scholarly journals A Class of PDEs with Nonlinear Superposition Principles

2012 ◽  
Vol 2012 ◽  
pp. 1-15
Author(s):  
Li Peng ◽  
Liu Keying ◽  
Pan Zuliang ◽  
Zhong Weizhou
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Lie Group ◽  
Nonlinear Pde ◽  
Partial Differential ◽  
Nonlinear Superposition ◽  
Invariant Group ◽  
Group A ◽  
Superposition Principles

Through assuming that nonlinear superposition principles (NLSPs) are embedded in a Lie group, a class of 3rd-order PDEs is derived from a general determining equation that determine the invariant group. The corresponding NLSPs and transformation to linearize the nonlinear PDE are found, hence the governing PDE is provedC-integrable. In the end, some applications of the PDEs are explained, which shows that the result has very subtle relations with linearization of partial differential equation.

scholarly journals Generating Functions for Bessel Functions

1959 ◽  
Vol 11 ◽  
pp. 148-155 ◽  
Author(s):  
Louis Weisner
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Generating Function ◽  
Lie Group ◽  
Generating Functions ◽  
Bessel Functions ◽  
Partial Differential ◽  
Systematic Determination ◽  
Image Position

On replacing the parameter n in Bessel's differential equation1.1by the operator y(∂/∂y), the partial differential equation Lu = 0 is constructed, where1.2This operator annuls u(x, y) = v(x)yn if, and only if, v(x) satisfies (1.1) and hence is a cylindrical function of order n. Thus every generating function of a set of cylindrical functions is a solution of Lu = 0.It is shown in § 2 that the partial differential equation Lu = 0 is invariant under a three-parameter Lie group. This group is then applied to the systematic determination of generating functions for Bessel functions, following the methods employed in two previous papers (4; 5).

Group Theoretical Analysis and Invariant Solutions for Unsteady Flow of a Fourth-Grade Fluid over an Infinite Plate Undergoing Impulsive Motion in a Darcy Porous Medium

2015 ◽  
Vol 70 (7) ◽  
pp. 483-497 ◽  
Author(s):  
Taha Aziz ◽  
Aeeman Fatima ◽  
Asim Aziz ◽  
Fazal M. Mahomed
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Lie Group ◽  
Group Analysis ◽  
Fourth Grade ◽  
Invariant Solutions ◽  
Conditional Symmetry ◽  
Partial Differential ◽  
Grade Fluid ◽  
Fourth Grade Fluid

AbstractIn this study, an incompressible time-dependent flow of a fourth-grade fluid in a porous half space is investigated. The flow is generated due to the motion of the flat rigid plate in its own plane with an impulsive velocity. The partial differential equation governing the motion is reduced to ordinary differential equations by means of the Lie group theoretic analysis. A complete group analysis is performed for the governing nonlinear partial differential equation to deduce all possible Lie point symmetries. One-dimensional optimal systems of subalgebras are also obtained, which give all possibilities for classifying meaningful solutions in using the Lie group analysis. The conditional symmetry approach is also utilised to solve the governing model. Various new classes of group-invariant solutions are developed for the model problem. Travelling wave solutions, steady-state solution, and conditional symmetry solutions are obtained as closed-form exponential functions. The influence of pertinent parameters on the fluid motion is graphically underlined and discussed.

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