# Algebraic solutions of ODE using p-adic numbers by Katz N.M.

By Katz N.M.

Similar differential equations books

Numerical Methods for Ordinary Differential Equations

Researchers and scholars from numerical tools, engineering and different sciences will locate this ebook offers an available and self-contained creation to numerical equipment for fixing traditional differential equations. It sticks out among different books at the topic end result of the author's lucid writing sort, and the built-in presentation of thought, examples, and routines.

Sobolev Spaces in Mathematics II: Applications in Analysis and Partial Differential Equations (International Mathematical Series)

Sobolev areas turn into the verified and common language of partial differential equations and mathematical research. between a major number of difficulties the place Sobolev areas are used, the next vital themes are the point of interest of this quantity: boundary worth difficulties in domain names with singularities, greater order partial differential equations, neighborhood polynomial approximations, inequalities in Sobolev-Lorentz areas, functionality areas in mobile domain names, the spectrum of a Schrodinger operator with destructive capability and different spectral difficulties, standards for the total integration of platforms of differential equations with purposes to differential geometry, a few points of differential kinds on Riemannian manifolds on the topic of Sobolev inequalities, Brownian movement on a Cartan-Hadamard manifold, and so on.

Extra info for Algebraic solutions of ODE using p-adic numbers

Sample text

25) TJ a) 0 n (1 8~ 12 +~a·z~~ ~ at j,l=l -Re~ ~ j,l=l J ax. ax J l a·1 au µ·au_ Jax l 1 at 18 12 n +2~h·µ ~ ~ J J j=l at 40 2. EXISTENCE OF A SOLUTION FOR A HYPERBOLIC EQUATION . ;':-... J 1 j,l=l J ax . at In the above, we set n n M = L ajlµjµi, N = Lhjµj. 25) ~ (ry + I~~ 1 2N) 2 - 2£ 112 M 112 \~~I+ ryL. Regarding the right-hand side of the above as a quadratic expres- . 25) is also non-negative. 29) ry 2 + 2Nry - M ~ 0. But, the left-hand side is the characteristic equation Po(t, x, ry, µ) of P.

On the other hand, a2 ( at 2 - a2 ) :2::: ax2 u(t,p) = 2 j=l (a2u at2 - J :2::: a2u ax2 2 j=l a2u) az2 (t,p) = J and so u(t,p)--+ o, au at (t,p)--+ 'l/;(p) = 'l/;(x) Ifwe define u(t,x) E C 2 ((0,oo) x JR2 ) by u(t,x) = u(t,x,O), (t---+ 0). o, 26 1. :: ax2 u = o, j=l -----+ J 0, au at (t, x) -----+ '1/J(x) (t --t 0). Now, we write u(t,x) = M[~](t,x,O) = - 1- 47rt { }l(y,z)-(x,O)l=t ~(y,z)dSy,z· We can express the part of the sphere of radius t and center at (x,O) with z > 0 as {(x1 + s1, x2 + s2, Jt 2 - si - s§ ); si + s§ < t 2}.

0. 47) M[g](t, x) = { g(x + tf,)dSe, 4t7r Js2 (t > 0), where g is a function defined on ~ 3 . It follows immediately from the definition that if g E cm(~ 3 ), then M[g] E cm((O, oo) x ~ 3 ). Therefore, if cp E C 3 (~ 2 ), then M[cp] E C 3 ((0,oo) x ~ 3 ). 48) - A) M[cp] = 0, { M[cp](t,x)------+ 0, :tM[cp](t,x)------+ cp(x). The convergence, in the above, is generalized and uniform in ~ 3 . Now, since M[cp] E C 3 ((0, oo) x ~ 3 ), the following holds: EP2 - A ) ot a M[cp] = 8ta ( 8t2 a2 - A ) M[cp] = O.