scholarly journals Localized method of particular solutions using polynomial basis functions for solving two-dimensional nonlinear partial differential equations

2021 ◽  
Vol 4 ◽  
pp. 100114
Author(s):  
T. Dangal ◽  
B. Khatri Ghimire ◽  
A.R. Lamichhane
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Basis Functions ◽  
Polynomial Basis ◽  
Two Dimensional ◽  
Partial Differential ◽  
Particular Solutions ◽  
Method Of Particular Solutions

Two-dimensional bifurcation phenomena in thermal convection in horizontal, concentric annuli containing saturated porous media

1988 ◽  
Vol 187 ◽  
pp. 267-300 ◽  
Author(s):  
K. Himasekhar ◽  
Haim H. Bau
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Real Axis ◽  
Thermal Convection ◽  
Saturated Porous Medium ◽  
Nonlinear Partial Differential Equations ◽  
Perturbation Expansion ◽  
Two Dimensional ◽  
Partial Differential ◽  
The Real

A saturated porous medium confined between two horizontal cylinders is considered. As a result of a temperature difference between the cylinders, thermal convection is induced in the medium. The flow structure is investigated in a parameter space (R, Ra) where R is the radii ratio and Ra is the Darcy-Rayleigh number. In particular, the cases of R = 2, 2½, 21/4 and 2½ are considered. The fluid motion is described by the two-dimensional Darcy-Oberbeck-Boussinesq's (DOB) equations, which we solve using regular perturbation expansion. Terms up to O(Ra60) are calculated to obtain a series presentation for the Nusselt number Nu in the form \[ Nu(Ra^2) = \sum_{s=0}^{30} N_sRa^{2s}. \] This series has a limited range of utility due to singularities of the function Nu(Ra). The singularities lie both on and off the real axis in the complex Ra plane. For R = 2, the nearest singularity lies off the real axis, has no physical significance, and unnecessarily limits the range of utility of the aforementioned series. For R = 2½, 2¼ and 21/8, the singularity nearest to the origin is real and indicates that the function Nu(Ra) is no longer unique beyond the singular point.Depending on the radii ratio, the loss of uniqueness may occur as a result of either (perfect) bifurcations or the appearance of isolated solutions (imperfect bifurcations). The structure of the multiple solutions is investigated by solving the DOB equations numerically. The nonlinear partial differential equations are converted into a truncated set of ordinary differential equations via projection. The steady-state problem is solved using Newton's technique. At each step the determinant of the Jacobian is evaluated. Bifurcation points are identified with singularities of the Jacobian. Linear stability analysis is used to determine the stability of various solution branches. The results we obtained from solving the DOB equations using perturbation expansion are compared with those we obtained from solving the nonlinear partial differential equations numerically and are found to agree well.

scholarly journals ALGORITHM FOR CONSTRUCTING A LAX PAIR AND A RECURSION OPERATOR FOR INTEGRABLE EQUATIONS

2019 ◽  
Vol 47 (1) ◽  
pp. 123-126
Author(s):  
I.T. Habibullin ◽  
A.R. Khakimova
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Invariant Manifold ◽  
Nonlinear Partial Differential Equations ◽  
Recursion Operators ◽  
Lax Pairs ◽  
Partial Differential ◽  
Differential Constraint ◽  
Integrable Equations ◽  
Particular Solutions

The method of constructing particular solutions to nonlinear partial differential equations based on the notion of differential constraint (or invariant manifold) is well known in the literature, see (Yanenko, 1961; Sidorov et al., 1984). The matter of the method is to add a compatible equation to a given equation and as a rule, the compatible equation is simpler. Such technique allows one to find particular solutions to a studied equation. In works (Pavlova et al., 2017; Habibullin et al., 2017, 2018; Khakimova, 2018; Habibullin et al., 2016, 2017, 2018) there was proposed a scheme for constructing the Lax pairs and recursion operators for integrable partial differential equations based on the use of similar idea. A suitable generalization is to impose a differential constraint not on the equation, but on its linearization. The resulting equation is referred to as a generalized invariant manifold. In works (Pavlova et al., 2017; Habibullin et al., 2017, 2018; Khakimova, 2018; Habibullin et al., 2016, 2017, 2018) it is shown that generalized invariant varieties allow efficient construction of Lax pairs and recursion operators of integrable equations. The research was supported by the RAS Presidium Program «Nonlinear dynamics: fundamental problems and applications».

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