• Title, Summary, Keyword: finite difference method

### A Generalized Finite Difference Method for Solving Fokker-Planck-Kolmogorov Equations

• Zhao, Li;Yun, Gun Jin
• International Journal of Aeronautical and Space Sciences
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• v.18 no.4
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• pp.816-826
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• 2017
• In this paper, a generalized discretization scheme is proposed that can derive general-order finite difference equations representing the joint probability density function of dynamic response of stochastic systems. The various order of finite difference equations are applied to solutions of the Fokker-Planck-Kolmogorov (FPK) equation. The finite difference equations derived by the proposed method can greatly increase accuracy even at the tail parts of the probability density function, giving accurate reliability estimations. Compared with exact solutions and finite element solutions, the generalized finite difference method showed increasing accuracy as the order increases. With the proposed method, it is allowed to use different orders and types (i.e. forward, central or backward) of discretization in the finite difference method to solve FPK and other partial differential equations in various engineering fields having requirements of accuracy or specific boundary conditions.

### A fourth order finite difference method applied to elastodynamics: Finite element and boundary element formulations

• Souza, L.A.;Carrer, J.A.M.;Martins, C.J.
• Structural Engineering and Mechanics
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• v.17 no.6
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• pp.735-749
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• 2004
• This work presents a direct integration scheme, based on a fourth order finite difference approach, for elastodynamics. The proposed scheme was chosen as an alternative for attenuating the errors due to the use of the central difference method, mainly when the time-step length approaches the critical time-step. In addition to eliminating the spurious numerical oscillations, the fourth order finite difference scheme keeps the advantages of the central difference method: reduced computer storage and no requirement of factorisation of the effective stiffness matrix in the step-by-step solution. A study concerning the stability of the fourth order finite difference scheme is presented. The Finite Element Method and the Boundary Element Method are employed to solve elastodynamic problems. In order to verify the accuracy of the proposed scheme, two examples are presented and discussed at the end of this work.

### ANALYSIS OF A ONE-DIMENSIONAL FIN USING THE ANALYTIC METHOD AND THE FINITE DIFFERENCE METHOD

• Han, Young-Min;Cho, Joo-Suk;Kang, Hyung-Suk
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.9 no.1
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• pp.91-98
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• 2005
• The straight rectangular fin is analyzed using the one-dimensional analytic method and the finite difference method. For the finite difference method, the numbers of nodes vary from 20 to 100. The relative errors of heat loss and temperature between the analytic method and the finite difference method are represented as a function of Biot Number and dimensionless fin length. One of the results shows that the relative error between the analytic method and the finite difference method decreases as the numbers of nodes for finite difference method increase.

### Efficient 3D Acoustic Wave Propagation Modeling using a Cell-based Finite Difference Method (셀 기반 유한 차분법을 이용한 효율적인 3차원 음향파 파동 전파 모델링)

• Park, Byeonggyeong;Ha, Wansoo
• Geophysics and Geophysical Exploration
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• v.22 no.2
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• pp.56-61
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• 2019
• In this paper, we studied efficient modeling strategies when we simulate the 3D time-domain acoustic wave propagation using a cell-based finite difference method which can handle the variations of both P-wave velocity and density. The standard finite difference method assigns physical properties such as velocities of elastic waves and density to grid points; on the other hand, the cell-based finite difference method assigns physical properties to cells between grid points. The cell-based finite difference method uses average physical properties of adjacent cells to calculate the finite difference equation centered at a grid point. This feature increases the computational cost of the cell-based finite difference method compared to the standard finite different method. In this study, we used additional memory to mitigate the computational overburden and thus reduced the calculation time by more than 30 %. Furthermore, we were able to enhance the performance of the modeling on several media with limited density variations by using the cell-based and standard finite difference methods together.

### FRACTIONAL CHEBYSHEV FINITE DIFFERENCE METHOD FOR SOLVING THE FRACTIONAL BVPS

• Journal of applied mathematics & informatics
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• v.31 no.1_2
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• pp.299-309
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• 2013
• In this paper, we introduce a new numerical technique which we call fractional Chebyshev finite difference method (FChFD). The algorithm is based on a combination of the useful properties of Chebyshev polynomials approximation and finite difference method. We tested this technique to solve numerically fractional BVPs. The proposed technique is based on using matrix operator expressions which applies to the differential terms. The operational matrix method is derived in our approach in order to approximate the fractional derivatives. This operational matrix method can be regarded as a non-uniform finite difference scheme. The error bound for the fractional derivatives is introduced. The fractional derivatives are presented in terms of Caputo sense. The application of the method to fractional BVPs leads to algebraic systems which can be solved by an appropriate method. Several numerical examples are provided to confirm the accuracy and the effectiveness of the proposed method.

