Appendix C: The hybrid equations

2013 ◽  
pp. 273-276
Keyword(s):  
Hybrid Equations

Finite Difference Schemes for Diffusion Problems Based on a Hybrid Perturbation Galerkin Method

2006 ◽  
Author(s):  
James Geer ◽  
John Fillo
Keyword(s):  
Finite Difference ◽  
Diffusion Equation ◽  
Galerkin Method ◽  
Difference Equations ◽  
Space Variable ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Hybrid Technique ◽  
Finite Difference Equations ◽  
Hybrid Equations

A new technique for the development of finite difference schemes for diffusion equations is presented. The model equations are the one space variable advection diffusion equation and the two space variable diffusion equation, each with Dirichlet boundary conditions. A two-step hybrid technique, which combines perturbation methods, based on the parameter ρ = Δt / (Δx)2, with the Galerkin method, provides a systematic way to develop new finite difference methods, referred to as hybrid equations. The main contributions of this paper include: 1) recovery of classical explicit or implicit finite difference schemes using only the perturbation terms; 2) development of new finite difference schemes, referred to as hybrid equations, which have better stability properties than the classical finite difference equations, permitting the use of larger values of the parameter ρ; and 3) higher order accurate methods, with either O((Δx)4) or O((Δx)6) truncation error, formed by convex linear combinations of the classical and hybrid equations. The solution of the hybrid finite difference equations requires only a tridiagonal equation solver and, hence, does not lead to excessive computational effort.

scholarly journals Proposal of the hybrid solution to determining the selected fracture parameters for SEN(B) specimens dominated by plane strain

2017 ◽  
Vol 65 (4) ◽  
pp. 523-532
Author(s):  
M. Graba
Keyword(s):  
Plane Strain ◽  
Analytical Methods ◽  
Limit Load ◽  
J Integral ◽  
Strain Condition ◽  
Load Line ◽  
Load Line Displacement ◽  
Hybrid Solution ◽  
Analytical Formulae ◽  
Hybrid Equations

AbstractIn the paper, new hybrid (numerical-analytical) methods to calculate the J-integral, the CTOD, and the load line displacement are presented. The proposed solutions are based on FEM calculations which were done for SEN(B) specimens dominated by plane strain condition. The paper includes the verification of the existing limit load solution for SEN(B) specimen with proposal of the new analytical formulae, which were used for building hybrid equations for determining three selected fracture mechanics parameters.

scholarly journals On Fractional Order Hybrid Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Mohamed A. E. Herzallah ◽  
Dumitru Baleanu
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
Fixed Point Theorems ◽  
Mild Solutions ◽  
Nonlinear Perturbations ◽  
Linear And Nonlinear ◽  
Hybrid Differential Equations ◽  
Hybrid Equations

We develop the theory of fractional hybrid differential equations with linear and nonlinear perturbations involving the Caputo fractional derivative of order0<α<1. Using some fixed point theorems we prove the existence of mild solutions for two types of hybrid equations. Examples are given to illustrate the obtained results.

Two time-scale approach implemented on hybrid equations

2009 ◽  
Author(s):  
Mihai Iordache ◽  
Lucia Dumitriu ◽  
Lucian Mandache
Keyword(s):  
Time Scale ◽  
Hybrid Equations

A New Method to Solve Static Voltage Collapse Point with Hybrid Power Network Equations

2013 ◽  
Vol 732-733 ◽  
pp. 796-799
Author(s):  
Tao Yi ◽  
Jian Peng Li ◽  
Song Zhang ◽  
Ying Bin Zhang
Keyword(s):  
Power Flow ◽  
Jacobian Matrix ◽  
Voltage Collapse ◽  
Power Network ◽  
Analysis Model ◽  
Power Flow Calculation ◽  
Flow Equations ◽  
Node Voltage ◽  
Hybrid Equations ◽  
Extended Power

Electric power network components can be simulated with equivalent circuit and establish the hybrid node voltage and branch current electricity network analysis model based on it. We form the equations reflected the voltage collapse conditions, and establish the extended power flow equations with the hybrid equations, the static voltage collapse point can be calculated by solving them. The advantage of this method is that the power flow calculation can be convergence smoothly near the collapse point because of the extended power flow equations when the Jacobian matrix tends to be singular. The convergence can be achievable just because the joining of the equation reflecting the voltage collapse conditions has changed the structure of the Jacobian matrix. The simulation results show that this method is very effective.

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