• Bulletin of the Korean Mathematical Society
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• v.60 no.3
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• pp.593-610
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• 2023

In view of Nevanlinna theory, we investigate the meromorphic solutions of q-difference differential equations and our results give the estimates about counting function and proximity function of meromorphic solutions to these equations. In addition, some interesting results are obtained for two general equations and a class of system of q-difference differential equations.

• Journal of applied mathematics & informatics
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• v.27 no.1_2
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• pp.441-452
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• 2009

In this paper, a fifth order numerical method is presented for solving singularly perturbed differential-difference equations with negative shift. In recent papers the term negative shift has been using for delay. Similar boundary value problems are associated with expected first exit time problem of the membrane, potential in models for neuron and in variational problems in control theory. In the numerical treatment for such type of boundary value problems, first we use Taylor approximation to tackle terms containing small shifts which converts it to a boundary value problem for singularly perturbed differential equation. The two point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a fifth order compact difference scheme is presented for the first order system and is solved using the boundary conditions. Several numerical examples are solved and compared with exact solution. It is observed that present method approximates the exact solution very well.

• Journal of applied mathematics & informatics
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• v.40 no.3_4
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• pp.409-431
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• 2022

In this paper, we deal with the existence of solutions of the following system of nonlinear rational difference equations with order five $x_{n+1}=\frac{y_{n-3}x_{n-4}}{y_n(a+by_{n-3}x_{n-4})}$, $y_{n+1}=\frac{x_{n-3}y_{n-4}}{x_n(c+dx_{n-3}y_{n-4})}$, n = 0, 1, ⋯, where parameters a, b, c and d are not executed at the same time and initial conditions x_-4, x_-3, x_-2, x_-1, x₀, y_-4, y_-3, y_-2, y_-1 and y₀ are non zero real numbers.

• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.689-706
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• 2008

In this paper, a fifth order numerical method is presented for solving singularly perturbed two point boundary value problems with a boundary layer at one end point. The two point boundary value problem is transformed into general first order ordinary differential equation system. A discrete approximation of a fifth order compact difference scheme is presented for the first order system. An asymptotically equivalent first order equation of the original singularly perturbed two point boundary value problem is obtained from the theory of singular perturbations. It is used in the fifth order compact difference scheme to get a two term recurrence relation and is solved. Several linear and non-linear singular perturbation problems have been solved and the numerical results are presented to support the theory. It is observed that the present method approximates the exact solution very well.

• Nuclear Engineering and Technology
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• v.41 no.10
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• pp.1347-1360
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• 2009

A multi-dimensional component for the thermal-hydraulic system analysis code, MARS, was developed for a more realistic three-dimensional analysis of nuclear systems. A three-dimensional and two-fluid model for a two-phase flow in Cartesian and cylindrical coordinates was employed. The governing equations and physical constitutive relationships were extended from those of a one-dimensional version. The numerical solution method adopted a semi-implicit and finite-difference method based on a staggered-grid mesh and a donor-cell scheme. The relevant length scale was very coarse compared to commercial computational fluid dynamics tools. Thus a simple Prandtl's mixing length turbulence model was applied to interpret the turbulent induced momentum and energy diffusivity. Non drag interfacial forces were not considered as in the general nuclear system codes. Several conceptual cases with analytic solutions were chosen and analyzed to assess the fundamental terms. RPI air-water and UPTF 7 tests were simulated and compared to the experimental data. The simulation results for the RPI air-water two-phase flow experiment showed good agreement with the measured void fraction. The simulation results for the UPTF downcomer test 7 were compared to the experiment data and the results from other multi-dimensional system codes for the ECC delivery flow.

• Journal of KIISE:Computer Systems and Theory
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• v.32 no.11_12
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• pp.692-704
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• 2005

With the improvement of computer hardware, GPUs(Graphics Processor Units) have tremendous memory bandwidth and computation power. This leads GPUs to use in general purpose computation. Especially, GPU implementation of compute-intensive physics based simulations is actively studied. In the solution of differential equations which are base of physics simulations, tridiagonal matrix systems occur repeatedly by finite-difference approximation. From the point of view of physics based simulations, fast solution of tridiagonal matrix system is important research field. We propose a fast GPU implementation for the solution of tridiagonal matrix systems. In this paper, we implement the cyclic reduction(also known as odd-even reduction) algorithm which is a popular choice for vector processors. We obtained a considerable performance improvement for solving tridiagonal matrix systems over Thomas method and conjugate gradient method. Thomas method is well known as a method for solving tridiagonal matrix systems on CPU and conjugate gradient method has shown good results on GPU. We experimented our proposed method by applying it to heat conduction, advection-diffusion, and shallow water simulations. The results of these simulations have shown a remarkable performance of over 35 frame-per-second on the 1024x1024 grid.

