Numerical solutions of strongly non-linear generalized Burgers–Fisher equation via meshfree spectral technique

2020 ◽  
pp. 1-22
Author(s):  
Manzoor Hussain ◽  
Sirajul Haq
Keyword(s):  
Numerical Solutions ◽  
Fisher Equation ◽  
Spectral Technique ◽  
Non Linear

scholarly journals Non-Linear Langevin and Fractional Fokker–Planck Equations for Anomalous Diffusion by Lévy Stable Processes

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 760 ◽  
Author(s):  
Johan Anderson ◽  
Sara Moradi ◽  
Tariq Rafiq
Keyword(s):  
Anomalous Diffusion ◽  
Transport Coefficient ◽  
Numerical Solutions ◽  
Distribution Functions ◽  
Space Distribution ◽  
Non Linear ◽  
Non Local ◽  
Local Transport ◽  
Fokker Planck Equations ◽  
Fokker Planck

The numerical solutions to a non-linear Fractional Fokker–Planck (FFP) equation are studied estimating the generalized diffusion coefficients. The aim is to model anomalous diffusion using an FFP description with fractional velocity derivatives and Langevin dynamics where Lévy fluctuations are introduced to model the effect of non-local transport due to fractional diffusion in velocity space. Distribution functions are found using numerical means for varying degrees of fractionality of the stable Lévy distribution as solutions to the FFP equation. The statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy and modified transport coefficient. The transport coefficient significantly increases with decreasing fractality which is corroborated by analysis of experimental data.

scholarly journals A Nonlinear Inverse Design Problem for a Pipe Type Heat Exchanger Equipped with Internal Z-Shape Lateral Fins and Ribs

Energies ◽  
2020 ◽  
Vol 13 (23) ◽  
pp. 6424
Author(s):  
Cheng-Hung Huang ◽  
Chih-Yang Kuo
Keyword(s):  
Heat Exchanger ◽  
Thermal Performance ◽  
Design Problem ◽  
Direct Problem ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Design Variables ◽  
Non Linear ◽  
Inverse Design Problem ◽  
Η Value

A non-linear three-dimensional inverse shape design problem was investigated for a pipe type heat exchanger to estimate the design variables of continuous lateral ribs on internal Z-shape lateral fins for maximum thermal performance factor η. The design variables were considered as the positions, heights, and number of ribs while the physical properties of air were considered as a polynomial function of temperature; this makes the problem non-linear. The direct problem was solved using software package CFD-ACE+, and the Levenberg–Marquardt method (LMM) was utilized as the optimization tool because it has been proven to be a powerful algorithm for solving inverse problems. Z-shape lateral fins were found to be the best thermal performance among Z-shape, S-shape, and V-shape lateral fins. The objective of this study was to include continuous lateral ribs to Z-shape lateral fins to further improve η. Firstly, the numerical solutions of direct problem were solved using both polynomial and constant air properties and then compared with the corrected solutions to verify the necessity for using polynomial air properties. Then, four design cases, A, B, C and D, based on various design variables were conducted numerically, and the resultant η values were computed and compared. The results revealed that considering continuous lateral ribs on the surface of Z-shape lateral fins can indeed improve η value at the design working condition Re = 5000. η values of designs A, B and C were approximately 13% higher than that for Z-shape lateral fins, however, when the rib numbers were increased, i.e., design D, the value of η became only 11.5 % higher. This implies that more ribs will not guarantee higher η value.

scholarly journals Milstein-type semi-implicit split-step numerical methods for nonlinear stochastic differential equations with locally Lipschitz drift terms

2019 ◽  
Vol 23 (Suppl. 1) ◽  
pp. 1-12 ◽  
Author(s):  
Burhaneddin Izgi ◽  
Coskun Cetin
Keyword(s):  
Numerical Methods ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Solutions ◽  
Rates Of Convergence ◽  
Ginzburg Landau ◽  
Linear Growth Condition ◽  
Locally Lipschitz ◽  
Non Linear ◽  
Ginzburg Landau Equations

We develop Milstein-type versions of semi-implicit split-step methods for numerical solutions of non-linear stochastic differential equations with locally Lipschitz coefficients. Under a one-sided linear growth condition on the drift term, we obtain some moment estimates and discuss convergence properties of these numerical methods. We compare the performance of multiple methods, including the backward Milstein, tamed Milstein, and truncated Milstein procedures on non-linear stochastic differential equations including generalized stochastic Ginzburg-Landau equations. In particular, we discuss their empirical rates of convergence.

