Hermite Polynomial Chaos Expansion for a Scalar ODE

PCE_ODE_HERMITE is a FORTRAN77 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.

The deterministic equation is

        du/dt = - alpha * u,
        u(0) = u0
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.

The exact expected value of the stochastic equation is known to be

        u(t) = u0 * exp ( t^2/2)
This should be matched by the first component of the polynomial chaos expansion.


The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.


PCE_ODE_HERMITE is available in a C version and a C++ version and a FORTRAN77 version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

BLACK_SCHOLES, a FORTRAN77 library which implements some simple approaches to the Black-Scholes option valuation theory, by Desmond Higham.

HERMITE_POLYNOMIAL, a FORTRAN90 library which evaluates the physicist's Hermite polynomial, the probabilist's Hermite polynomial, the Hermite function, and related functions.

ORNSTEIN_UHLENBECK, a FORTRAN77 library which approximates solutions of the Ornstein-Uhlenbeck stochastic differential equation (SDE) using the Euler method and the Euler-Maruyama method.

PCE_BURGERS, a FORTRAN77 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.

SDE, a FORTRAN77 library which illustrates the properties of stochastic differential equations (SDE's), and common algorithms for their analysis, by Desmond Higham;

STOCHASTIC_RK, a FORTRAN77 library which applies a Runge Kutta (RK) scheme to a stochastic differential equation.


  1. Roger Ghanem, Pol Spanos,
    Stochastic Finite Elements: A Spectral Approach,
    Revised Edition,
    Dover, 2003,
    ISBN: 0486428184,
    LC: TA347.F5.G56.
  2. Dongbin Xiu,
    Numerical Methods for Stochastic Computations: A Spectral Method Approach,
    Princeton, 2010,
    ISBN13: 978-0-691-14212-8,
    LC: QA274.23.X58.

Source Code:

Examples and Tests:

List of Routines:

You can go up one level to the FORTRAN77 source codes.

Last modified on 16 September 2012.