taylor expansion u(x) for s(u, x, E)

place Link to heading

Estuary is coastal water body where streams from rivers mixes with streams from the ocean.

concentration Link to heading

Concentration of single point source conservative pollutant in an estuary is as follow

$$\tag{1} s = s_0 \exp \left( \frac{xu}{E} \right) $$

for $x \le 0$, where $s$ is final centration $\rm (mg/L)$, $u$ is velocity $\rm (miles/day)$, $x$ is distance from point source $\rm (miles)$. There are coefficients for initial concentration $s_0 = 350 \ {\rm mg/L}$ and tide wave dispersion coefficients $E = 5 \ {\rm miles^2/day}$.

velocity Link to heading

In a certain estuary the velocity can be formulated as

$$\tag{2} u(x) = | \sin 4.8x|. $$

task Link to heading

Use Taylor series to approximate the value of velociy for given distance $u(x)$ until 5% error result.
Use the aproximation value of velocity to calculate the concentration pollutant in particular distance in the estuary $s(x)$

output Link to heading

If $x > 0$, repeat ask for $x$
$x = -1 \ {\rm miles}$
$N = 15$
$u = 0.98452 \ {\rm miles/day}$
$\epsilon = 1.1692 \ \%$
$s = 287.4445 \ {\rm mg/L}$

taylor series Link to heading

In general a function $f(x)$ can be approximated to $N+1$ terms

$$\tag{3} f(x) \approx \sum_{n=0}^N \frac{(x - x_0)^n}{n!} \left. \frac{d^n f(x)}{dx^n} \right|_{x=x_0} = \sum _{n=0}^N g_n(x, x_0), $$

which is known as Taylor series. And for $f(x)$ in Eqn (2) it would be similar to

$$\tag{4} f(x) = \sin ax. $$

$n = 0$ Link to heading

$$\tag{5.0} g_0(x, x_0) = \sin a x_0. $$

$n = 1$ Link to heading

$$\tag{5.1} g_1(x, x_0) = (x - x_0) \cdot a \cos a x_0. $$

$n = 2$ Link to heading

$$\tag{5.2} g_2(x, x_0) = \tfrac12(x - x_0)^2 \cdot -a^2 \sin a x_0. $$

$n = 3$ Link to heading

$$\tag{5.3} g_3(x, x_0) = \tfrac16 (x - x_0)^3 \cdot -a^3 \cos a x_0. $$

general form Link to heading

Eqns (5.0) - (5.3) can be represented in

$$ g_n(x, x_0) = (-1)^{\lfloor \frac12 n \rfloor} \frac{a^n (x - x_0)^n}{n!} \left\{ \begin{array}{rc} \sin a x_0, & n = 0, 2, 4, \dots \newline \cos a x_0, & n = 1, 3, 5, \dots \end{array} \right. $$ which holds for all $n$.

python Link to heading

code Link to heading

import math

def concentration(x, u):
  s0 = 350
  E = 5
  return s0 * math.exp(x * u / E)

def velocity(x, a):
  return abs(math.sin(a * x))

def taylorterm(n, x, x0, a):
  sign = (-1)**math.floor(0.5 * n)
  an = a**n
  dxn = (x - x0)**n
  facn = math.factorial(n)
  if n % 2 == 0:
    fx = math.sin(a * x0)
  else:
    fx = math.cos(a * x0)
  
  h = (sign * an * dxn / facn) * fx
  return h

a = 4.8
x = -1
x0 = 0

print("n  utaylor error s")

N = 20
velotaylor = 0
for n in range(N):
  u = velocity(x, a)
  velotaylor += taylorterm(n, x, x0, a)
  err = abs(u - abs(velotaylor))
  
  conc = concentration(x, velotaylor)
  
  str1 = f'{n:02d}'
  str2 = f'{velotaylor:0.5f}'
  str3 = f'{(err*100):0.4f}%'
  str4 = f'{conc:0.4f}'
  print(str1, str2, str3, str4)

Code is available https://onecompiler.com/python/3zn4x6dnc.

result Link to heading

n  utaylor error s
00 0.00000 99.6165% 350.0000
01 -4.80000 380.3835% 914.0938
02 -4.80000 380.3835% 914.0938
03 13.63200 1263.5835% 22.9091
04 13.63200 1263.5835% 22.9091
05 -7.60166 660.5499% 1600.8115
06 -7.60166 660.5499% 1600.8115
07 4.04652 305.0353% 155.8088
08 4.04652 305.0353% 155.8088
09 0.31910 67.7065% 328.3609
10 0.31910 67.7065% 328.3609
11 1.09982 10.3659% 280.8915
12 1.09982 10.3659% 280.8915
13 0.98452 1.1648% 287.4445
14 0.98452 1.1648% 287.4445
15 0.99717 0.1003% 286.7181
16 0.99717 0.1003% 286.7181
17 0.99610 0.0068% 286.7796
18 0.99610 0.0068% 286.7796
19 0.99617 0.0004% 286.7754

matlab Link to heading

code Link to heading

% input
for i =1:inf
    x = input("x = ");
    if x <= 0
        break
    else
        disp("x should <= 0");
    end
end


% parameters
a = 4.8;
x0 = 0;
emin = 0.05;

% iteration
vtay = 0;
err = 1;
N = 20;
for n = 0:N
    vtay = vtay + tayt(n, x, x0, a);
    vact = velo(x, a);
    err = abs(vact - abs(vtay));
    if err < emin
        break
    end
end

% concentration of pollutan
s = conc(x, vtay)

% results
fprintf('n = %d\n', n);
fprintf('err = %.4f\n', err * 100);
fprintf('vtay = %.5f\n', vtay);
fprintf('s = %.4f\n', s);

% n-th term of taylor series
function t = tayt(n, x, x0, a)
    sign = (-1)^floor(0.5 * n);
    an = a^n;
    dxn = (x - x0)^n;
    facn = factorial(n);
    
    if mod(n, 2) == 0
        fx = sin(a * x0);
    else
        fx = cos(a * x0);
    end

    t = (sign * an * dxn / facn) * fx;
end

% actual velocity
function v = velo(x, a)
    v = abs(sin(a * x));
end

% concentration of pollutant
function c = conc(x, u)
    s0 = 350;
    E = 5;
    c = s0 * exp(x * u / E);
end

result Link to heading

x = 
1
x should <= 0
x = 
3
x should <= 0
x = 
-1

s =

  287.4445

n = 13
err = 1.1648
vtay = 0.98452
s = 287.4445

comparison Link to heading

Parameters	Assignment	Python	Matlab
$N$	15	14	13
$u$	0.98452	0.98452	0.98452
$\epsilon$	1.1692	1.1648	1.1648
$s$	287.4445	287.4445	287.4445

Values for $N$ and $\epsilon$ are not the same for Assignment and Program.