State evolution laws and earthquake nucleation

2021 ◽  
Author(s):  
Robert Viesca
Keyword(s):  
Fixed Point ◽  
Stability Analysis ◽  
Direct Effect ◽  
Numerical Solutions ◽  
Slip Rate ◽  
Instability Development ◽  
State Evolution ◽  
Earthquake Nucleation ◽  
Non Linear ◽  
Evolution Laws

<p>In models of faults as elastic continua with a frictional interface, earthquake nucleation is the initiation of a propagating dynamic fault rupture nucleated by a localized slip instability. A mechanism capturing both the weakening process leading to nucleation as well as fault healing between events, is a slip rate- and state-dependent friction, with so-called direct effect and evolution effects [Dieterich, JGR 1979; Ruina, JGR 1983]. While the constitutive representation of the direct effect is theoretically supported [e.g., Nakatani, JGR 2001; Rice et al., JMPS 2001], that of the evolution effect remains empirical and a number of state-evolution laws have been proposed to fit lab rock friction data [Ruina, JGR 1983; Kato and Tullis, GRL 2001; Bar-Sinai et al., GRL 2012; Nagata et al., JGR 2012]. These laws may share a common linearization about steady-state, such that a linear stability analysis of steady, uniform sliding yields a single critical wavelength for unstable growth of perturbations [Rice and Ruina, JAM 1983]. However, the laws’ differences are apparent at later, non-linear stages of instability development.</p><div>Previously, we showed that instability development under aging-law state evolution could be understood in terms of dynamical systems [Viesca, PR-E 2016, PRS-A 2016]: the non-linear acceleration of slip occurs as the attraction of a fault’s slip rate to a fixed point, corresponding to slip rate diverging with a fixed spatial distribution and rate of acceleration. Here we show that this framework can also be applied to understand slip instability development under all commonly used evolution laws, including the so-called slip and Nagata laws. To do so, we develop an intermediate state evolution law that transitions between the slip and aging laws with the adjustment of a single parameter. We show that, to within a variable transformation, the intermediate law is equivalent to the Nagata law and that fixed-point blow-up solutions exist for any value of the transition parameter. We assess these fixed-points’ stability via a linear stability analysis and provide an explanation for previously observed behavior in numerical solutions for slip rate and state evolution under various evolution laws [Ampuero and Rubin, JGR 2008; Kame et al., 2013; Bar-Sinai et al., PR-E 2013; Bhattacharya and Rubin, JGR 2014].</div>

On Selecting Assumed Modes in a Controlled Articulated Flexible Multibody Dynamics System

2007 ◽  
Author(s):  
Theodore G. Mordfin ◽  
Sivakumar Tadikonda
Keyword(s):  
Exact Solutions ◽  
Numerical Solutions ◽  
Structural Characteristics ◽  
Flexible Multibody Dynamics ◽  
Control Laws ◽  
Simulation Efficiency ◽  
Multibody Dynamic ◽  
Non Linear ◽  
Flexible Multibody ◽  
Constituent Component

The modeling and simulation of controlled-articulation flexible multibody dynamic systems often involves the use of approximating functions, or assumed modes, to represent the structural characteristics of the constituent component bodies. However, clear and complete guidance on appropriate component body modeling techniques is lacking. As a result, researchers and applications engineers encounter severe and unexplained numerical problems when simulating such systems. Earlier studies demonstrated these problems, explained their causes, and developed modeling guidelines from the perspective of accuracy, robustness, and simulation efficiency. In this study, the guidelines are tested and confirmed for a controlled-articulation flexible multibody dynamic system. In support of this effort, exact closed-form and numerical solutions are developed for the small elastic motions of a planar, flexible, two-link system in which each link is represented by an Euler-Bernoulli bar in transverse vibration. The inboard link is pinned to the ground, and the outboard link is pinned to the outboard end of the first link in an arbitrary configuration. Articulation is controlled by proportional and proportional/derivative feedback control laws. The exact solutions are “truth models” for the linear characteristics of an analogous non-linear large articulation model in which link deformations are represented by assumed modes. Using a linearized version of the non-linear large-articulation model as an assumed modes testbed, the modeling guidelines are tested against the exact solutions. The numerical results conform with expectation, and the efficacy of the guidelines is successfully confirmed.

scholarly journals Analysis of fractional non-linear diffusion behaviors based on Adomian polynomials

2017 ◽  
Vol 21 (2) ◽  
pp. 813-817 ◽  
Author(s):  
Guo-Cheng Wu ◽  
Dumitru Baleanu ◽  
Wei-Hua Luo
Keyword(s):  
Taylor Expansion ◽  
Numerical Solutions ◽  
Fractional Diffusion ◽  
Diffusion Equations ◽  
Linear Diffusion ◽  
Adomian Polynomials ◽  
Fractional Diffusion Equations ◽  
Non Linear ◽  
Recurrence Formulae ◽  
Diffusion Behaviors

A time-fractional non-linear diffusion equation of two orders is considered to investigate strong non-linearity through porous media. An equivalent integral equation is established and Adomian polynomials are adopted to linearize non-linear terms. With the Taylor expansion of fractional order, recurrence formulae are proposed and novel numerical solutions are obtained to depict the diffusion behaviors more accurately. The result shows that the method is suitable for numerical simulation of the fractional diffusion equations of multi-orders.

Sign in / Sign up

Export Citation Format

Share Document