### A Comparison between 3-D Analytical and Finite Difference Method for a Trapezoidal Profile Fin

• Lee, Sung-Joo;Kang, Hyung-Suk
• Journal of Industrial Technology
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• v.21 no.A
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• pp.41-50
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• 2001
• A comparison is made of the temperature distribution and heat loss from a trapezoidal profile fin using two different 3-dimensional methods. These two methods are analytical and finite difference methods. In the finite difference method 78 nodes are used for a fourth of the fin. A trapezoidal profile fin being the height of the fin tip is half of that of the fin base is chosen arbitrarily as the model. One of the results shows that the relative error in the total convection heat loss obtained by using 78 nodes in the finite difference method as compared to the heat conduction through the fin root obtained by analytic method seems to be good (i.e., -3.5%

### One-node and two-node hybrid coarse-mesh finite difference algorithm for efficient pin-by-pin core calculation

• Song, Seongho;Yu, Hwanyeal;Kim, Yonghee
• Nuclear Engineering and Technology
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• v.50 no.3
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• pp.327-339
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• 2018
• This article presents a new global-local hybrid coarse-mesh finite difference (HCMFD) method for efficient parallel calculation of pin-by-pin heterogeneous core analysis. In the HCMFD method, the one-node coarse-mesh finite difference (CMFD) scheme is combined with a nodal expansion method (NEM)-based two-node CMFD method in a nonlinear way. In the global-local HCMFD algorithm, the global problem is a coarse-mesh eigenvalue problem, whereas the local problems are fixed source problems with boundary conditions of incoming partial current, and they can be solved in parallel. The global problem is formulated by one-node CMFD, in which two correction factors on an interface are introduced to preserve both the surface-average flux and the net current. Meanwhile, for accurate and efficient pin-wise core analysis, the local problem is solved by the conventional NEM-based two-node CMFD method. We investigated the numerical characteristics of the HCMFD method for a few benchmark problems and compared them with the conventional two-node NEM-based CMFD algorithm. In this study, the HCMFD algorithm was also parallelized with the OpenMP parallel interface, and its numerical performances were evaluated for several benchmarks.

### Numerical analysis of FGM plates with variable thickness subjected to thermal buckling

• Bouguenina, Otbi;Belakhdar, Khalil;Tounsi, Abdelouahed;Adda Bedia, El Abbes
• Steel and Composite Structures
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• v.19 no.3
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• pp.679-695
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• 2015
• A numerical solution using finite difference method to evaluate the thermal buckling of simply supported FGM plate with variable thickness is presented in this research. First, the governing differential equation of thermal stability under uniform temperature through the plate thickness is derived. Then, the governing equation has been solved using finite difference method. After validating the presented numerical method with the analytical solution, the finite difference formulation has been extended in order to include variable thickness. The accuracy of the finite difference method for variable thickness plate has been also compared with the literature where a good agreement has been found. Furthermore, a parametric study has been conducted to analyze the effect of material and geometric parameters on the thermal buckling resistance of the FGM plates. It was found that the thickness variation affects isotropic plates a bit more than FGM plates.

### A FINITE DIFFERENCE SCHEME FOR RLW-BURGERS EQUATION

• Zhao, Xiaohong;Li, Desheng;Shi, Deming
• Journal of applied mathematics & informatics
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• v.26 no.3_4
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• pp.573-581
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• 2008
• In this paper, a finite difference method for a Cauchy problem of RLW-Burgers equation was considered. Although the equation is not energy conservation, we have given its the energy conservative finite difference scheme with condition. Convergence and stability of the difference solution were proved. Numerical results demonstrate that the method is efficient and reliable.

### Time-Dependent Optimal Heater Control Using Finite Difference Method

• Li, Zhen-Zhe;Heo, Kwang-Su;Choi, Jun-Hoo;Seol, Seoung-Yun
• Proceedings of the KSME Conference
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• pp.2254-2255
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• 2008
• Thermoforming is one of the most versatile and economical process to produce polymer products. The drawback of thermoforming is difficult to control thickness of final products. Temperature distribution affects the thickness distribution of final products, but temperature difference between surface and center of sheet is difficult to decrease because of low thermal conductivity of ABS material. In order to decrease temperature difference between surface and center, heating profile must be expressed as exponential function form. In this study, Finite Difference Method was used to find out the coefficients of optimal heating profiles. Through investigation, the optimal results using Finite Difference Method show that temperature difference between surface and center of sheet can be remarkably minimized with satisfying Temperature of Forming Window.