Kalo-gold Law

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Gamma method)It was written by Russian mathematician Borz gregorrievich Kalo-kin (RUSSIAN: Russian: boris Galerkin) a numerical analysis method. This method can be used to simplify the problem of solving the differential equations (by using the variational principle of the functional corresponding to the equation) into the problem of solving the linear equations. A high-dimensional (multi-variable) linear equations can be simplified by using linear algebra to solve the differential equations.

The Gamma method uses the weak form corresponding to the differential equation. Its principle is to select a limited number of test functions (also knownBasic functionsOrShape function), And then require the Weighted Integral (The weight function is the test function itself.) To meet the original equation, you can obtain a set of linear algebra equations that are easy to solve, and the natural boundary conditions can automatically meet.

It must be emphasized that, as a form of selecting a test function of the weighted margin method, the method obtained only an approximate solution in the original solution domain (only the weighted average satisfies the original equation, ).

Because the wonderful thing about the garliaojin method is to study their abstract methods, we first give their abstract derivation. Finally, we will give an example of the application.

Weak Form of a problem

We use an abstract question to introduce the Gamma method and express the question as a weak form in a Hilbert space. That is, the solution enables all

Yes. Here, a bilinear expression is a linear expression.

Discretization

SelectNDimension sub-spaces, and then solve the problem projection in the sub-spaces: To make all

We call this equation the kalo-Lakin equation. Note that the equation form is not changed, but the solution domain is changed.

Galoogin orthogonal

This is a key property that makes the Gamma method very effective. Because we can take a trial vector of the original equation. Bring and subtract to obtain the Cartesian relationship of the Error

Here is the error between the solution of the real solution and the solution of the Gamma equation.

Matrix Form

Because the objective of the garliaojin method is to simplify the problem into a linear equations, we construct its matrix form for numerical solutions using computers.

It is a group of bases in a space. Then it is clear that the base vectors are selected in sequence as the trial vectors of the gamma-gold equation.Adequate, That is, the solution enables

Expressed by the base vector above:, the above equation is obtained.

In this way, we get the above linear equations

Matrix Symmetry

Due to the definition of matrix items, the coefficient matrix of the gamma-liaojin equation is a required and sufficient condition of the symmetric matrix. The bilinear expression is symmetric.

Kalo-gold Law