Methods of Mathematical Physics

1
Methods of Mathematical Physics

• Integral Transformation

2
Integral Transformation

• General concept of the integral transformation
• Fundamental ideas of The method of Integral
  Transformation
• Fourier Transforms of wave problems
• Fourier Transforms of heat problems
• Fourier Transforms of steady problems
• Conclusion of this charter

3
General Concepts

• Integral transformation
• Definition
• Turn the oragl function into one respect to new
  coefficients
• General form
• f(x) ? F(k) ?f(x) K(x,k) dx
• General properties
• linearity? f(x) ? g(x) ? ? F(k) ? G(k)
• Inverse transformationf(x) ?? F(k)
• derivative
• f(x) ? f(x) K(x,k)bound - ?f(x) Kx(x,k) dx

4
General Concepts

• Typical integral transformations

• Fourier Transformation

• Laplace Transformation

5
Fundamental Idea

• Programmer
• Turn the original equation into an ordinary
  differential equation.
• Solve the ordinary differential equation.
• Charge the solution of the ordinary differential
  equation into the solution of the original.
• Applying range
• Fourier Transformation
• Unbounded region
• Laplace Transformation
• Semi- unbounded region ?

6
Fourier Transform for the wave problem
Solving the wave problems
7
Fourier Transform for the heat problems
Solving the heat problems
8
Fourier Transform for the steady problems
Solving the steady problems
9
Conclusion for this charter

• In principle, Fourier Transform can be applied to
  any unbounded problems
• The steady problem
• The heat problem
• The wave problem
• The key to solving the Eq. by use of Fourier
  Transform method is inverse transfrom
• Generally, the results from Fourier Transform
  method are integral formulas.