Higher-Order Partial Differential Equations
From Scholarpedia
| Andrei D. Polyanin et al. (2009), Scholarpedia, 4(1):4605. | revision #55839 [link to/cite this article] | |||||||||||||||||||
Curator: Dr. Andrei D. Polyanin, Institute for Problems in Mechanics, Moscow, Russia
Curator: Dr. William E. Schiesser, Lehigh University and University of Pennsylvania, USA
Curator: Dr. Alexei I. Zhurov, Cardiff University, UK, and Institute for Problems in Mechanics, Moscow, Russia.
First-Order Partial Differential Equations
Second-Order Partial Differential Equations
Higher-Order Partial Differential Equations
Apart from second-order PDEs, higher-order equations also quite often arise in applications. Below are only a few important examples of such equations with some of their solutions.
Higher-Order Linear Partial Differential Equations
Equation of transverse vibration of elastic rod:
The equation has the following particular solutions:
![\begin{array}{l} w(x,t)=\bigl[A\sin(\lambda x)+B\cos(\lambda x)+C\sinh(\lambda x)+D\cos(\lambda x)\bigr]\sin(\lambda^2at),\\[3pt] w(x,t)=\bigl[A_1\sin(\lambda x)+B_1\cos(\lambda x)+C_1\sinh(\lambda x)+ D_1\cos(\lambda x)\bigr]\cos(\lambda^2at), \end{array}](/wiki/images/math/8dc370a0b7f1522dacbcd63dfbcf4c3b.png)
where 
, 
, 
, 
,
, 
, 
, 
, and
are arbitrary constants.
For solutions to associated Cauchy problems and boundary value problems, see Equation of transverse vibration of elastic rods at EqWorld.
Biharmonic equation:
- (1)
where 
is the biharmonic operator,
The biharmonic equation (1) is encountered in plane problems of elasticity
(
is the Airy stress function). It is also used to describe slow flows of
viscous incompressible fluids ( is the stream function).
Various representations of the general solution to equation (1) in terms of harmonic functions include
![\!\!\!\begin{array}{l} w(x,y)=xu_1(x,y)+u_2(x,y),\\[3pt] w(x,y)=yu_1(x,y)+u_2(x,y),\\[3pt] w(x,y)=(x^2+y^2)u_1(x,y)+u_2(x,y), \end{array}](/wiki/images/math/1564ff5f6e119fe8d9dfa84765ccddf3.png)
where 
and 
are arbitrary functions satisfying the Laplace
equation 
(
).
Complex form of representation of the general solution:
![w(x,y)=\mbox{Re}\bigl[\overline z f(z)+g(z)\bigr],](/wiki/images/math/f9a2b0539a60cb9183ef297209e73dce.png)
where 
and 
are arbitrary analytic functions of the complex
variable 
; 
, 
. The symbol ![\mbox{Re}[A]](/wiki/images/math/6628b20b8f403d654eef653551a165b6.png)
stands
for the real part of a complex quantity .
For solutions to associated boundary value problems, see Biharmonic equation at EqWorld.
Higher-Order Nonlinear Partial Differential Equations
Korteweg–de Vries equation:
It is used in many sections of nonlinear mechanics and theoretical physics for describing one-dimensional nonlinear dispersive nondissipative waves. In particular, the mathematical modeling of moderate-amplitude shallow-water surface waves is based on this equation. For exact solutions to this equation, see Korteweg–de Vries equation at EqWorld.
Equation of a steady laminar boundary layer on a flat plate:
where is the stream function. For exact solutions, see Boundary layer equations at EqWorld.
Boussinesq equation:
This equation arises in several physical applications: propagation of long waves in shallow water, one-dimensional nonlinear lattice-waves, vibrations in a nonlinear string, and ion sound waves in a plasma. For exact solutions, see Boussinesq equation at EqWorld.
Equation of motion of a viscous fluid:
This is a two-dimensional stationary equation of motion of a viscous
incompressible fluid—it is obtained from the Navier–Stokes equations
by the introduction of the stream function . For exact solutions to this equation, see Navier–Stokes equations at EqWorld.
