A Boundary Integral Equation Formulation for an Unsteady Anisotropic-Diffusion Convection Equation of Exponentially Variable Coefficients and Compressible Flow

Mohammad Ivan Azis 경북대학교 자연과학대학 수학과 2022 Kyungpook mathematical journal Vol.62 No.3

The anisotropic-diffusion convection equation with exponentially variable co efficients is discussed in this paper. Numerical solutions are found using a combined Laplace transform and boundary element method. The variable coefficients equation is usually used to model problems of functionally graded media. First the variable coeffi cients equation is transformed to a constant coefficients equation. The constant coefficients equation is then Laplace-transformed so that the time variable vanishes. The Laplace transformed equation is consequently written as a boundary integral equation which in volves a time-free fundamental solution. The boundary integral equation is therefore employed to find numerical solutions using a standard boundary element method. Finally the results obtained are inversely transformed numerically using the Stehfest formula to get solutions in the time variable. The combined Laplace transform and boundary element method are easy to implement and accurate for solving unsteady problems of anisotropic exponentially graded media governed by the diffusion convection equation.

Diffusion synthetic acceleration with the fine mesh rebalance of the subcell balance method with tetrahedral meshes for S_N transport calculations

Muhammad, Habib,Hong, Ser Gi Korean Nuclear Society 2020 Nuclear Engineering and Technology Vol.52 No.3

A diffusion synthetic acceleration (DSA) technique for the S_N transport equation discretized with the linear discontinuous expansion method with subcell balance (LDEM-SCB) on unstructured tetrahedral meshes is presented. The LDEM-SCB scheme solves the transport equation with the discrete ordinates method by using the subcell balances and linear discontinuous expansion of the flux. Discretized DSA equations are derived by consistently discretizing the continuous diffusion equation with the LDEM-SCB method, however, the discretized diffusion equations are not fully consistent with the discretized transport equations. In addition, a fine mesh rebalance (FMR) method is devised to accelerate the discretized diffusion equation coupled with the preconditioned conjugate gradient (CG) method. The DSA method is applied to various test problems to show its effectiveness in speeding up the iterative convergence of the transport equation. The results show that the DSA method gives small spectral radii for the tetrahedral meshes having various minimum aspect ratios even in highly scattering dominant mediums for the homogeneous test problems. The numerical tests for the homogeneous and heterogeneous problems show that DSA with FMR (with preconditioned CG) gives significantly higher speedups and robustness than the one with the Gauss-Seidel-like iteration.

자성 입자의 확산 방정식에 대한 알려진 해에 관한 검토

김학진 ( Hackjin Kim ) 충남대학교 기초과학연구소 2016 忠南科學硏究誌 Vol.33 No.1

Magnetic particles of solution sedimentate by gravitational or magnetic field and the sedimentation dynamics is described by a kind of diffusion equation. Since the diffusion equation is the second order differential equation, it is not easy to find the solutions of the equations under specific boundary conditions. In this study, a reported solution for the diffusion under a special boundary condition, [Y. L. Raikher and M. I. Shliomis, J. Magn. Magn, Mater. 122 (1993) 93] is examined and a way to find the solution of the diffusion equation is suggested.

TRANSFORMATION OF DIMENSIONLESS HEAT DIFFUSION EQUATION FOR THE SOLUTION OF DYNAMIC DOMAIN IN PHASE CHANGE PROBLEMS

MUHAMMAD ASHRAF,R. AVILA,S. S. RAZA 한국산업응용수학회 2009 Journal of the Korean Society for Industrial and A Vol.13 No.1

In the present work transformation of dimensionless heat diffusion equation for the solution of moving boundary problems have been formulated. The formulation is based on 1-D, 2-D and 3-D, unsteady heat diffusion equations. These equations are first turned into dimensionless form by using dimensionless quantities and their transformation was formulated in liquid and solid phases. The salient feature of this work is that during the transformation of dimensionless heat diffusion equation there arises a convective term v which is responsible for the motion of interface in liquid as well as solid phase. In the transformed heat equation, a correction factor β also arises naturally which gives the correct transformed flux at interface.

