The mathematical simulation of water contaminant measurement is often used to assess the water quality. The monitoring point placement for water quality measurement in an opened-closed reservoir can give accurate or inaccurate assessment. In this research, the mathematical model of the approximated water quality in an opened-closed reservoir with removal mechanism system is proposed. The water quality model consists of the hydrodynamic model and the dispersion model. The hydrodynamic model is used to describe the water current in the opened-closed reservoir. The transient advection-diffusion equation with removal mechanism provides the water pollutant concentration. The water velocity from the hydrodynamic model is plugged into the dispersion model. The finite difference techniques are used to approximate the solution of the water quality model. The proposed numerical simulations give a suitable area of zonal removal mechanism placement. The proposed simulations also give the overall and specified approximated water quality for each point and time when the exit gate is opened on the different periods of time. In addition, the proposed techniques can give a suitable period of time to open the exit gate to achieve a good agreement water quality by using contaminant removal mechanism.

The hydrodynamic model is used to determine the water wave flow. In this research, a nondimensional form of a two-dimensional hydrodynamic model with generalized boundary condition g(x, t) and initial conditions for describing the elevation of water wave in an open uniform reservoir is proposed. The separation of variables method with mathematical induction is employed to find an analytical solution to the model. An example of flow calculations in an open uniform reservoir is also demonstrated. MSC: 35L05; 35L04; 35L40

The purpose of this research is to present the steps of one-dimensional differential transformation method (DTM) to find the series solutions for the vibrations of a circular membrane under the specified initial and boundary conditions. The problems will be studied in the both cases of vibrations depending only on radius and of the vibrations depending on both radius and angle. We illustrate four examples of problems which the exact solutions can be solve analytically and compare them to the DTM results, to show that the DTM is reliable and of high accuracy. This work shows that the DTM is easier to use than the analytical method from the point of view of programming.

This paper is concerned with a finding of an approximate solution to the Duffing oscillator with a linear external force using a modified differential transform method (MDTM) and Runge-Kutta-Nystrom (RKN) method. The results obtained from the two methods are then numerically and graphically compared. We found that both results are close to one another when an amplitude and frequency of external forces are very small. However, both results are quite different when the amplitude and frequency of external forces are large. Their results are in agreement with the phase plane diagram which demonstrates a stable equilibrium point for both methods in case of small amplitude and frequency of external force. However, for large amplitude and frequency of external forces, only MDTM shows the stable equilibrium point.