From: israel@math.ubc.ca (Robert Israel) Subject: Re: PDE: u_t = c x^2(1-x)^2 u_xx Date: 10 Jun 1999 22:14:10 GMT Newsgroups: sci.math To: econ-theorist@earthling.net Keywords: Reducing to Heat Equation In article <375E9AFD.2C5E1DB2@umich.edu>, Lones Smith writes: > I have the PDE of the subject header for t >= 0, and 0 <= x< = 1, > with time boundary conditions u(0,x) = f(x), where f is a given > convex piecewise linear function, AND terminal condition > u(infinity,x) = x f(1)+(1-x)f(0). It helps to use the change of variables: x = 1/(1+exp(-y + c t)), u = U/(1+exp(-y + c t)) under which your PDE becomes U_t = c U_yy (-infinity < y < infinity) which is the heat equation. But your terminal condition can't work. Note that U(0,y) = f(x)/x -> f(1) as y -> infinity and is asymptotic to f(0) exp(-y) as y -> -infinity. But since the heat equation has solutions 1 and exp(-y) exp(c t) and is positivity-preserving, any solution with U(0,y) >= A exp(-y) + B for some constants A and B will have U(t,y) >= A exp(-y) exp(c t) + B, and in particular if A > 0 and c > 0 it will have U(t,y) -> infinity as t -> infinity. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ============================================================================== From: israel@math.ubc.ca (Robert Israel) Subject: Re: PDE: u_t = c x^2(1-x)^2 u_xx Date: 10 Jun 1999 22:53:30 GMT Newsgroups: sci.math In article <7jpdbi$35b$1@nntp.ucs.ubc.ca>, israel@math.ubc.ca (Robert Israel) writes: > In article <375E9AFD.2C5E1DB2@umich.edu>, > Lones Smith writes: > > I have the PDE of the subject header for t >= 0, and 0 <= x< = 1, > > with time boundary conditions u(0,x) = f(x), where f is a given > > convex piecewise linear function, AND terminal condition > > u(infinity,x) = x f(1)+(1-x)f(0). > > It helps to use the change of variables: > > x = 1/(1+exp(-y + c t)), u = U/(1+exp(-y + c t)) > > under which your PDE becomes U_t = c U_yy (-infinity < y < infinity) > which is the heat equation. But your terminal condition can't work. Oops... I forgot about the exp(c t), which saves you:it doesn't matter if U(t,y) -> infinity, you can still have u(t,y) -> constant. Of course, u(t,x) = x f(1) + (1-x) f(0) is a steady-state solution (corresponding to U(t,y) = f(1) + f(0) exp(-y + c t)). If you subtract that, and your f is piecewise linear (or more generally, Lipschitz), you have a heat equation problem where U(0,y) is bounded. Therefore the solution U(t,y) is bounded, which means u(t,y) -> 0 as t -> infinity for each y. The conclusion is that your terminal condition will always be satisfied. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ============================================================================== From: israel@math.ubc.ca (Robert Israel) Subject: Re: PDE: u_t = c x^2(1-x)^2 u_xx Date: 11 Jun 1999 20:40:06 GMT Newsgroups: sci.math In article <37615596.9F4430F3@yale.edu>, Hansen Chen writes: > Would you care sharing the insight that led you to this transformation? I would if I could remember... I think what it started with, was that I wanted to get rid of the x^2 (1-x)^2. If you write x = g(y), then you see that what you want is g' = g (1-g) which leads to g = 1/(1+exp(-y)). Then somehow I noticed that the equation becomes simpler if you look at u/(1+exp(-y)). That led to a heat equation with drift, u_t = c (u_xx + u_x), and the last step to eliminate the drift was quite standard. I'm not an expert on it, but there's a general theory of how one PDE can be transformed to another one, using analysis of the symmetries of the equation. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ============================================================================== From: israel@math.ubc.ca (Robert Israel) Subject: Re: PDE: u_t = c x^2(1-x)^2 u_xx Date: 12 Jun 1999 00:27:25 GMT Newsgroups: sci.math In article <3761973C.13A22A66@yale.edu>, Hansen Chen writes: > Robert Israel wrote: > > I'm not an expert on it, but there's a general theory of how one PDE can be > > transformed to another one, using analysis of the symmetries of the equation. > > Are you talking about Lie group on differential equations? Yes. For example, see G.W. Bluman and S. Kumei, Symmetry-based algorithms to relate partial differential equations. I. Local symmetries. European J. Appl. Math. 1 (1990), no. 3, 189--216. Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2