Bongsoo Jang

Solving linear and nonlinear initial value problems by the projected differential transform method

In this work, we propose a novel computational algorithm for solving linear and nonlinear initial value problems by using the modified version of differential transform method (DTM), which is called the projected differential transform method (PDTM). The PDTM can be easily applied to the initial value problems with less computational work. For the several illustrative examples, the computational results are compared with those obtained by many other methods; the Adomian decomposition, the variational iteration and the spline method. For all examples, the PDTM provides exact solutions. It has been shown that the PDTM is a reliable algorithm in obtaining analytic as well as approximate solution for the initial value problems.

Comments on Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method

Tari et al. [A. Tari, M.Y. Rahimi, S. Shahmorad, F. Talati, Solving a class of two-dimensional linear and nonlinear Volterra integral equations by the differential transform method, J. Comput. Appl. Math. 228 (2009) 70-76], presented some fundamental properties of TDTM for the kernel functions in two-dimensional Volterra integral equations. Here, we suggest simple proofs of those fundamental properties by using the basic properties of TDTM. Furthermore, we present some fundamental properties of TDTM for the kernel functions of a quotient type in two-dimensional Volterra integral equations. Numerical illustrations are demonstrated to show the effectiveness of the TDTM for solving two-dimensional Volterra integral equations.

Exact solutions to one dimensional non-homogeneous parabolic problems by the homogeneous Adomian decomposition method

In this paper, we propose a reliable modification of Adomian decomposition method, namely the homogeneous Adomian decomposition method (HADM), that solves one dimensional non-homogeneous parabolic equations with a variable coefficient. The effectiveness of this method is verified through illustrative examples.

Enhanced Multistage Differential Transform Method: Application to the Population Models

We present an efficient computational algorithm, namely, the enhanced multistage differential transform method (E-MsDTM) for solving prey-predator systems. Since the differential transform method (DTM) is based on the Taylor series, it is difficult to obtain accurate approximate solutions in large domain. To overcome this difficulty, the multistage differential transform method (MsDTM) has been introduced and succeeded to have reliable approximate solutions for many problems. In MsDTM, it is the key to update an initial condition in each subdomain. The standard MsDTM utilizes the approximate solution directly to assign the new initial value. Because of local convergence of the Taylor series, the error is accumulated in a large domain. In E-MsDTM, we propose the new technique to update an initial condition by using integral operator. To demonstrate efficiency of the proposed method, several numerical tests are performed and compared with ones obtained by other numerical methods such as MsDTM, multistage variational iteration method (MVIM), and fourth-order Runge-Kutta method (RK4).

Emergence of unusual coexistence states in cyclic game systems

Evolutionary games of cyclic competitions have been extensively studied to gain insights into one of the most fundamental phenomena in nature: biodiversity that seems to be excluded by the principle of natural selection. The Rock-Paper-Scissors (RPS) game of three species and its extensions [e.g., the Rock-Paper-Scissors-Lizard-Spock (RPSLS) game] are paradigmatic models in this field. In all previous studies, the intrinsic symmetry associated with cyclic competitions imposes a limitation on the resulting coexistence states, leading to only selective types of such states. We investigate the effect of nonuniform intraspecific competitions on coexistence and find that a wider spectrum of coexistence states can emerge and persist. This surprising finding is substantiated using three classes of cyclic game models through stability analysis, Monte Carlo simulations and continuous spatiotemporal dynamical evolution from partial differential equations. Our finding indicates that intraspecific competitions or alternative symmetry-breaking mechanisms can promote biodiversity to a broader extent than previously thought.

Solutions to the non-homogeneous parabolic problems by the extended HADM

In this paper, we extend the homogeneous Adomain decomposition method (HADM) introduced by Jang [Bongsoo Jang, Exact solutions to one-dimensional non-homogeneous parabolic problems by the homogeneous decomposition method, Appl. Math. Comput. 186 (2) (2007) 969-979] to solve non-homogeneous parabolic partial differential equations with variable coefficients. The new modified technique provides the less computational work as well as easy calculation in obtaining the accurate approximations compared with the standard ADM. Illustrative examples show the excellent performance for the new modified decomposition method.

Cooling of Heat Sources by Natural Convection Heat Transfer in a Vertical Annulus

This article reports convection heat transfer in a short and tall annular enclosure with two discrete isoflux heat sources of different lengths. The discrete heat sources are mounted at the inner wall and the outer wall is maintained at a lower temperature, whereas the top and bottom walls and the unheated portions of the inner wall are kept at adiabatic. An implicit finite-difference method is employed to solve the vorticity–stream function formulations of the governing equations. The significant influence of the discrete heaters on the flow and heat transfer is analyzed for a wide range of modified Rayleigh numbers, aspect ratio, and length ratio (ϵ) of heat sources. Our numerical results reveal that the average Nusselt number decreases with aspect ratio, whereas the magnitude of maximum temperature increases with the aspect ratio. For most of the parametric cases considered in the present study, the heat transfer rate is found to be higher at the bottom heater than at the top heater except for ϵ = 0.5. The effect of heater length ratio on the heat transfer rate is noticeable for unit aspect ratio, whereas its effect is insignificant as the aspect ratio increases. Furthermore, it was found that the maximum temperature is found generally at the top heater except for the case ϵ = 0.5, where the maximum temperature is found at the bottom heater.

Analysis of least squares pseudo-spectral method for the interface problem of the Navier-Stokes equations

The aim of this paper is to propose and analyze the first order system least squares method for the incompressible Navier-Stokes equation with discontinuous viscosity and singular force along the interface as the earlier work of the first author on Stokes interface problem (Hessari, 2014). Interface conditions are derived, and the Navier-Stokes equation transformed into a first order system of equations by introducing velocity gradient as a new variable. The least squares functional is defined based on L 2 norm applied to the first order system. Both discrete and continuous least squares functionals are put into the canonical form and the existence and uniqueness of branch of nonsingular solutions are shown. The spectral convergence of the proposed method is given. Numerical studies of the convergence are also provided.

A novel semi-analytical approach for solving nonlinear Volterra integro-differential equations

In this work, we present an efficient semi-analytical method based on the Taylor series for solving nonlinear Volterra integro-differential equations, namely the differential transform method (DTM). The DTM provides a recursive relation for the coefficients of the Taylor series that is derived from the given equations. We provide a new recursive relation for the nonlinear Volterra integro-differential equations with complex nonlinear kernels. Since the DTM is based on the Taylor series, it is difficult to obtain accurate approximate solutions in a large domain. To overcome this difficulty, the standard DTM is applied in each subdomain, called the multistage differential transform method (MsDTM). We also present an convergence analysis for the proposed method. To demonstrate the efficiency of the proposed method, several numerical examples are performed and support the results in our analysis.

Multistage homotopy perturbation method for nonlinear reaction networks

In this paper, we present the multistage homotopy perturbation method for finding the solution of the chemical kinetics with nonlinear reactions. We develop a general scheme for finding the analytic solution of chemical reaction networks and apply it to motivating chemical examples such as the enzyme kinetics model and the Brusselator model. We illustrate the numerical result for the models and show the accuracy of the method.