proliferation and cell loss determines whether the colony expands or progress. We focus on the chemical inhibition of mitosis within multicell spheroids. The main assumption of modeling the diffusion of a growth inhibitory factor (GIF) within a multicell spheroid and its possible effects on cell mitosis and proliferation

Numerical Studies for Fractional-Order Logistic Differential Equation with Two Different Delays

A numerical method for solving the fractional-order logistic differential equation with two different delays (FOLE) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based upon Chebyshev approximations. The properties of Chebyshev polynomials are utilized to reduce FOLE to a system of algebraic equations. Special attention is given to study the convergence and the error estimate of the presented method. Numerical illustrations are presented to demonstrate utility of the proposed method. Chaotic behavior is observed and the smallest fractional order for the chaotic behavior is obtained. Also, FOLE is studied using variational iteration method (VIM) and the fractional complex transform is introduced to convert fractional Logistic equation to its differential partner, so that its variational iteration algorithm can be simply constructed. Numerical experiment is presented to illustrate the validity and the great potential of both proposed techniques

Non-Standard Crank-Nicholson Method for Solving the Variable Order Fractional Cable Equation

In this paper, a non-standard Crank-Nicholson finite difference method (NSCN) is presented. NSCN is used to study numerically the variable-order fractional Cable equation, where the variable order fractional derivatives are described in the Riemann- Liouville and the Gr¨unwald-Letnikov sense. The stability analysis of the proposed methods is given by a recently proposed procedure similar to the standard John von Neumann stability analysis. The reliability and efficiency of the proposed approach are demonstrated by some numerical experiments. It is found that NSCN is preferable than the standard Crank-Nicholson finite difference method (SCN)