User:Leo Trottier/PDE/Higher-order partial differential equations
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Higher-Order Linear Partial Differential Equations
Equation of transverse vibration of elastic rod:
\[ \frac{\partial^2w}{\partial t^2}+a^2\frac{\partial^4w}{\partial x^4}=0. \]
The equation has the following particular solutions: \[ \begin{array}{l} w(x,t)=\bigl[A\sin(\lambda x)+B\cos(\lambda x)+C\sinh(\lambda x)+D\cos(\lambda x)\bigr]\sin(\lambda^2at),\\ w(x,t)=\bigl[A_1\sin(\lambda x)+B_1\cos(\lambda x)+C_1\sinh(\lambda x)+ D_1\cos(\lambda x)\bigr]\cos(\lambda^2at), \end{array} \] where \(A\), \(B\), \(C\), \(D\ ,\) \(A_1\), \(B_1\), \(C_1\), \(D_1\), and \(\lambda\) 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:
\[\tag{1} \Delta\Delta w=0, \]
where \(\Delta\Delta\) is the biharmonic operator, \[ \Delta\Delta \equiv\Delta^{\!2}= \frac{\partial^4}{\partial x^4}+ 2\frac{\partial^4}{\partial x^2\,\partial y^2}+ \frac{\partial^4}{\partial y^4}. \]
The biharmonic equation (1) is encountered in plane problems of elasticity (\(w\)is the Airy stress function). It is also used to describe slow flows of viscous incompressible fluids (\(w\) is the stream function).
Various representations of the general solution to equation (1) in terms of harmonic functions include \[ \!\!\!\begin{array}{l} w(x,y)=xu_1(x,y)+u_2(x,y),\\ w(x,y)=yu_1(x,y)+u_2(x,y),\\ w(x,y)=(x^2+y^2)u_1(x,y)+u_2(x,y), \end{array} \] where \(u_1\) and \(u_2\) are arbitrary functions satisfying the Laplace equation \(\Delta u_k=0\,\) (\(k=1,\,2\)).
Complex form of representation of the general solution: \[ w(x,y)=\mbox{Re}\bigl[\overline z f(z)+g(z)\bigr], \] where \(f(z)\) and \(g(z)\) are arbitrary analytic functions of the complex variable \(z=x+iy\ ;\) \(\overline z=x-iy\), \(i^2=-1\ .\) The symbol \(\mbox{Re}[A]\) stands for the real part of a complex quantity \(A\ .\)
For solutions to associated boundary value problems, see Biharmonic equation at EqWorld.
Higher-Order Nonlinear Partial Differential Equations
Korteweg–de Vries equation:
\[ \frac{\partial w}{\partial t}+\frac{\partial^3w}{\partial x^3} -6w\frac{\partial w}{\partial x}=0. \] 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:
\[ \frac{\partial w}{\partial y}\frac{\partial^2w}{\partial x\,\partial y}- \frac{\partial w}{\partial x}\frac{\partial^2w}{\partial y^2}=a \frac{\partial^3w}{\partial y^3}. \] where \(w\) is the stream function. For exact solutions, see Boundary layer equations at EqWorld.
Boussinesq equation:
\[ \frac{\partial^2w}{\partial t^2}+\frac{\partial}{\partial x} \biggl(w\frac{\partial w}{\partial x}\biggr)+\frac{\partial^4w}{\partial x^4}=0. \] 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:
\[ \frac{\partial w}{\partial y}\frac{\partial}{\partial x}(\Delta w)- \frac{\partial w}{\partial x}\frac{\partial}{\partial y}(\Delta w)=a\,\Delta\Delta w,\qquad \Delta w=\frac{\partial^2w}{\partial x^2}+\frac{\partial^2w}{\partial y^2}. \] 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 \(w\ .\) For exact solutions to this equation, see Navier–Stokes equations at EqWorld.
