Numerical Solution of Parabolic Equation on a 2-D Infinite Strip with Composite Hermite-Legendre Spectral Method

2013 ◽  
Vol 647 ◽  
pp. 875-879
Author(s):  
Ting Jing Zhao
Keyword(s):  
Differential Equation ◽  
Parabolic Equation ◽  
Model Problem ◽  
Convergence Result ◽  
Parabolic Differential Equation ◽  
Infinite Strip ◽  
Two Dimensional ◽  
Numerical Tests ◽  
Legendre Spectral Method ◽  
Legendre Galerkin Method

For solving numerically parabolic differential equation on a two-dimensional infinite strip, composite Hermite-Legendre Galerkin method is proposed in this article. By making use of stabilised scaled factor, the proposed method achieves stability. We also establish the convergence result for the proposed method. Numerical tests conduct for the model problem. It is shown that the proposed method is efficient.

A REACTION INFILTRATION PROBLEM: EXISTENCE, UNIQUENESS, AND REGULARITY OF SOLUTIONS IN TWO SPACE DIMENSIONS

1995 ◽  
Vol 05 (05) ◽  
pp. 599-618 ◽  
Author(s):  
JOHN CHADAM ◽  
XINFU CHEN ◽  
ROBERTO GIANNI ◽  
RICCARDO RICCI
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Parabolic Equation ◽  
Elliptic Equation ◽  
Partial Regularity ◽  
Regularity Of Solutions ◽  
Infinite Strip ◽  
Two Dimensional ◽  
Finite Strip ◽  
Two Space Dimensions

In this paper, we consider a reaction infiltration problem consisting of a parabolic equation for the concentration, an elliptic equation for the pressure, and an ordinary differential equation for the porosity. We establish global existence, uniqueness and regularity of the solution in a two-dimensional finite strip (−M, M)×(0, 1) and the existence and partial regularity of solutions in an infinite strip (−∞, ∞)×(0, 1).

scholarly journals On the Cauchy problem for a degenerate parabolic differential equation

1998 ◽  
Vol 21 (3) ◽  
pp. 555-558
Author(s):  
Ahmed El-Fiky
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Parabolic Equation ◽  
Degenerate Parabolic Equation ◽  
Parabolic Differential Equation ◽  
The Cauchy Problem ◽  
Uniqueness Of The Solution ◽  
Degenerate Parabolic

The aim of this work is to prove the existence and the uniqueness of the solution of a degenerate parabolic equation. This is done using H. Tanabe and P.E. Sobolevsldi theory.

scholarly journals Exact variational representations for the solution of the simple parabolic equation

1971 ◽  
Vol 5 (3) ◽  
pp. 305-314
Author(s):  
R.S. Anderssen
Keyword(s):  
Differential Equation ◽  
Fourier Series ◽  
Parabolic Equation ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Parabolic Differential Equation ◽  
Initial Boundary Value Problems ◽  
Series Representations ◽  
Initial Boundary Value

By constructing a special set of A-orthonormal functions, it is shown that, under certain smoothness assumptions, the variational and Fourier series representations for the solution of first initial boundary value problems for the simple parabolic differential equation coincide. This result is then extended in order to construct a variational representation for the solution of a very general first initial boundary value problem for this equation.

A nonlinear instability burst in plane parallel flow

1972 ◽  
Vol 51 (4) ◽  
pp. 705-735 ◽  
Author(s):  
L. M. Hocking ◽  
K. Stewartson ◽  
J. T. Stuart ◽  
S. N. Brown
Keyword(s):  
Differential Equation ◽  
Reynolds Number ◽  
Critical Value ◽  
Three Dimensions ◽  
Parabolic Differential Equation ◽  
Nonlinear Instability ◽  
Two Dimensional ◽  
Slowly Varying Function ◽  
Plane Poiseuille Flow ◽  
Long Time

An infinitesimal centre disturbance is imposed on a fully Ldveloped plane Poiseuille flow at a Reynolds numberRslightly greater than the critical valueRcfor instability. After a long time,t, the disturbance consists of a modulated wave whose amplitudeAis a slowly varying function of position and time. In an earlier paper (Stewartson & Stuart 1971) the parabolic differential equation satisfied byAfor two-dimensional disturbances was found; the theory is here extended to three dimensions. Although the coefficients of the equation are coinples, a start is made on elucidating the properties of its solutions by assuming that these coefficients are real. It is then found numerically and confirmed analytically that, for a finite value of (R-Rc)t, the amplitudeAdevelops an infinite peak at the wave centre. The possible relevance of this work to the phenomenon of transition is discussed.

