Somie info about this PDE please ?

[zetafunction]

Somie info about this PDE please ??

$$\frac{\partial u}{\partial t} = r u - (1+\nabla^2)^2u + f(u)$$

where f(u) is an smooth function , u=U(x,t) is the solution of the PDE

this is the Swift-Hohenberg equation, my teacher has asked me to solve it or look some info about it specially

- How it can be solved by Analytic methods in 1-D (x,t)

- Special cases: spherical, cilindric coordinates

i heard it was compeltely known how to solve (approximately) this equation, could someone give me some info about how the S-H is deduced mathematically and how to find the solutions for several f(u) ?? .. thanks in advance.

 

[gtoguy97]

The Swift-Hohenberg equation is a fourth order nonlinear partial differential equation (PDE) that is widely used to model the destabilizing effects of thermal convection. It was first proposed by J.M. Swift and P.C. Hohenberg in 1977 as a model for Rayleigh-Benard convection. The equation is written in the general form:\frac{\partial u}{\partial t} = r u - (1+\nabla^2)^2u + f(u) where u=U(x,t) is the solution of the PDE, r is the Rayleigh number, \nabla^2 is the Laplacian operator and f(u) is an arbitrary smooth function. The equation can be solved analytically in one dimension using separation of variables. In this case, it can be shown that the solutions of the S-H equation are of the form u(x,t) = A sin(kx + ωt) + B cos(kx + ωt) where k and ω are constants determined by the boundary conditions. For special cases such as spherical and cylindrical coordinates, the S-H equation can be solved using finite difference methods. These methods are more computationally intensive than analytic methods, but they are more accurate and easier to implement. Finally, the S-H equation can also be solved numerically using various numerical methods such as the finite element method or the finite volume method. These methods are usually more accurate than analytical methods, but they are also more computationally intensive.