PCE_ODE_HERMITE
Hermite Polynomial Chaos Expansion for a Scalar ODE

PCE_ODE_HERMITE 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.

The deterministic equation is

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

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

        u(t) = u0 * exp ( t^2/2)
      

Licensing:

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

Languages:

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

Related Data and Programs:

BLACK_SCHOLES, a FORTRAN90 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 FORTRAN90 library which approximates solutions of the Ornstein-Uhlenbeck stochastic differential equation (SDE) using the Euler method and the Euler-Maruyama method.

PCE_BURGERS, 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.

SDE, a FORTRAN90 library which illustrates the properties of stochastic differential equations, and common algorithms for their analysis, by Desmond Higham;

STOCHASTIC_RK, a FORTRAN90 library which applies a Runge Kutta (RK) scheme to a stochastic differential equation.

Reference:

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:

• pce_ode_hermite.f90, the source code.

Examples and Tests:

• pce_ode_hermite_test.f90, a sample calling program.
• pce_ode_hermite_test.txt, the output file.

List of Routines:

• HE_DOUBLE_PRODUCT_INTEGRAL: integral of He(i,x)*He(j,x)*e^(-x^2/2).
• HE_TRIPLE_PRODUCT_INTEGRAL: integral of He(i,x)*He(j,x)*He(k,x)*e^(-x^2/2).
• PCE_ODE_HERMITE applies the polynomial chaos expansion to a scalar ODE.
• R8_FACTORIAL computes the factorial of N.
• TIMESTAMP prints the current YMDHMS date as a time stamp.

You can go up one level to the FORTRAN90 source codes.