적응 가중 미디언 필터를 이용한 영상 확산 알고리즘

황인호,이경훈,김웅희,Hwang, In-Ho,Lee, Kyung-Hoon,Kim, Woong-Hee 한국통신학회 2007 韓國通信學會論文誌 Vol.32 No.5c

Recently, many research activities in the image processing area are concentrated on developing new algorithms by finding the solution of the 'diffusion equation'. The diffusion algorithms are expected to be utilized in numerous applications including noise removal and image restoration, edge detection, segmentation, etc. In this paper, at first, it will be shown that the anisotropic diffusion algorithms have the similar structure with the adaptive FIR filters with cross-shaped 5-tap kernel, and this relatively small-sized kernel causes many iterating procedure for satisfactory filtering effects. Moreover, it will also be shown that lots of modifications which are adopted to the conventional Gaussian diffusion method in order to weaken the edge blurring nature of the linear filtering process increases another computational burden. We propose a new Median diffusion scheme by replacing the adaptive linear filters in the diffusion process with the AWM (Adaptive Weighted Median) filters. A diffusion-equation-based adaptation scheme is also proposed. With the proposed scheme, the size of the diffusion kernel can be increased, and thus diffusion speed greatly increases. Simulation results shows that the proposed Median diffusion scheme outperforms in noise removal (especially impulsive noise), and edge preservation. 편미분 방정식을 도입하여 새로운 영상처리 기술을 개발하려는 연구가 활발히 진행 중이며, 특히 확산 방정식을 풀어 잡음 제거, 영상 복원, 에지 검출 및 영상 분할 등에 응용할 수 있는 이미지 확산 알고리즘에 관심이 높다. 본 논문에서는 기존의 비등방성 확산 방식이 결국은 커널 크기가 작은 적응 필터링 방식과 동일한 효과를 낸다는 것을 보이고, 확산 과정에서 선형 필터의 단점을 보완할 수 있도록 가중 미디언(WM, Weighted Median) 필터를 적용한 새로운 확산 기법을 제안하였다. 제안된 WM 필터가 비등방성 커널을 갖도록 필터계수에 대응하는 가중치들을 이미지의 국부적인 변화량에 따라 적응적으로 가변할 수 있는 기법을 제안하였다. 뿐만 아니라 반복 과정에서의 확산 속도를 증가할 수 있도록 커널의 크기를 증가시키기 위한 방안도 제시하였다. 실제 영상을 사용한 실험을 통하여 제안된 방식이 기존의 방식에 비해 잡음 제거 (특히 임펄스성 잡음) 특성이나 에지 보존 특성이 더 우수하다는 것을 보였다. 또한 기존의 방식에 비해 확장된 크기를 갖는 커널을 이용함으로써 확산 속도를 높일 수 있다는 것을 보였다.

단순 우성 중성자 수송방정식을 이용한 노달 수송해법

노태완 한국방사성폐기물학회 2018 방사성폐기물학회지 Vol.16 No.2

Nodal transport methods are proposed for solving the simplified even-parity neutron transport (SEP) equation. These new methods are attributed to the success of existing nodal diffusion methods such as the Polynomial Expansion Nodal and the Analytic Function Expansion Nodal Methods, which are known to be very effective for solving the neutron diffusion equation. Numerical results show that the simplified even-parity transport equation is a valid approximation to the transport equation and that the two nodal methods developed in this study also work for the SEP transport equation, without conflict. Since accuracy of methods is easily increased by adding node unknowns, the proposed methods will be effective for coarse mesh calculation and this will also lead to computation efficiency. 중성자 확산방정식에 대해 개발된 노달 확산이론을 단순 우성 중성자 수송방정식에 적용할 수 있는 노달 수송이론을 제시한다. 노달이론으로 다항식전개 노달법과 해석함수전개 노달법을 채택하였고 단순 우성 수송방정식은 수송방정식에 대한 합리적 근사이며 기존의 노달해법이 방향 차분된 단순 우성 수송방정식에 정확히 적용될 수 있음을 수치적으로 확인하였다. 본 연구에서는 방법론 개발이 목적이므로 노드 당 최소한의 미지수를 정의하여 사용했지만 미지수를 추가함으로써 정확도를 증가시킬 수 있음은 기존의 노달 확산이론의 경우와 같다. 즉 중성자 수송방정식에 대해 노달이론을 적용하여 소격격자에 대해 계산 정확성이 확보되고 이는 결국 계산 효율성 증대로 나타난다.

