mathmatic project 1

Who can do this urgent project within two days. I will give a new document when pm [login to view URL] also have a simlar project 2 . 1. Derive the following equation for u: ut &#56256;&#56320; kuxx + (vu)x + cu = 0; x 2 R; 0 < t < T: (1) 2. Suppose that the eects of diusion and degradation are neglectably small and the water ow is constant with velocity a > 0. Then we can approximate the problem (1) by studying the following equation, ut + aux = 0; x 2 R; 0 < t < T: The initial condition shall be given by u(x; 0) = u0(x); x 2 R; where u0 is twice dierentiable with bounded derivatives. This equation can be approximated by the following numerical scheme: un+1 j &#56256;&#56320; un j
t + a un j &#56256;&#56320; un j&#56256;&#56320;1 h = 0; j 2 Z; 0  n   T
t  ; u0 j = u0(xj); j 2 Z: Prove that there exists a constant C which depends on u0, a, and T such that sup j2Z ju(xj ; tn) &#56256;&#56320; un j j  C(
t + h) for all 0  n   T
t  ; (2) provided that a
t=h  1: You can proceed as follows: (a) Let Ln j := u(xj ; tn+1) &#56256;&#56320; u(xj ; tn)
t + a u(xj ; tn) &#56256;&#56320; u(xj&#56256;&#56320;1; tn) h and use Taylor expansion to show that jLn j j  CL(
t+h); where CL may depend on a and sup juxxj; sup juxtj; sup juttj in R  [0; T]: 1 (b) Show that the error en j = u(xj ; tn) &#56256;&#56320; un j satises en+1 j &#56256;&#56320; en j
t + a en j &#56256;&#56320; en j&#56256;&#56320;1 h = Ln j and use induction on n to prove sup j2Z jen j j  n
tCL(
t + h) for all 0  n   T
t  and then conclude (2) holds. 3. To remove one of the simplications from the previous exercise assume now that the water ows with velocity v 2 C1(R) where v(x)  0 and v0(x)  0 for all x 2 R. Then we can approximate the original problem (1) by studying ut + (vu)x = 0; x 2 R; t > 0; with initial condition u(x; 0) = u0(x); x 2 R: (a) Prove that supx2R ju(x; t)j  supx2R ju0(x)j for all t > 0: (b) Discuss the stability of the following scheme to approximate the problem: un+1 j &#56256;&#56320; un j
t + vj un j &#56256;&#56320; un j&#56256;&#56320;1 h + un j vj &#56256;&#56320; vj&#56256;&#56320;1 h = 0; j 2 Z; n 2 Z; n  0; u0 j = u0(xj); j 2 Z; with vj = v(xj ). 4. (a) Assume constant velocity of the water a > 0: Find a transformation of u of the form ~u = f(t)u with a suitable function f(t) to reduce the original problem (1) to ~ut &#56256;&#56320; k~uxx + a~ux = 0; x 2 R; 0 < t < T: (3) (b) Use the closed-form solution v(x; t) = 1 p 4kt Z R exp  &#56256;&#56320; (x &#56256;&#56320; y)2 4kt  v0(y) dy for the heat equation vt &#56256;&#56320; kvxx = 0; x 2 R; 0 < t < T; v(x; 0) = v0(x); x 2 R; (where v0 is continuous with R R jv0(y)j dy < 1) and the change of variable formula v(x; t) = ~u(x + at; t) to derive a formula for the solution of (3) supplemented with initial condition ~u(x; 0) = ~u0(x); x 2 R: 2