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mathmatic project 1

Who can do this urgent project within two days.

I will give a new document when pm [url removed, login to view] 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 e ects of di usion 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 di erentiable

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 at=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  ntCL(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

Kỹ năng: Toán học, Matlab and Mathematica

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