Question

Please write neatly

Let f(x) be a Cl(R) function and u(t, x) be a solution to the inviscid Burger's IVP Ut + uuz = 0, 4(0,x) = f(x). (a) Use the formula ult, x) = f (x – ut) to derive the following explicit identity for the partial derivative Ou du Ət f($)f'(5) 1+tf'(E) with $ = X – ut. (b) Assume that there exists a number wo E R such that f'(20) < 0 and f(x) > 0. Use part (a) to prove that + + as t + - Filmo) along the characteristic line going through the point Xo.

Answer 1

ANSWER Given that oco,x) = fre) langranges equation at = dd - du 3) 0 = constant =4, 0 and de = dt . a-cit = 2 - 0 from (1) and (2) we have Ø -cit j =U - © OCO,x) = fou 0 Cx) = fcx) - Then a cx-cit) =0= f(x- cit) u=fce-ut) and out) = f'(x-ut) [44) -0) =) Sy [1+++'(x-ut)) = -UP' (x-ut) o = -Uf'ex-Ut)/1++%(-ut) odp) = -Uf'(x-ut) Http/-06)

= -of'x-ut) itt plcx-ut) = -f(x-ut) f'(X-ut) itt f'(x-ut) 5-f1&Jf'cę) Here $ = x-ut. H++ plie) b) XER, such that flexo) 20, and flexo)>0 H + L f (53f'ls)-f() + t f'(4) ott flood = -f(8) Itt fire) so denominator will become Zero. I can) = -f(8) tt-1 floo) ett fral