By Andrew Russell Forsyth
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Extra resources for Theory of differential equations
Example text
In fact, the Jacobian matrix of the map (y, s) H x at (0,0) is given by "' ax D(y,s) y=0,s=0 Where 0) = F(0, u(0), V'uo(0), ao). 12). By the Hence det J(0) implicit function theorem, for any x e 1[8n sufficiently small, we can solve 2. First-Order Differential Equations 28 x = x(y, s) uniquely for y E I[8"-1 and s E ll8 sufficiently small. Then define u(x) = o(y, s). We will prove that this is the desired solution and pz(y, s) _ Uxi(x(y, s)) for i = 1,... , n. Step 2. We claim that F'(x(y,S),(P(y,S),p(y,S)) = 0, for any y and s sufficiently small.
Consider the linear functional F : L*Cl(IEBn x (0, T)) -+ ][8 given by F(L*v) = for any v E C1 (W x (0, T)). We note that F acting on L*v in the left-hand side is defined in terms of v itself in the right-hand side. Hence we need to verify that such a definition makes sense. In other words, we need to prove that L*vl = L*v2 implies (f, vl) L2 (fin x (O,T)) (f, v2) L2 (fin x (O,T) )' for any v, v2 E C1(RTh x (0, T)). By linearity, it suffices to prove that L*v = 0 implies v = 0 for any v e Cl (][8n x (0, T) ).
1. With t as time, the graph of the solution represents a wave propagating to the right with velocity 1 without changing shape. It is clear that u exists globally in ll82. 2. Quasilinear Equations. Next, we discuss initial-value problems for first-order quasilinear PDEs. Let 1 C Jn be a domain containing the origin and a2 and f be smooth functions in SZ x J. 5) a2 (x, u)u= f(x,u), z=1 u(x', 0) = uo(x'). , an(0,uo(0)) 0. 5) admits a smooth solution u. 2. The Method of Characteristics 21 function we intend to find, is present.
Theory of differential equations by Andrew Russell Forsyth
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