MATH528

MATH52808/24/2018

Lab01: An introduction to TeXmacs, Mathematica

1TeXmacs

TeXmacs is an editor especially well suited for scientific work:

Mathematical text is easily written, in legible form, e.g. $y^{'} = f (x)$ is a differential equation and

${\begin{cases} y^{'} = f (x) \\ y (x_{0}) = y_{0} \end{cases} .,$ (1)

is an initial value problem.
The text can be exported to LaTeX, HTML, PDF formats.

Computations can be interspersed, such that one obtains a “living” document that contains the code used to produce results.

Mathematica solution of the ODE $y^{'} = x + y$

In[1]:=

ODE = y'[x]==x+y[x]

$y^{'} (x) = y (x) + x$

In[2]:=

DSolve[ODE,y[x],x]

${{y (x) \to c_{1} e^{x} - x - 1}}$

In[3]:=

2Mathematica

Mathematica is a computational platform that combines symbolic, numerical and graphical calculations. Sample computations:

In[3]:=

10!

$3628800$

In[5]:=

p4=Expand[(x+y)^4]

$x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4}$

In[6]:=

Factor[p4]

${(x + y)}^{4}$

In[8]:=

Integrate[1/(1+x^2),x]

$\tan^{- 1} (x)$

In[10]:=

LaplaceTransform[Sin[t],t,s]

$\frac{1}{s^{2} + 1}$

In[13]:=

FourierTransform[Exp[-x^2],x,k]

$\frac{e^{- \frac{k^{2}}{4}}}{\sqrt{2}}$

In[14]:=

3Euler's method

An analytical solution to an ODE may be impossible. This usually happens for non-linear, higher-order ODEs, e.g.,

In[7]:=

ODE = y”[x] + x y[x] y'[x] == 0

$y^{''} (x) + x y (x) y^{'} (x) = 0$

In[8]:=

DSolve[ODE,y[x],x]

$DSolve [y^{''} (x) + x y (x) y^{'} (x) = 0, y (x), x]$

Even if an analytical solution is obtainable, it might be too complicated to work with

In[24]:=

f[x_,y_] = (x+y)^5;

ODE = y'[x] == f[x,y[x]]

$y^{'} (x) = {(y (x) + x)}^{5}$

In[25]:=

DSolve[ODE,y[x],x]

$Solve [\frac{1}{20} (4 \log (y (x) + x + 1) + (\sqrt{5} - 1) \log ({(y (x) + x)}^{2} + \frac{1}{2} (\sqrt{5} - 1) (y (x) + x) + 1) - (1 + \sqrt{5}) \log ({(y (x) + x)}^{2} - \frac{1}{2} (1 + \sqrt{5}) (y (x) + x) + 1) + 2 \sqrt{10 - 2 \sqrt{5}} \tan^{- 1} (\frac{4 y (x) + 4 x - \sqrt{5} - 1}{\sqrt{10 - 2 \sqrt{5}}}) + 2 \sqrt{2 (5 + \sqrt{5})} \tan^{- 1} (\frac{4 y (x) + 4 x + \sqrt{5} - 1}{\sqrt{2 (5 + \sqrt{5})}}) - 20 x) = c_{1}, y (x)]$

In[26]:=

In such cases an approximation may be obtained numerically. Software systems like Mathematica contain sophisticated procedures that combine multiple algorithms to find a solution

In[42]:=

IniCond = y[0]==0

$y (0) = 0$

In[43]:=

Y[x_] = y[x] /. NDSolve[{ODE,IniCond},y[x],{x,0,1}][[1,1]];

YN = Table[{x,Y[x]},{x,0,1,0.2}]

$(\begin{array}{cc} 0 . & 0 . \\ 0.2 & 0.000010759 \\ 0.4 & 0.00068596 \\ 0.6 & 0.00806314 \\ 0.8 & 0.0516308 \\ 1 . & 0.348729 \end{array})$

In[44]:=

The simplest numerical algorithm to approximate an ODE is solution is Euler's method

Y_{i + 1} = Y_{i} + h f (x_{i}, Y_{i}),

where $x_{i} = x_{0} + i h$ , $h$ is the step size, and $Y_{i} ≅ y (x_{i})$ . Euler's method is applied repeatedly within a loop to obtain the approximation

In[29]:=

h=0.2; x0=0.; y0=0.; xfinal=1.; nsteps=Floor[(xfinal-x0)/h]

$5$

In[45]:=

YE = {y0,0,0,0,0,0};

For[i=1, i<=nsteps, i++,

YE[[i+1]] = YE[[i]] + h f[x0 + i h, YE[[i]]]

];

{YN,YE}

$(\begin{array}{cccccc} {0 ., 0 .} & {0.2, 0.000010759} & {0.4, 0.00068596} & {0.6, 0.00806314} & {0.8, 0.0516308} & {1 ., 0.348729} \\ 0 . & 0.000064 & 0.00211364 & 0.0179415 & 0.0911634 & 0.400534 \end{array})$

In[46]:=

Note that the approximate values from Euler's method are not the same as those from the Mathematica built-in NDSolve. In this lab we'll investigate why this happens.