scholarly journals A NOTE ON THE NUMERICAL APPROACH FOR THE REACTION–DIFFUSION PROBLEM WITH A FREE BOUNDARY CONDITION

2010 ◽  
Vol 51 (3) ◽  
pp. 317-330
Author(s):  
E. ÖZUĞURLU
Keyword(s):  
Differential Equation ◽  
Free Boundary ◽  
Newton’S Method ◽  
Newton's Method ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Liquid Films ◽  
Diffusion Problem ◽  
Nicolson Scheme ◽  
Crank Nicolson

AbstractThe equation modelling the evolution of a foam (a complex porous medium consisting of a set of gas bubbles surrounded by liquid films) is solved numerically. This model is described by the reaction–diffusion differential equation with a free boundary. Two numerical methods, namely the fixed-point and the averaging in time and forward differences in space (the Crank–Nicolson scheme), both in combination with Newton’s method, are proposed for solving the governing equations. The solution of Burgers’ equation is considered as a special case. We present the Crank–Nicolson scheme combined with Newton’s method for the reaction–diffusion differential equation appearing in a foam breaking phenomenon.

scholarly journals Finite Difference Method for Two-Sided Two Dimensional Space Fractional Convection-Diffusion Problem with Source Term

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1878
Author(s):  
Eyaya Fekadie Anley ◽  
Zhoushun Zheng
Keyword(s):  
Diffusion Equation ◽  
Source Term ◽  
Diffusion Problem ◽  
Stability And Convergence ◽  
Difference Approximation ◽  
Finite Domain ◽  
Two Dimensional ◽  
Convection Diffusion ◽  
Nicolson Scheme ◽  
Crank Nicolson

In this paper, we have considered a numerical difference approximation for solving two-dimensional Riesz space fractional convection-diffusion problem with source term over a finite domain. The convection and diffusion equation can depend on both spatial and temporal variables. Crank-Nicolson scheme for time combined with weighted and shifted Grünwald-Letnikov difference operator for space are implemented to get second order convergence both in space and time. Unconditional stability and convergence order analysis of the scheme are explained theoretically and experimentally. The numerical tests are indicated that the Crank-Nicolson scheme with weighted shifted Grünwald-Letnikov approximations are effective numerical methods for two dimensional two-sided space fractional convection-diffusion equation.

scholarly journals Modelling of Non–Linear Distortion in Vacuum Triodes Using Trans–Characteristics Inverse and Newton’s Method

2013 ◽  
Vol 64 (4) ◽  
pp. 244-249
Author(s):  
Martin Dadić
Keyword(s):  
Fourier Transform ◽  
Newton’S Method ◽  
Newton's Method ◽  
Harmonic Distortion ◽  
Numerical Approach ◽  
Intermodulation Distortion ◽  
Brute Force ◽  
Input Output ◽  
Audio Amplifiers ◽  
Vacuum Triodes

The increased interest in vacuum tube audio amplifiers led to an increased interest in mathematical modelling of such kind of amplifiers. The main purpose of this paper is to develop a novel global numerical approach in calculation of the harmonic distortion (HD) and intermodulation distortion (IM) of vacuum-triode audio amplifiers, suitable for applications using brute-force of modern computers. Since the 3/2 power law gives only the transcharacteristic inverse of a vacuum triode amplifier, unknown plate currents are determined in this paper iteratively using Newton’s method. Using the resulting input/output pairs, harmonic distortions and intermodulations are calculated using discrete Fourier transform and three different analytical methods.

A series solution of the Falkner–Skan equation using the crocco–wang transformation

2017 ◽  
Vol 28 (11) ◽  
pp. 1750139 ◽  
Author(s):  
Asai Asaithambi
Keyword(s):  
Differential Equation ◽  
Friction Coefficient ◽  
Newton’S Method ◽  
Newton's Method ◽  
Series Solution ◽  
Skin Friction Coefficient ◽  
Point Boundary ◽  
Third Order ◽  
Order Problem ◽  
The Third

A direct series solution for the Falkner–Skan equation is obtained by first transforming the problem using the Crocco–Wang transformation. The transformation converts the third-order problem to a second-order two-point boundary value problem. The method first constructs a series involving the unknown skin-friction coefficient [Formula: see text]. Then, [Formula: see text] is determined by using the secant method or Newton’s method. The derivative needed for Newton’s method is also computed using a series derived from the transformed differential equation. The method is validated by solving the Falkner–Skan equation for several cases reported previously in the literature.

scholarly journals A FREE BOUNDARY PROBLEM MODELING A FOAM DRAINAGE

2000 ◽  
Vol 10 (06) ◽  
pp. 945-961 ◽  
Author(s):  
THIERRY COLIN ◽  
PIERRE FABRIE
Keyword(s):  
Free Boundary ◽  
Burgers Equation ◽  
Reaction Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Diffusion Problem ◽  
A Free Boundary Problem ◽  
Initial Boundary Value ◽  
Well Posed ◽  
Foam Drainage

We study a reaction-diffusion problem with a free boundary governing the evolution of a foam. We show that the problem is globaly well-posed and that the solution converges, when the viscosity tends to zero to the solution of an initial-boundary value problem for Burgers equation.

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