Nodal Transport Methods Using the Simplified Even-Parity Neutron Transport Equation

Taewan Noh 한국방사성폐기물학회 2018 방사성폐기물학회지 Vol.16 No.2

Nodal transport methods are proposed for solving the simplified even-parity neutron transport (SEP) equation. These new methods are attributed to the success of existing nodal diffusion methods such as the Polynomial Expansion Nodal and the Analytic Function Expansion Nodal Methods, which are known to be very effective for solving the neutron diffusion equation. Numerical results show that the simplified even-parity transport equation is a valid approximation to the transport equation and that the two nodal methods developed in this study also work for the SEP transport equation, without conflict. Since accuracy of methods is easily increased by adding node unknowns, the proposed methods will be effective for coarse mesh calculation and this will also lead to computation efficiency.

Generalized Tikhonov methods for an inverse source problem of the time-fractional diffusion equation

Ma, Yong-Ki,Prakash, P.,Deiveegan, A. Elsevier 2018 Chaos, solitons, and fractals Vol.108 No.-

<P><B>Abstract</B></P> <P>In this paper, we identify the unknown space-dependent source term in a time-fractional diffusion equation with variable coefficients in a bounded domain where additional data are consider at a fixed time. Using the generalized and revised generalized Tikhonov regularization methods, we construct regularized solutions. Convergence estimates for both methods under an <I>a-priori</I> and <I>a-posteriori</I> regularization parameter choice rules are given, respectively. Numerical example shows that the proposed methods are effective and stable.</P> <P><B>Highlights</B></P> <P> <UL> <LI> This paper investigates the problem of determining a space-dependent source term in a time-fractional diffusion equation with variable coefficients in a bounded domain where additional data are consider at a fixed time. Using the generalized and revised generalized Tikhonov regularization methods, we construct regularized solutions. </LI> <LI> One important feature of our paper is that the Convergence estimates for both methods under a-priori and a-posteriori regularization parameter choice rules are given, respectively. </LI> <LI> In this work we have extended the revised generalized Tikhonov regularization method for the inverse problem of determining the source term in time-fractional diffusion equation. The obtained result shows a new contribution in the field of fractional diffusion equation. </LI> <LI> However it should be emphasized that the revised generalized Tikhonov regularization method is mainly concerned with inverse source problems for the heat equation and there have been no attempts made for studying the time-fractional diffusion problem. </LI> </UL> </P>

An Asymptotic finite element method for singularly perturbed higher order ordinary differential equations of convection-diffusion type with discontinuous source term

A. Ramesh Babu,N. Ramanujam 한국전산응용수학회 2008 Journal of applied mathematics & informatics Vol.26 No.5

We consider singularly perturbed Boundary Value Problems (BVPs) for third and fourth order Ordinary Differential Equations(ODEs) of convection-diffusion type with discontinuous source term and a small positive parameter multiplying the highest derivative. Because of the type of Boundary Conditions(BCs) imposed on these equations these problems can be transformed into weakly coupled systems. In this system, the first equation does not have the small parameter but the second contains it. In this paper a computational method named as " An asymptotic finite element method " for solving these systems is presented. In this method we first find an zero order asymptotic approximation to the solution and then the system is decoupled by replacing the first component of the solution by this approximation in the second equation. Then the second equation is independently solved by a fitted mesh Finite Element Method (FEM). Numerical experiments support our theoritical results. We consider singularly perturbed Boundary Value Problems (BVPs) for third and fourth order Ordinary Differential Equations(ODEs) of convection-diffusion type with discontinuous source term and a small positive parameter multiplying the highest derivative. Because of the type of Boundary Conditions(BCs) imposed on these equations these problems can be transformed into weakly coupled systems. In this system, the first equation does not have the small parameter but the second contains it. In this paper a computational method named as " An asymptotic finite element method " for solving these systems is presented. In this method we first find an zero order asymptotic approximation to the solution and then the system is decoupled by replacing the first component of the solution by this approximation in the second equation. Then the second equation is independently solved by a fitted mesh Finite Element Method (FEM). Numerical experiments support our theoritical results.

