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Quantitative Analysis
Parallel Processing
Numerical Analysis
C++ Multithreading
Python for Excel
Python Utilities
Services
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I. Basic math.
II. Pricing and Hedging.
III. Explicit techniques.
IV. Data Analysis.
V. Implementation tools.
VI. Basic Math II.
1. Real Variable.
2. Laws of large numbers.
3. Characteristic function.
4. Central limit theorem (CLT) II.
5. Random walk.
6. Conditional probability II.
7. Martingales and stopping times.
8. Markov process.
9. Levy process.
10. Weak derivative. Fundamental solution. Calculus of distributions.
11. Functional Analysis.
12. Fourier analysis.
13. Sobolev spaces.
14. Elliptic PDE.
15. Parabolic PDE.
A. Galerkin approximation for parabolic Dirichlet problem.
B. Energy estimates for Galerkin approximate solution.
C. Existence of weak solution for parabolic Dirichlet problem.
D. Parabolic regularity.
VII. Implementation tools II.
VIII. Bibliography
Notation. Index. Contents.

Galerkin approximation for parabolic Dirichlet problem.


e are considering the problem ( Parabolic Dirichlet problem ).

Let MATH is the basis of MATH formed by the eigenfunctions of the operator $-\Delta$ with the following normalization:

MATH (Definition of Galerkin basis 1)
The existence of such basis follows from the proposition ( Eigenvalues of symmetric elliptic operator ). To derive the orthogonality in MATH , write MATH and integrate by parts on the LHS.

We form the sum MATH and select the functions MATH to satisfy the conditions MATH We integrate by parts the term MATH and write the equations for $u_{m}$ in the form

MATH (Galerkin problem)
We substitute the definition of $u_{m}$ and use the orthonormality of MATH . We obtain
MATH (Galerkin coefficients)
This is a Cauchy system of ODEs and it has a form that insures that there is always a solution MATH .





Notation. Index. Contents.


















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