scholarly journals Sharp Interface Limit of a Stokes/Cahn–Hilliard System, Part II: Approximate Solutions

2021 ◽  
Vol 23 (2) ◽  
Author(s):  
Helmut Abels ◽  
Andreas Marquardt
Keyword(s):  
Fractional Order ◽  
Approximate Solutions ◽  
Sharp Interface ◽  
Main Step ◽  
Two Dimensional ◽  
Matched Asymptotics ◽  
Sharp Interface Limit ◽  
System Part ◽  
Proof Of Convergence ◽  
Problematic Feature

AbstractWe construct rigorously suitable approximate solutions to the Stokes/Cahn–Hilliard system by using the method of matched asymptotics expansions. This is a main step in the proof of convergence given in the first part of this contribution, [3], where the rigorous sharp interface limit of a coupled Stokes/Cahn–Hilliard system in a two dimensional, bounded and smooth domain is shown. As a novelty compared to earlier works, we introduce fractional order terms, which are of significant importance, but share the problematic feature that they may not be uniformly estimated in $$\epsilon $$ ϵ in arbitrarily strong norms. As a consequence, gaining necessary estimates for the error, which occurs when considering the approximations in the Stokes/Cahn–Hilliard system, is rather involved.

scholarly journals Numerical Solution of the Fractional Partial Differential Equations by the Two-Dimensional Fractional-Order Legendre Functions

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Fukang Yin ◽  
Junqiang Song ◽  
Yongwen Wu ◽  
Lilun Zhang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Order ◽  
Approximate Solutions ◽  
Operational Matrices ◽  
Legendre Functions ◽  
Two Dimensional ◽  
Algebraic Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential

A numerical method is presented to obtain the approximate solutions of the fractional partial differential equations (FPDEs). The basic idea of this method is to achieve the approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions (2D-FLFs). The operational matrices of integration and derivative for 2D-FLFs are first derived. Then, by these matrices, a system of algebraic equations is obtained from FPDEs. Hence, by solving this system, the unknown 2D-FLFs coefficients can be computed. Three examples are discussed to demonstrate the validity and applicability of the proposed method.

scholarly journals (Non-)convergence of solutions of the convective Allen–Cahn equation

2021 ◽  
Vol 3 (1) ◽  
Author(s):  
Helmut Abels
Keyword(s):  
Transport Equation ◽  
Mean Curvature ◽  
Convergence Result ◽  
Normal Direction ◽  
Sharp Interface ◽  
Navier Stokes ◽  
Matched Asymptotics ◽  
Sharp Interface Limit ◽  
Convergence Of Solutions ◽  
Cahn Equation

AbstractWe consider the sharp interface limit of a convective Allen–Cahn equation, which can be part of a Navier–Stokes/Allen–Cahn system, for different scalings of the mobility $$m_\varepsilon =m_0\varepsilon ^\theta $$ m ε = m 0 ε θ as $$\varepsilon \rightarrow 0$$ ε → 0 . In the case $$\theta >2$$ θ > 2 we show a (non-)convergence result in the sense that the concentrations converge to the solution of a transport equation, but they do not behave like a rescaled optimal profile in normal direction to the interface as in the case $$\theta =0$$ θ = 0 . Moreover, we show that an associated mean curvature functional does not converge to the corresponding functional for the sharp interface. Finally, we discuss the convergence in the case $$\theta =0,1$$ θ = 0 , 1 by the method of formally matched asymptotics.

scholarly journals NUMERICAL SOLUTION OF THE PARTIAL DIFFERENTIAL EQUATION USING RANDOMLY GENERATED FINITE GRIDS AND TWO-DIMENSIONAL FRACTIONAL-ORDER LEGENDRE FUNCTION

2021 ◽  
Author(s):  
Sanaullah Mastoi
Keyword(s):  
Finite Difference ◽  
Numerical Solution ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Polar Coordinates ◽  
Legendre Functions ◽  
Two Dimensional ◽  
Cartesian Coordinates ◽  
Uniform Grids

There are various methods to solve the physical life problem involving engineering, scientific and biological systems. It is found that numerical methods are approximate solutions. In this way, randomly generated finite difference grids achieve an approximation with fewer iterations. The idea of randomly generated grids in cartesian coordinates and polar form are compared with the exact, iterative method, uniform grids, and approximate solutions in a generalized expansion form of two-dimensional fractional-order Legendre functions. The most ideal and benchmarking method is the finite difference method over randomly generated grids on Cartesian coordinates, polar coordinates used for numerical solutions. This concept motivates the investigation of the effects of the randomly generated meshes. The two-dimensional equation is solved over randomly generated meshes to test randomly generated grids and the implementation. The feasibility of the numerical solution is analyzed by comparing simulation profiles.

scholarly journals Sharp Interface Limit of Stochastic Cahn-Hilliard Equation with Singular Noise

2022 ◽  
Author(s):  
Ľubomír Baňas ◽  
Huanyu Yang ◽  
Rongchan Zhu
Keyword(s):  
White Noise ◽  
Space Time ◽  
Sharp Interface ◽  
Two Dimensional ◽  
Sharp Interface Limit ◽  
Hilliard Equation ◽  
Stochastic Problems ◽  
Divergence Type ◽  
Cahn Hilliard Equation

AbstractWe study the sharp interface limit of the two dimensional stochastic Cahn-Hilliard equation driven by two types of singular noise: a space-time white noise and a space-time singular divergence-type noise. We show that with appropriate scaling of the noise the solutions of the stochastic problems converge to the solutions of the determinisitic Mullins-Sekerka/Hele-Shaw problem.

scholarly journals Application of Asymptotic Homotopy Perturbation Method to Fractional Order Partial Differential Equation

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2215
Author(s):  
Haji Gul ◽  
Sajjad Ali ◽  
Kamal Shah ◽  
Shakoor Muhammad ◽  
Thanin Sitthiwirattham ◽  
...  
Keyword(s):  
Perturbation Method ◽  
Fractional Order ◽  
Homotopy Perturbation Method ◽  
Approximate Solutions ◽  
Order Partial Differential Equation ◽  
Two Dimensional ◽  
Numerical Treatment ◽  
Helmholtz Equations ◽  
Homotopy Perturbation ◽  
Linear Differential

In this article, we introduce a new algorithm-based scheme titled asymptotic homotopy perturbation method (AHPM) for simulation purposes of non-linear and linear differential equations of non-integer and integer orders. AHPM is extended for numerical treatment to the approximate solution of one of the important fractional-order two-dimensional Helmholtz equations and some of its cases . For probation and illustrative purposes, we have compared the AHPM solutions to the solutions from another existing method as well as the exact solutions of the considered problems. Moreover, it is observed that the symmetry or asymmetry of the solution of considered problems is invariant under the homotopy definition. Error estimates for solutions are also provided. The approximate solutions of AHPM are tabulated and plotted, which indicates that AHPM is effective and explicit.

scholarly journals High performance self-powered photodetector based on 1D Te-2D WS2 mixed-dimensional heterostructure

2021 ◽  
Author(s):  
Lixiang Han ◽  
Mengmeng Yang ◽  
Peiting Wen ◽  
Wei Gao ◽  
nengjie huo ◽  
...  
Keyword(s):  
High Performance ◽  
Van Der Waals ◽  
Sharp Interface ◽  
Two Dimensional ◽  
High Quality ◽  
One Dimensional ◽  
Self Powered

One dimensional (1D)-two dimensional (2D) van der Waals (vdWs) mixed-dimensional heterostructures with advantages of atomically sharp interface, high quality and good compatibility have attracted tremendous attention in recent years. The...

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