• Journal of the Korea Convergence Society
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• v.8 no.12
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• pp.283-289
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• 2017

In this study, a general program has been developed to calculate the static design factor of a vehicle suspension system. The partial derivatives of Jacobians for constraint equations are calculated using the symbolic technique. In the commercial program, finite difference method is used to calculate the Jacobian matrix of Jacobian. But in this study, it is calculated by using the symbol calculation method to precisely consider it. The calculated Jacobian matrix for the system has proved its accuracy through the solution of the numerical example. A simulation was performed for a double wishbone suspension of a 1/4 vehicle. The result can be used to calculate the static design factor of the suspension, and also add a convergence module that can perform virtual tests.

• Journal of Biosystems Engineering
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• v.3 no.2
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• pp.35-61
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• 1978

To find out the power tiller's travel and tractive characteristics on the general slope land, the tractive p:nver transmitting system was divided into the internal an,~ external power transmission systems. The performance of power tiller's engine which is the initial unit of internal transmission system was tested. In addition, the mathematical model for the tractive force of driving wheel which is the initial unit of external transmission system, was derived by energy and force balance. An analytical solution of performed for tractive forces was determined by use of the model through the digital computer programme. To justify the reliability of the theoretical value, the draft force was measured by the strain gauge system on the general slope land and compared with theoretical values. The results of the analytical and experimental performance of power tiller on the field may be summarized as follows; (1) The mathematical equation of rolIing resistance was derived as $$Rh=\frac {W_z-AC \[1+ \frac{sl}{K} \(\varrho ^{-\frac{sl}{K}-1\)\] sin\theta_1}} {tan\phi \[1+ \frac{sl}{K} \(\varrho ^{-\frac{sl}{K}-1\)\]+\frac{tan\theta_1}{1}$$ and angle of rolling resistance as $$\theta _1 - tan^1\[ \frac {2T(AcrS_0 - T)+\sqrt (T-AcrS_0)^2(2T)^2-4(T^2-W_2^2r^2)\times (T-AcrS_0)^2 W_z^2r^2S_0^2tan^2\phi} {2(T^2-W_z^2r^2)S_0tan\phi}\] $$and the equation of frft force was derived as$$P=(AC+Rtan\phi)\[1+ \frac{sl}{K} \(\varrho ^{-\frac{sl}{K}-1\)\]cos\phi_1 \ulcorner \frac {W_z \ulcorner{AC\[ [1+ \frac{sl}{K} \(\varrho ^{-\frac{sl}{K}-1\)\]sin\phi_1 {tan\phi[1+ \frac{sl}{K} \(\varrho ^{-\frac{sl}{K}-1\]+ \frac {tan\phi_1} { 1} \ulcorner W_1sin\alpha $$The slip coefficient K in these equations was fitted to approximately 1. 5 on the level lands and 2 on the slope land. (2) The coefficient of rolling resistance Rn was increased with increasing slip percent 5 and did not influenced by the angle of slope land. The angle of rolling resistance Ol was increasing sinkage Z of driving wheel. The value of Ol was found to be within the limits of Ol =2\ulcorner "'16\ulcorner. (3) The vertical weight transfered to power tiller on general slope land can be estim ated by use of th~ derived equation: $$R_pz= \frac {\sum_{i=1}^{4}{W_i}} {l_T} { (l_T-l) cos\alpha cos\beta \ulcorner \bar(h) sin \alpha - W_1 cos\alpha cos\beta$$The vertical transfer weight $R_pz$ was decreased with increasing the angle of slope land. The ratio of weight difference of right and left driving wheel on slop eland,$\lambda= \frac { {W_L_Z} - {W_R_Z}} {W_Z} $, was increased from ,$\lambda$=0 to$\lambda$=0.4 with increasing the angle of side slope land ($\beta = 0^\circ~20^\circ) (4) In case of no draft resistance, the difference between the travelling velocities on the level and the slope land was very small to give 0.5m/sec, in which the travelling velocity on the general slope land was decreased in curvilinear trend as the draft load increased. The decreasing rate of travelling velocity by the increase of side slope angle was less than that by the increase of hill slope angle a, (5) Rate of side slip by the side slope angle was defined as $ S_r=\frac {S_s}{l_s} \times$ 100( %), and the rate of side slip of the low travelling velocity was larger than that of the high travelling velocity. (6) Draft forces of power tiller did not affect by the angular velocity of driving wheel, and maximum draft coefficient occurred at slip percent of S=60% and the maximum draft power efficiency occurred at slip percent of S=30%. The maximum draft coefficient occurred at slip percent of S=60% on the side slope land, and the draft coefficent was nearly constant regardless of the side slope angle on the hill slope land. The maximum draft coefficient occurred at slip perecent of S=65% and it was decreased with increasing hill slope angle $\alpha$. The maximum draft power efficiency occurred at S=30 % on the general slope land. Therefore, it would be reasonable to have the draft operation at slip percent of S=30% on the general slope land. (7) The portions of the power supplied by the engine of the power tiller which were used as the source of draft power were 46.7% on the concrete road, 26.7% on the level land, and 13~20%; on the general slope land ($\alpha = O~ 15^\circ ,\beta = 0 ~ 10^\circ$) , respectively. Therefore, it may be desirable to develope the new mechanism of the external pO'wer transmitting system for the general slope land to improved its performance.l slope land to improved its performance.

