Return to computing page for the first course APMA0330
Return to computing page for the second course APMA0340
Return to computing page for the fourth course APMA0360
Return to Mathematica tutorial for the first course APMA0330
Return to Mathematica tutorial for the second course APMA0340
Return to Mathematica tutorial for the fourth course APMA0360
Return to the main page for the course APMA0330
Return to the main page for the course APMA0340
Return to the main page for the course APMA0360
Introduction to Linear Algebra with Mathematica

Glossary

Preface

Monte-Carlo computations often yield numerical answers of limited accuracy, and are therefore employed as a last resort. It has been found, however, that some of the limitations of Monte-Carlo methods can be overcome through a judicious useof orthogonal expansions. When a numerical answeris obtained asthe expected value of an estimator, expansion of that estimator in a seriesof orthogonal func- tions (or functionals) can reduce the variance of the estimate. Expansion of the estimand in orthogonal polynomials can increase accuracy and efficiency and simplify the solution of nonlinear problems.

Solution of Heat Equation

It is known that the solution to the initial value problem for the heat equation
\[ u_t = u_{xx}, \qquad u(x,0) = f(x) , \]
is represented as the convolution integral:
\[ u(x,t) = \frac{1}{\sqrt{4\pi t}} \int f(\xi )\, e^{- (x- \xi )^2/(4t)} {\text d}\xi = f \star \frac{1}{\sqrt{4\pi t}} e^{-x^2/(4t)} . \]

 

  1. Chorin, A.J., Hermite Expansions in Monte-Carlo Computation, Journal of Computational Physics, 1971, Vol. 8, pp. 472--482.
  2. R. E. A. C. Paley and N. Wiener,“Fourier Transforms in the Complex Domain,” Colloquium Publication Vol. 19, American Mathematical Society, Providence, R. I., 1934.

 

Return to Mathematica page
Return to the main page (APMA0360)
Return to the Part 1 Basic Concepts
Return to the Part 2 Fourier Series
Return to the Part 3 Integral Transformations
Return to the Part 4 Parabolic PDEs
Return to the Part 5 Hyperbolic PDEs
Return to the Part 6 Elliptic PDEs
Return to the Part 7 Numerical Methodss