Chebyshev-Legendre Spectral Domain Decomposition Method for Two-Dimensional Vorticity Equations

2016 ◽  
Vol 19 (5) ◽  
pp. 1221-1241 ◽  
Author(s):  
Hua Wu ◽  
Jiajia Pan ◽  
Haichuan Zheng
Keyword(s):  
Domain Decomposition Method ◽  
Stability And Convergence ◽  
Two Dimensional ◽  
One Dimensional ◽  
Collocation Points ◽  
The Matrix ◽  
Vorticity Equations ◽  
Legendre Spectral Method ◽  
Legendre Galerkin Method ◽  
The Stability

AbstractWe extend the Chebyshev-Legendre spectral method to multi-domain case for solving the two-dimensional vorticity equations. The schemes are formulated in Legendre-Galerkin method while the nonlinear term is collocated at Chebyshev-Gauss collocation points. We introduce proper basis functions in order that the matrix of algebraic system is sparse. The algorithm can be implemented efficiently and in parallel way. The numerical analysis results in the case of one-dimensional multi-domain are generalized to two-dimensional case. The stability and convergence of the method are proved. Numerical results are given.

scholarly journals Exact Solutions and Conserved Vectors of the Two-Dimensional Generalized Shallow Water Wave Equation

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1439
Author(s):  
Chaudry Masood Khalique ◽  
Karabo Plaatjie
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Shallow Water ◽  
Water Wave ◽  
Elliptic Integral ◽  
Momentum Conservation ◽  
Two Dimensional ◽  
Order Ordinary Differential Equation ◽  
Closed Form Solutions ◽  
Shallow Water Wave

In this article, we investigate a two-dimensional generalized shallow water wave equation. Lie symmetries of the equation are computed first and then used to perform symmetry reductions. By utilizing the three translation symmetries of the equation, a fourth-order ordinary differential equation is obtained and solved in terms of an incomplete elliptic integral. Moreover, with the aid of Kudryashov’s approach, more closed-form solutions are constructed. In addition, energy and linear momentum conservation laws for the underlying equation are computed by engaging the multiplier approach as well as Noether’s theorem.

scholarly journals DIFFERENTIAL EQUATION APPROACH FOR ONE- AND TWO-DIMENSIONAL LATTICE GREEN'S FUNCTION

2007 ◽  
Vol 21 (02n03) ◽  
pp. 139-154 ◽  
Author(s):  
J. H. ASAD
Keyword(s):  
Differential Equation ◽  
Green’S Function ◽  
Recurrence Relation ◽  
Green's Function ◽  
Two Dimensional ◽  
Dimensional Lattice ◽  
One Dimensional ◽  
The One ◽  
Lattice Green's Function ◽  
Lattice Green’S Function

A first-order differential equation of Green's function, at the origin G(0), for the one-dimensional lattice is derived by simple recurrence relation. Green's function at site (m) is then calculated in terms of G(0). A simple recurrence relation connecting the lattice Green's function at the site (m, n) and the first derivative of the lattice Green's function at the site (m ± 1, n) is presented for the two-dimensional lattice, a differential equation of second order in G(0, 0) is obtained. By making use of the latter recurrence relation, lattice Green's function at an arbitrary site is obtained in closed form. Finally, the phase shift and scattering cross-section are evaluated analytically and numerically for one- and two-impurities.

Generalized plane stress in a semi-infinite strip under arbitrary end-load

1964 ◽  
Vol 281 (1385) ◽  
pp. 184-206 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Boundary Conditions ◽  
Fourth Order ◽  
Plane Stress ◽  
Stress Function ◽  
Infinite Strip ◽  
Order Ordinary Differential Equation ◽  
Orthogonal Functions

The problem involves the determination of a biharmonic generalized plane-stress function satisfying certain boundary conditions. We expand the stress function in a series of non-orthogonal eigenfunctions. Each of these is expanded in a series of orthogonal functions which satisfy a certain fourth-order ordinary differential equation and the boundary conditions implied by the fact that the sides are stress-free. By this method the coefficients involved in the biharmonic stress function corresponding to any arbitrary combination of stress on the end can be obtained directly from two numerical matrices published here The method is illustrated by four examples which cast light on the application of St Venant’s principle to the strip. In a further paper by one of the authors, the method will be applied to the problem of the finite rectangle.

Sign in / Sign up

Export Citation Format

Share Document