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).open4

Nonlinear Klein-Gordon and Schrodinger Equations by the Projected Differential Transform Method

The differential transform method (DTM) is based on the Taylor series for all variables, but it differs from the traditional Taylor series in calculating coefficients. Even if the DTM is an effective numerical method for solving many nonlinear partial differential equations, there are also some difficulties due to the complex nonlinearity. To overcome difficulties arising in DTM, we present the new modified version of DTM, namely, the projected differential transform method (PDTM), for solving nonlinear partial differential equations. The proposed method is applied to solve the various nonlinear Klein-Gordon and Schrodinger equations. Numerical approximations performed by the PDTM are presented and compared with the results obtained by other numerical methods. The results reveal that PDTM is a simple and effective numerical algorithm.open1

An efficient numerical approach for solving two-point fractional order nonlinear boundary value problems with Robin boundary conditions

This article proposes new strategies for solving two-point Fractional order Nonlinear Boundary Value Problems (FNBVPs) with Robin Boundary Conditions (RBCs). In the new numerical schemes, a two-point FNBVP is transformed into a system of Fractional order Initial Value Problems (FIVPs) with unknown Initial Conditions (ICs). To approximate ICs in the system of FIVPs, we develop nonlinear shooting methods based on Newton's method and Halley's method using the RBC at the right end point. To deal with FIVPs in a system, we mainly employ High-order Predictor-Corrector Methods (HPCMs) with linear interpolation and quadratic interpolation (Nguyen and Jang in Fract. Calc. Appl. Anal. 20(2):447-476, 2017) into Volterra integral equations which are equivalent to FIVPs. The advantage of the proposed schemes with HPCMs is that even though they are designed for solving two-point FNBVPs, they can handle both linear and nonlinear two-point Fractional order Boundary Value Problems (FBVPs) with RBCs and have uniform convergence rates of HPCMs, O(h(2)) and O(h(3)) for shooting techniques with Newton's method and Halley's method, respectively. A variety of numerical examples are demonstrated to confirm the effectiveness and performance of the proposed schemes. Also we compare the accuracy and performance of our schemes with another method

Development of a 2-Channel Embedded Infrared Fiber-Optic Temperature Sensor Using Silver Halide Optical Fibers

A 2-channel embedded infrared fiber-optic temperature sensor was fabricated using two identical silver halide optical fibers for accurate thermometry without complicated calibration processes. In this study, we measured the output voltages of signal and reference probes according to temperature variation over a temperature range from 25 to 225 °C. To decide the temperature of the water, the difference between the amounts of infrared radiation emitted from the two temperature sensing probes was measured. The response time and the reproducibility of the fiber-optic temperature sensor were also obtained. Thermometry with the proposed sensor is immune to changes if parameters such as offset voltage, ambient temperature, and emissivity of any warm object. In particular, the temperature sensing probe with silver halide optical fibers can withstand a high temperature/pressure and water-chemistry environment. It is expected that the proposed sensor can be further developed to accurately monitor temperature in harsh environments

A semi-analytic method with an effect of memory for solving fractional differential equations

In this paper, we propose a new modification of the multistage generalized differential transform method (MsGDTM) for solving fractional differential equations. In MsGDTM, it is the key how to impose an initial condition in each sub-domain to obtain an accurate approximate solution. In several literature works (Odibat et al. in Comput. Math. Appl. 59:1462-1472, 2010; Alomari in Comput. Math. Appl. 61:2528-2534, 2011; Gokdoğan et al. in Math. Comput. Model. 54:2132-2138, 2011), authors have updated an initial condition in each sub-domain by using the approximate solution in the previous sub-domain. However, we point out that this approach is hard to apply an effect of memory which is the basic property of fractional differential equations. Here we provide a new algorithm to impose the initial conditions by using the integral operator that enhances accuracy. Several illustrative examples are demonstrated, and it is shown that the proposed technique is robust and accurate for solving fractional differential equations.close0

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.close1