• Journal of Biosystems Engineering
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• v.3 no.2
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• pp.34-34
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• 1978

To find out the power tiller's travel and tractive characteristics on the general slope land, the tractive p:nver transmitting system was divided into the internal an,~ external power transmission systems. The performance of power tiller's engine which is the initial unit of internal transmission system was tested. In addition, the mathematical model for the tractive force of driving wheel which is the initial unit of external transmission system, was derived by energy and force balance. An analytical solution of performed for tractive forces was determined by use of the model through the digital computer programme. To justify the reliability of the theoretical value, the draft force was measured by the strain gauge system on the general slope land and compared with theoretical values. The results of the analytical and experimental performance of power tiller on the field may be summarized as follows; (1) The mathematical equation of rolIing resistance was derived as $$Rh=\frac {W_z-AC \[1+ \frac{sl}{K} \(\varrho ^{-\frac{sl}{K}-1\)\] sin\theta_1}} {tan\phi \[1+ \frac{sl}{K} \(\varrho ^{-\frac{sl}{K}-1\)\]+\frac{tan\theta_1}{1}$$ and angle of rolling resistance as $$\theta _1 - tan^1\[ \frac {2T(AcrS_0 - T)+\sqrt (T-AcrS_0)^2(2T)^2-4(T^2-W_2^2r^2)\times (T-AcrS_0)^2 W_z^2r^2S_0^2tan^2\phi} {2(T^2-W_z^2r^2)S_0tan\phi}\] $$and the equation of frft force was derived as$$P=(AC+Rtan\phi)\[1+ \frac{sl}{K} \(\varrho ^{-\frac{sl}{K}-1\)\]cos\phi_1 ? \frac {W_z ?{AC\[ [1+ \frac{sl}{K} \(\varrho ^{-\frac{sl}{K}-1\)\]sin\phi_1 {tan\phi[1+ \frac{sl}{K} \(\varrho ^{-\frac{sl}{K}-1\]+ \frac {tan\phi_1} { 1} ? W_1sin\alpha $$The slip coefficient K in these equations was fitted to approximately 1. 5 on the level lands and 2 on the slope land. (2) The coefficient of rolling resistance Rn was increased with increasing slip percent 5 and did not influenced by the angle of slope land. The angle of rolling resistance Ol was increasing sinkage Z of driving wheel. The value of Ol was found to be within the limits of Ol =2? "'16?. (3) The vertical weight transfered to power tiller on general slope land can be estim ated by use of th~ derived equation: $$R_pz= \frac {\sum_{i=1}^{4}{W_i}} {l_T} { (l_T-l) cos\alpha cos\beta ? \bar(h) sin \alpha - W_1 cos\alpha cos\beta$$The vertical transfer weight $R_pz$ was decreased with increasing the angle of slope land. The ratio of weight difference of right and left driving wheel on slop eland,$\lambda= \frac { {W_L_Z} - {W_R_Z}} {W_Z} $, was increased from ,$\lambda$=0 to$\lambda$=0.4 with increasing the angle of side slope land ($\beta = 0^\circ~20^\circ) (4) In case of no draft resistance, the difference between the travelling velocities on the level and the slope land was very small to give 0.5m/sec, in which the travelling velocity on the general slope land was decreased in curvilinear trend as the draft load increased. The decreasing rate of travelling velocity by the increase of side slope angle was less than that by the increase of hill slope angle a, (5) Rate of side slip by the side slope angle was defined as $ S_r=\frac {S_s}{l_s} \times$ 100( %), and the rate of side slip of the low travelling velocity was larger than that of the high travelling velocity. (6) Draft forces of power tiller did not affect by the angular velocity of driving wheel, and maximum draft coefficient occurred at slip percent of S=60% and the maximum draft power efficiency occurred at slip percent of S=30%. The maximum draft coefficient occurred at slip percent of S=60% on the side slope land, and the draft coefficent was nearly constant regardless of the side slope angle on the hill slope land. The maximum draft coefficient occurred at slip perecent of S=65% and it was decreased with increasing hill slope angle $\alpha$. The maximum draft power efficiency occurred at S=30 % on the general slope land. Therefore, it would be reasonable to have the draft operation at slip percent of S=30% on the general slope land. (7) The portions of the power supplied by the engine of the power tiller which were used as the source of draft power were 46.7% on the concrete road, 26.7% on the level land, and 13~20%; on the general slope land ($\alpha = O~ 15^\circ ,\beta = 0 ~ 10^\circ$) , respectively. Therefore, it may be desirable to develope the new mechanism of the external pO'wer transmitting system for the general slope land to improved its performance.