General Software for Two-Dimensional Nonlinear Partial Differential Equations

1981 ◽  
Vol 7 (1) ◽  
pp. 106-125 ◽  
Author(s):  
David K. Melgaard ◽  
Richard F. Sincovec
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Partial Differential Equations ◽  
Two Dimensional ◽  
Partial Differential

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.

Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Lyapunov Inequalities ◽  
Funct Anal

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

scholarly journals Analytical mathematical schemes: Circular rod grounded via transverse Poisson’s effect and extensive wave propagation on the surface of water

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 545-554
Author(s):  
Asghar Ali ◽  
Aly R. Seadawy ◽  
Dumitru Baleanu
Keyword(s):  
Wave Propagation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Longitudinal Wave ◽  
Symbolic Computation ◽  
Boussinesq Equation ◽  
Nonlinear Partial Differential Equations ◽  
Wave Model ◽  
Simple Equation ◽  
Partial Differential

AbstractThis article scrutinizes the efficacy of analytical mathematical schemes, improved simple equation and exp(-\text{Ψ}(\xi ))-expansion techniques for solving the well-known nonlinear partial differential equations. A longitudinal wave model is used for the description of the dispersion in the circular rod grounded via transverse Poisson’s effect; similarly, the Boussinesq equation is used for extensive wave propagation on the surface of water. Many other such types of equations are also solved with these techniques. Hence, our methods appear easier and faster via symbolic computation.

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