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Equation data
Category:3. Linear Partial Differential Equations
Subcategory:3.1. Second-Order Parabolic Equations
Equation(s):$\displaystyle \frac{\partial w}{\partial t}-\dfrac{1}{2}\frac{\partial^2 w}{\partial x^2}=0$.
Solution(s),
Transformation(s),
Integral(s)
:
1. Solution:\hfill\break
$$
w=N(\theta(x,t))\dfrac{c}{\sqrt {t-t_0}} \exp\left\lbrace -\dfrac{(x-\lambda)^2}{2(t-t_0)}\right\rbrace,
$$
where 
$\displaystyle N(\theta)=\frac 1{\sqrt{2\pi}}\int^\theta_{-\infty}\exp\left(-\frac12u^2\right)\,du$  
is the normal distribution function, 
$$ 
\theta(x,t)=\varepsilon\dfrac {(x-\lambda )+p(t-t_0)}{\sqrt{(t-t_0)[a(t-t_0) - 1]}}\quad
 (\varepsilon= +1,-1),
$$
and $a$, $c$, $p$, $t_0$, $\lambda$ are arbitrary constants $(a>0, \ c \neq 0)$.
\medskip

2. Solution:\hfill\break
$$
w=N(\theta(x,t))c \exp\left\lbrace n(x-\lambda)+\dfrac {n^2}{2}(t-t_0) \right\rbrace,
$$
where $N(\theta)$  as the normal distribution function,  
$$
\theta(x,t)=\varepsilon\dfrac {(x-\lambda )+n(t-t_0)}{\sqrt{t-t_0}}\quad
 (\varepsilon= +1,-1),
$$
and $c$, $n$, $\lambda$, $t_0$ are arbitrary constants.
Remarks:The first solution has various interesting (paradoxical) properties.  
First,  only under the condition  $t>t_0 +1/a$, function $\theta (x,t)$ is real 
and can have a logical interpretation. This property guarantees that this solution
will not share the interpretation difficulty of the Green function of the diffusion equation: 
this solution does not describe a motion with infinite speed.

We can also state that under conditions $t \rightarrow t_0 +1/a,\  a\rightarrow \infty$, 
this solution tends to Dirac delta function $\delta (x-\lambda)$. 
Also for $t\rightarrow \infty,$\  $\theta  \rightarrow\varepsilon p/\sqrt{a}. $

Note that this solution can be normalized because   $0\leq N(\theta)\leq 1$   
for any value of its argument.  Graphically, the presence of  $N(\theta)$   
creates an asymmetry of the Gauss curve.  The selection of   $\varepsilon$'s  sign 
allows displacement of this deformation to the left or to the right side of the curve.
Novelty:Material has been fully published elsewhere
References:N. Sukhomlin and J. Ortiz, New exact solutions for the Black Scholes equation and diffusion equation, Applied Mathematics E-Notes, 2007, Vol. 7, pp. 206--213 (http://www.emis.ams.org/journals/AMEN/2007/060606-2.pdf).
Author/Contributor's Details
Last name:Sukhomlin
First name:Nikolay
Country:Dominican Republic
Statistic information
Submission date:Mon 16 Jun 2008 16:52
Edits by author:0

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