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<html>
<head>
<title>
PCE_ODE_HERMITE - Hermite Polynomial Chaos Expansion for a Scalar ODE
</title>
</head>
<body bgcolor="#eeeeee" link="#cc0000" alink="#ff3300" vlink="#000055">
<h1 align = "center">
PCE_ODE_HERMITE <br> Hermite Polynomial Chaos Expansion for a Scalar ODE
</h1>
<hr>
<p>
<b>PCE_ODE_HERMITE</b>
is a FORTRAN90 library which
defines and solves a time-dependent scalar exponential decay ODE
with uncertain decay coefficient, using a polynomial chaos expansion,
in terms of Hermite polynomials.
</p>
<p>
The deterministic equation is
<pre>
du/dt = - alpha * u,
u(0) = u0
</pre>
In the stochastic version, it is assumed that the decay coefficient
ALPHA is a Gaussian random variable with mean value ALPHA_MU and variance
ALPHA_SIGMA^2.
</p>
<p>
The exact expected value of the stochastic equation is known to be
<pre>
u(t) = u0 * exp ( t^2/2)
</pre>
This should be matched by the first component of the polynomial chaos
expansion.
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>PCE_ODE_HERMITE</b> is available in
<a href = "../../c_src/pce_ode_hermite/pce_ode_hermite.html">a C version</a> and
<a href = "../../cpp_src/pce_ode_hermite/pce_ode_hermite.html">a C++ version</a> and
<a href = "../../f77_src/pce_ode_hermite/pce_ode_hermite.html">a FORTRAN77 version</a> and
<a href = "../../f_src/pce_ode_hermite/pce_ode_hermite.html">a FORTRAN90 version</a> and
<a href = "../../m_src/pce_ode_hermite/pce_ode_hermite.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../f_src/black_scholes/black_scholes.html">
BLACK_SCHOLES</a>,
a FORTRAN90 library which
implements some simple approaches to
the Black-Scholes option valuation theory,
by Desmond Higham.
</p>
<p>
<a href = "../../f_src/hermite_polynomial/hermite_polynomial.html">
HERMITE_POLYNOMIAL</a>,
a FORTRAN90 library which
evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial,
the Hermite function, and related functions.
</p>
<p>
<a href = "../../f_src/pce_burgers/pce_burgers.html">
PCE_BURGERS</a>,
a FORTRAN90 program which
defines and solves a version of the time-dependent viscous Burgers equation,
with uncertain viscosity, using a polynomial chaos expansion in terms
of Hermite polynomials,
by Gianluca Iaccarino.
</p>
<p>
<a href = "../../f_src/sde/sde.html">
SDE</a>,
a FORTRAN90 library which
illustrates the properties of stochastic differential equations, and
common algorithms for their analysis,
by Desmond Higham;
</p>
<p>
<a href = "../../f_src/stochastic_rk/stochastic_rk.html">
STOCHASTIC_RK</a>,
a FORTRAN90 library which
applies a Runge Kutta (RK) scheme to a stochastic differential equation.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Roger Ghanem, Pol Spanos,<br>
Stochastic Finite Elements: A Spectral Approach,<br>
Revised Edition,<br>
Dover, 2003,<br>
ISBN: 0486428184,<br>
LC: TA347.F5.G56.
</li>
<li>
Dongbin Xiu,<br>
Numerical Methods for Stochastic Computations: A Spectral Method Approach,<br>
Princeton, 2010,<br>
ISBN13: 978-0-691-14212-8,<br>
LC: QA274.23.X58.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "pce_ode_hermite.f90">pce_ode_hermite.f90</a>, the source code.
</li>
<li>
<a href = "pce_ode_hermite.sh">pce_ode_hermite.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "pce_ode_hermite_prb.f90">pce_ode_hermite_prb.f90</a>,
a sample calling program.
</li>
<li>
<a href = "pce_ode_hermite_prb.sh">pce_ode_hermite_prb.sh</a>,
BASH commands to compile and run the sample program.
</li>
<li>
<a href = "pce_ode_hermite_prb_output.txt">pce_ode_hermite_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>HE_DOUBLE_PRODUCT_INTEGRAL:</b> integral of He(i,x)*He(j,x)*e^(-x^2/2).
</li>
<li>
<b>HE_TRIPLE_PRODUCT_INTEGRAL:</b> integral of He(i,x)*He(j,x)*He(k,x)*e^(-x^2/2).
</li>
<li>
<b>PCE_ODE_HERMITE</b> applies the polynomial chaos expansion to a scalar ODE.
</li>
<li>
<b>R8_FACTORIAL</b> computes the factorial of N.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last modified on 17 March 2012.
</i>
<!-- John Burkardt -->
</body>
</html>