한국인 폐확산능 정상예측식의 임상적 유용성과 정확성

나승원 ( Seung Won Ra ),박태선 ( Tai Sun Park ),홍윤기 ( Yoon Ki Hong ),홍상범 ( Sang Bum Hong ),심태선 ( Tae Sun Shim ),임채만 ( Chae Man Lim ),이상도 ( Sang Do Lee ),고윤석 ( Youn Suck Koh ),김우성 ( Woo Sung Kim ),김동순 ( Dong 대한결핵 및 호흡기학회 2008 Tuberculosis and Respiratory Diseases Vol.64 No.2

연구배경: 폐확산능을 해석하는 데 필요한 정상예측식으로는 한국인을 대상으로 하여 박 등이 개발한 식(박 식)이 있으나 아직 외국 정상예측식을 많이 사용하고 있다. 이에 국내에서 많이 사용하는 외국 정상예측식인 Burrows 식과 박 식의 임상적 유용성과 정확성을 비교하고자 하였다. 방법: 1. 임상적 유용성 연구; 2006년 7월부터 12월까지 6개월간 폐확산능검사를 시행한 환자 중 두 식을 각각 정상예측식으로 적용하였을 때 폐확산능 해석이 다른 276명(대상군 A)을 대상으로 하였다. 대상군 A에게 두 식을 각각 적용하였을 때 폐확산능 해석과 임상적 판단과의 일치도를 비교하여 임상적 유용성을 평가하였다. 2. 간질성폐질환 진단의 정확성 비교; 2001년부터 2006년까지 폐조직검사를 시행하여 확진된 간질성폐질환군과 서울아산병원에서 모집한 정상군을 대상으로 하여 정상예측식으로 두 식을 각각 적용하였을 때 폐확산능 해석의 정확도를 비교하였고, 두 식이 차이가 나는지 맥니머의 카이스퀘어 검정을 하였다. 결과: 1. 임상적 판단과의 일치도 비교; 276명을 임상정보를 토대로 폐확산능을 예측하여 분류한 결과 정상 54명, 감소 220명, 불분명이 2명이었다. 예측식으로 박 식과 Burrows 식을 적용하였을 때 임상적 판단과 일치하는 환자는 각각 78%와 22%이었다(p<0.001). 2. 간질성폐질환 진단의 정확성 비교; 박 식은 민감도 90.1%, 특이도 100%이었고 Burrows 식은 민감도 64.2%, 특이도 100%로 민감도가 통계학적으로 유의하게 박 식이 높았다(p<0.001). 결론: 우리나라 정상예측식인 박 식이 외국 정상예측식인 Burrows 식을 정상예측식으로 적용하는 것보다 임상적 유용성이나 간질성폐질환 진단의 민감도에서 더 우월하였다. 향후 폐확산능검사의 정상예측식으로 박 식을 사용해야 할 것으로 사료된다. Background: Park et al. developed the Korean reference equation for the measurement of diffusing capacity in 1985. However, the equation has not been widely used in Korea and foreign reference equations have been popularly used. We intended to compare the clinical usefulness and the accuracy of the the Korean reference equation (Park`s equation) with that of the foreign equation (Burrows` equation) that is commonly used in Korea. Methods: 1. Evaluation of clinical usefulness; Among 1,584 patients who underwent diffusing capacity (DLCO) at the Asan Medical Center from July to December 2006, group A subjects included 276 patients who had different interpretations of DLCO in trials employing Burrows` equation and Park`s equation. Clinical assessment was decided by consensus of two respiratory physicians. In order to evaluate the clinical usefulness of Burrows` equation and Park`s equation, agreement of clinical assessment and DLCO interpretation were measured. 2. Evaluation of accuracy; Group B subjects were 81 patients with interstitial lung disease (ILD) and 39 normal subjects. The 81 ILD patients were diagnosed following a surgical lung biopsy. The accuracy of diagnosing ILD as well as sensitivity and specificity were evaluated according to the use of the reference equations (Burrows` equation and Park`s equation) for DLCO. Results: Agreement between clinical assessment and interpretation of DLCO was 22% for the use of Burrows` equation and 78% for the use of Park`s equation. The sensitivity and specificity of the Burrows` equation for diagnosing ILD were 64.2% and 100%. The sensitivity and specificity of the Park`s equation for diagnosing ILD were 90.1% and 100%. The sensitivity of the Park`s equation for diagnosing ILD was significantly higher than that of Burrows` equation (p<0.001). Conclusion: The Korean reference equation (Park`s equation) was more clinically useful and had higher sensitivity for diagnosing ILD than the foreign reference equation (Burrows` equation). (Tuberc Respir Dis 2008;64:80-86)