The Error Bounds for Three-Dimensional Nodal LTSN Method

2008 ◽  
Author(s):  
Eliete Biasotto Hauser ◽  
Ruben Panta Pazos ◽  
Marco T. M. B. Vilhena ◽  
Ricardo C. Barros
Keyword(s):  
Spectral Analysis ◽  
Analytical Solution ◽  
Laplace Transform ◽  
Error Bounds ◽  
Three Dimensional ◽  
Discrete Ordinates ◽  
Exponential Functions ◽  
Decay Constants ◽  
The Laplace Transform ◽  
Cartesian Geometry

In this paper we present a proof about the convergence of the 3D Nodal-LTSN Method in order to solve the transport problem in a parallelepiped domain. For that, we define functions associated to the errors, one in the approximated flux, another in the quadrature formula and establish a relation between them. We present a Nodal-LTSN method to generate an analytical solution for discrete ordinates problems in three-dimensional cartesian geometry. We first transverse integrate the SN equations and then we apply the Laplace transform. The essence of this method is the diagonalization of the LTSN transport matrices and the spectral analysis garantees this. The transverse leakage terms that appear in the transverse integrated SN equations are represented by exponential functions with decay constants that depend on the characteristics of the material of the medium the particles leave behind. We present numerical results generated by the offered method applied to typical shielding model problems.

Analytical solutions of citrate–phosphate coupled model of rice (Oryza sativa L.) roots

2020 ◽  
Vol 13 (07) ◽  
pp. 2050061
Author(s):  
Huiping Zhang ◽  
Shuyue Wang ◽  
Zhonghui Ou
Keyword(s):  
Oryza Sativa ◽  
Analytical Solution ◽  
Laplace Transform ◽  
Oryza Sativa L ◽  
Coupled Model ◽  
Inverse Laplace Transform ◽  
Citrate Solution ◽  
Separation Variable ◽  
The Laplace Transform ◽  
Citrate Phosphate

The citrate secreted by the rice (Oryza sativa L.) roots will promote the absorption of phosphate, and this process is described by the Kirk model. In our work, the Kirk model is divided into citrate sub-model and phosphate sub-model. In the citrate sub-model, we obtain the analytical solution of citrate with the Laplace transform, inverse Laplace transform and convolution theorem. The citrate solution is substituted into the phosphate sub-model, and the analytical solution of phosphate is obtained by the separation variable method. The existence of the solutions can be proved by the comparison test, the Weierstrass M-test and the Abel discriminating method.

A New Analytical Solution of the Triple-Porosity Model for History Matching and Performance Forecasting in Unconventional Oil Reservoirs

SPE Journal ◽  
2018 ◽  
Vol 23 (06) ◽  
pp. 2060-2079 ◽  
Author(s):  
Kaixuan Qiu ◽  
Heng Li
Keyword(s):  
Analytical Solution ◽  
Differential Equations ◽  
Laplace Transform ◽  
History Matching ◽  
Oil Reservoirs ◽  
Production Data ◽  
Numerical Inversion ◽  
Unconventional Oil ◽  
The Laplace Transform ◽  
Porosity Model

Summary Production data from hydraulically fractured horizontal wells in ultralow-permeability reservoirs have been observed to exhibit a half-slope straight line on a log-log plot of rate vs. time. Using these observations, many analytical models of linear flow have been developed to analyze the well performance in these reservoirs. However, it is relatively complicated to solve these models with the Laplace transform and numerical inversion. In this paper, a new analytical solution of a triple-porosity model is derived for unconventional oil reservoirs. The partial-differential equations (PDEs) are transformed into ordinary-differential equations (ODEs) by integration, instead of the Laplace transform. A rate-vs.-time solution in real-time space can be obtained, bypassing the numerical inversion for the Laplace transform. Our model is validated by comparison with both the numerical-model and Laplace-transform solutions. The physical meanings of the model parameters are analyzed through sensitivity analysis. The new model is then applied to field-production data for history matching and forecasting. Besides matching the production data, the pore volume (PV) of the matrix and fracture media can be estimated, which helps quantitative analysis of unconventional reservoirs.

Analytical solution of spray cooling characteristics on a hot surface using the Laplace transform

2015 ◽  
Vol 24 (6) ◽  
pp. 957-973 ◽  
Author(s):  
Han-Taw Chen ◽  
Jing-Hong Lin ◽  
Xiao-Jie Xu ◽  
Chein-Shan Liu ◽  
Jiang-Ren Chang
Keyword(s):  
Analytical Solution ◽  
Laplace Transform ◽  
Spray Cooling ◽  
The Laplace Transform ◽  
Hot Surface

scholarly journals Three-Dimensional Semianalytical Solutions for Piezoelectric Laminates Subjected to Underwater Shocks

