Methods of Mathematical Physics Integral transformation
Methods of Mathematical Physics Integral Transformation
Integral Transformation a General concept of the integra transformation Fundamental ideas of the method of Integral transformation Fourier Transforms of wave problems a Fourier Transforms of heat problems Fourier Transforms of steady problems Conclusion of this charter
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
General Concepts a Integral transformation Definition turn the oragl function into one respect to new coefficients e General form f(x)→F(k)=∫f(×)K(xk)dx General properties linearity:af()+βg(X)→αF(k)+βG(k) ● Inverse transformation:f(×)←→F(k) derivative f(x)→f(×)K(Xk)| bound-∫f(×)Kx(x,k)dx
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 transformation：f(x) ←→ F(k) • derivative： f’(x) → f(x) K(x,k)|bound - ∫f(x) Kx(x,k) dx
General Concepts Typical integral transformations F(k)= f(r) dx ■ Fourier 丌 Transfor mation f(x)= F(k)e dk a Laplace F(p)=Lf(x)e-p*dx Transfor O+10 mation f(x) 2ai do-ioo F(p)ep dp
General Concepts ◼ Typical integral transformations： ◼ Fourier Transfor mation f x F k e dk F k f x e dx ikx ikx − − − = = ( ) ( ) ( ) 2 1 ( ) ◼ Laplace Transfor mation + − − = = i i p x p x F p e dp i f x F p f x e dx ( ) 2 1 ( ) ( ) ( ) 0
Fundamental idea ■ Programmer: Turn the original equation into an ordinary differential equation Solve the ordinary differential equation a Charge the solution of the ordinary differential equation into the solution of the original a Applying range: Fourier Transformation Unbounded region aplace transformation ·Semi- unbounded region
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