2020 ◽  
Vol 2020 ◽  
pp. 1-20
Author(s):  
Xu Liang ◽  
Wenbin Lu ◽  
Ronghua Zhu ◽  
Changpeng Ye ◽  
Guohua Liu
Keyword(s):  
Dynamic Response ◽  
Boundary Conditions ◽  
Laplace Transform ◽  
Electric Potential ◽  
Differential Quadrature Method ◽  
Three Dimensional ◽  
Laminated Plates ◽  
Inversion Method ◽  
Variation Principle ◽  
The Laplace Transform

In this study, a piezoelectric laminate is analyzed by applying the Laplace transform and its numerical inversion, Fourier transform, differential quadrature method (DQM), and state space method. Based on the modified variation principle for the piezoelectric laminate, the fundamental equations for dynamic problems are derived. The solutions for the displacement, stress, electric potential, and dielectric displacement are obtained using the proposed method. Durbin’s inversion method for the Laplace transform is adopted. Four boundary conditions are discussed through the DQM. The proposed method is validated by comparing the results with those of the finite element method (FEM). Moreover, this semianalytical method is further extended to describe the dynamic response of piezoelectric laminated plates subjected to underwater shocks by introducing Taylor’s fluid-structure interaction algorithm. Both air-backed and water-backed laminated plates are investigated, and the effect of the fluid is examined. In the time domain, the electric potential and displacements of sample points are calculated under four boundary conditions. The present method is shown to be accurate and can be a useful method to calculate the dynamic response of underwater laminated sensors.

scholarly journals A new fractional derivative model for the anomalous diffusion problem

2019 ◽  
Vol 23 (Suppl. 3) ◽  
pp. 1005-1011
Author(s):  
Zhanqing Chen ◽  
Peitao Qiu ◽  
Xiao-Jun Yang ◽  
Yiying Feng ◽  
Jiangen Liu
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Anomalous Diffusion ◽  
Heat Conduction Problem ◽  
Diffusion Problem ◽  
The Laplace Transform ◽  
Complex Phenomena ◽  
First Time

In this paper, a new fractional derivative within the exponential decay kernel is addressed for the first time. A new anomalous diffusion model is proposed to describe the heat-conduction problem. With the use of the Laplace transform, the analytical solution is discussed in detail. The presented result is as an accurate and efficient approach proposed for the heat-conduction problem in the complex phenomena.

scholarly journals Analytical Solution of Deformations for Two-Layer Timoshenko Beams Glued by a Viscoelastic Interlayer

2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
Zhiyi Pei ◽  
Lin Wang ◽  
Peng Wu ◽  
Jiandong Zhang ◽  
Ding Zhou
Keyword(s):  
Analytical Solution ◽  
Laplace Transform ◽  
Memory Effect ◽  
Shear Deformation Theory ◽  
Beam Theory ◽  
Transverse Load ◽  
Numerical Comparison ◽  
Timoshenko Beams ◽  
The Laplace Transform ◽  
Viscoelastic Interlayer

An analytical solution of stresses and deformations for two-layer Timoshenko beams glued by a viscoelastic interlayer under uniform transverse load is presented. The standard linear solid model is employed to simulate the viscoelastic characteristics of the interlayer, in which the memory effect of strains is considered. The mechanical behavior of each layer is described by the first-order shear deformation theory (FSDT). By means of the principle of minimum potential energy, a group of equations for displacements and rotation angles are derived out. The final solution is obtained by conducting the Laplace transform and the inversion of Laplace transform to the equation group. Numerical comparison shows that the present solutions and the finite element results are in good agreement. It is shown that the present results are more accurate than those obtained from the Euler-Bernoulli beam theory, especially for thick beams. And the present solutions can accurately describe the variation of stresses and deformations of the beam with the time, compared with those ignoring the memory effect of strains. Finally, the effects of the geometric parameters and material properties of the interlayer on stresses and deformations of the beam are studied in detail.

scholarly journals On Analytical Solution of Time-Fractional Type Model of the Fisher’s Equation

2020 ◽  
pp. 1419-1425
Author(s):  
Shaheed N. Huseen
Keyword(s):  
Analytical Solution ◽  
Laplace Transform ◽  
Homotopy Analysis Method ◽  
Type Model ◽  
Analysis Method ◽  
Fisher’S Equation ◽  
Fisher's Equation ◽  
Homotopy Analysis ◽  
The Laplace Transform ◽  
Fractional Type

In this paper, the time-fractional Fisher’s equation (TFFE) is considered to exam the analytical solution using the Laplace q-Homotopy analysis method (Lq-HAM)”. The Lq-HAM is a combined form of q-homotopy analysis method (q-HAM) and Laplace transform. The aim of utilizing the Laplace transform is to outdo the shortage that is mainly caused by unfulfilled conditions in the other analytical methods. The results show that the analytical solution converges very rapidly to the exact solution.

Sign in / Sign up

Export Citation Format

Share Document